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Thermally fully developed, electro-osmotically generated convective transport has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. The exact solution for the fully developed, dimensionless temperature profile and corresponding Nusselt number have been determined analytically for both geometries and both thermal boundary conditions. The fully developed temperature profiles and Nusselt number are found to depend on the relative duct radius (ratio of the Debye length to duct radius or plate gap half-width) and the magnitude of the dimensionless volumetric source.

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... The DC EOF [6][7][8] and related heat [9,10] and mass transfer problems [11,12] in micochannel or nanochannel have been widely studied theoretically [6,[9][10][11][12] and experimentally [7,8]. Among these, Bautista et al [13] and Ramos et al [14] used the lubrication theory to study the Joule heating effects on the DC EOFs and heat transfers of viscoelastic and Newtonian fluids in a slit microchannels, where the fluid properties or the charge density of the fluid depend on non-uniform temperature which is caused by Joule heating. ...

... The DC EOF [6][7][8] and related heat [9,10] and mass transfer problems [11,12] in micochannel or nanochannel have been widely studied theoretically [6,[9][10][11][12] and experimentally [7,8]. Among these, Bautista et al [13] and Ramos et al [14] used the lubrication theory to study the Joule heating effects on the DC EOFs and heat transfers of viscoelastic and Newtonian fluids in a slit microchannels, where the fluid properties or the charge density of the fluid depend on non-uniform temperature which is caused by Joule heating. ...

... In the above mentioned references [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], the charge density on the channel wall is assumed to be independent of the solution properties. However, this is usually unrealistic for some nanochannels made of materials such as SiO 2 and Al 2 O 3 , where the charge density is strongly dependent upon hydrogen ions, and hence, dependent upon the pH of the electrolyte solution. ...

In this paper, the separation of variables method is applied to investigate the effects of solution pH, background salt concentration and AC electric field frequency on time periodic electroosmotic flow in a pH-regulated parallel-plate nanochannel. The surface charge is generated by the protonation and deprotonation of the functional group SiOH. The background salt is KCl. The pH value of the solution is adjusted by HCl and KOH. Analytical and semi-analytical solutions for electric potential and velocity distributions are obtained. The results show that the electric potential caused by the electric double layer depends greatly on the solution pH and background salt concentration. The amplitudes of the velocity and flow rate of the time periodic electroosmotic flow decrease with the background salt concentration and increase with the deviation of the solution pH from the isoelectric point. In a nanochannel having a height less than 100nm, the electroosmotic velocity amplitude is not affected by the AC electric field frequency because the oscillating Reynolds number is much less than unity.

... This is because the local charge density within the EDL is nonzero, and the applied voltage difference will exert an electric body force on the liquid to drive the fluid. The transport process of EOF has been widely studied and used in various microfluidic devices [24][25][26][27][28][29]. For example, Herr et al. [24] analytically and experimentally investigated EOF in cylindrical capillaries with nonuniform surface charge distributions. ...

... Husain and Kim [28] investigated the thermal performances of MCHS for EOF, PDF, and mixed EOF and PDF, and they found that application of an external electric field could enhance the flow rate, and consequently reduce the thermal resistance. Maynes and Webb [29] analytically studied the convective heat transfer of the thermally fully-developed EOF in a parallel plate microchannel and microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. ...

... It should be noted that, though some studies have been performed to analyze the fluidic and thermal performances of EOF within an MCHS, most of the previous studies focused on the MCHS with parallel channel layout [24][25][26][27][28][29], which has the problem of large fluidic resistance and large temperature gradient. Thus, the channel layout of MCHS needs further optimization to improve its fluidic and thermal performances. ...

This paper numerically studies the thermal performances of electroosmotic flow (EOF) in a symmetric Y-shaped microchannel heat sink (MCHS) having a constant total channel surface area, that is, constant convective heat transfer area. It is found that the average convective heat transfer coefficient of EOF increases with the increasing driven voltage, which is attributed to the increase of EOF flowrate with the increasing driven voltage. However, the maximum MCHS temperature shows an increasing after decreasing trend with the driven voltage owing to the dramatically increasing Joule heating when the voltage is large enough. Further, both the maximum MCHS temperature and average convective heat transfer coefficient are sensitive to the cross-sectional dimensions of the Y-shaped microchannels. The thermal performances of EOF in the Y-shaped MCHS show a strengthening to weakening trend with the increasing daughter-to-parent channel diameter ratio of the Y-shaped microchannel with circular cross-sectional shape, and show a similar strengthening to weakening trend with the increasing daughter-to-parent channel width ratio and the increasing microchannel height of the Y-shaped microchannel with rectangular cross-sectional shape. These cross-sectional dimension dependences of thermal performances are related to the increasing to decreasing trend of EOF flowrate changing with the microchannel cross-sectional dimensions.

... For instance, the analysis of electro-osmotic flows in microchannels can be altered by imposing nonisothermal conditions. In this direction, some analytical results for electro-osmotic flows in microchannels with different geometries can be found in Maynes and Webb [2], Su et al. [3] and Horiuchi and Dutta [4]. In parallel, Tang et al. [5,6] developed numerical studies under steady and transient conditions to determine the influence of the Joule effect on the electro-osmotic flow. ...

... In Eqs. (1)(2)(3)(4), u and v are the velocity components in the x and y directions, respectively; T and p represent the temperature of the electrolyte solution and the induced hydrodynamic pressure. Also, C p , k, η, σ, ρ, and ρ e are the specific heat, the thermal conductivity, the dynamic viscosity, electrical conductivity, the density, and the electric charge density of the electrolyte solution, respectively. ...

... In Eqs. (1), (2), and (6), the electric charge density of the fluid is given by ρ e −2z 2 e 2 n ∞ ψ∕k B Tx; y, where ψ is the electric potential generated into the thin electric double layer, n ∞ is the ionic number concentration in the bulk solution, z is the absolute value of a (z:z) electrolyte solution valency, e is the elementary charge, k B is the Boltzmann constant, and T is the absolute temperature. We must recognize that the charge density is clearly a function that depends on temperature field, and this simple fact can lead to new temperature gradients in the governing equations, as we will show below. ...

This paper presents new numerical predictions for an electro-osmotic flow circulating in a slit microchannel when the influence of local temperature field caused at least by the Joule effect can alter the flow pattern. Taking into account that the Debye length depends on temperature T, the above length should be used with caution because it appears in any electro-osmotic mathematical model. In fact, the dependence of the Debye length on the temperature field can generate additional temperature gradients along the microchannel, and the isothermal hypothesis is no longer valid, modifying the resulting governing equations. These terms are enough to change the electric potential and the flow field. Therefore, the Navier–Stokes equations, together with the energy, Poisson, and Ohmic current conservation equations, are solved using the finite element method. For this purpose, a dimensionless thermal parameter, α, which measures the temperature deviations of a reference temperature, has been introduced. The numerical predictions show that the volumetric flow rate decreases in comparison with a uniforme Debye length, and strong induced pressure gradients are sensibly altered by the existence of this parameter.

... On application of an external electric field, the mobile ions in the EDL region will move, resulting in a bulk liquid motion via viscous effect. This is known as the electroosmotic flow (EOF) [70][71]. Apart from electroosmotically driven transport, pressure-driven flows may also have interesting facets with regard to electrokinetic transport in narrow confinements. ...

... Studies on heat transfer characteristics of electrokinetic flows have assumed great importance primarily because of the inherent importance of thermal transport in micro-electro-mechanically actuated flows in modern age industrial applications. Maynes and Webb [70][71] investigated thermally fully developed flow situation in microchannels for pure electroosmotic and combined pressure-driven and electroosmotic flows. Horiuchi and Dutta [76] provided analytical solutions for temperature distribution and Nusselt number for thermally developing electroosmotic flows through straight microchannels. ...

... In an another study, the same authors[101] have discussed analytical solutions of velocity distribution and streaming potential with a consideration of dynamical interplay between interfacial electrokinetics and a combined dissipative and elastic behaviour of flow through narrow confinements. A few studies have been dedicated to understand the thermal transport phenomena in electroosmotically driven flow in narrow confinements ([70],[76][77],[106][107][108][109][110][111][112][113][114]) with and without considering the effects of axial pressure gradients. Such studies are important in many practical engineering applications dealing with the internal heat generation arising out of the temperature rise of the narrow fluidic passage. ...

... This is known as the electroosmotic flow (EOF) [16][17]. Several studies on the hydrodynamic and thermal transport in electroosmotically driven flow in narrow fluidic confinements have been reported in literature [18][19][20][21][22][23][24][25]. Horiuchi and Dutta [18] studied flow and heat transfer characteristics of electroosmotically driven flows through narrow confinement under Joule heating effects. ...

... They reported that in the absence of internal heat generation, the heat transfer between the channel and the environment takes place only in the entry region. In another work, Maynes, and Webb [19] investigated the heat transfer characteristics for thermally fully developed electroosmotic transport in a narrow confinement under constant wall heat flux and constant http wall temperature boundary conditions. They showed that increasing the heat generation due to the Joule heating results in a decrease in Nusselt number for constant wall heat flux boundary condition. ...

... In Eq. (19), the parameter } is termed as irreversibility distribution ratio and is a direct function of fluid properties and temperature. ...

... However, unlike the hydrodynamic characteristics, the study of the thermal features of electrokinetic flow is recent. Pioneering research works on heat transfer characteristics of electroosmotic flow were conducted by Myness and Webb's research group [28][29][30]. Their studies dealt with exploring the thermally fully developed features of electroosmotic flow at small [28] and high [30] zeta potentials as well as examining the viscous heating effects on the thermal features [29]. ...

... Pioneering research works on heat transfer characteristics of electroosmotic flow were conducted by Myness and Webb's research group [28][29][30]. Their studies dealt with exploring the thermally fully developed features of electroosmotic flow at small [28] and high [30] zeta potentials as well as examining the viscous heating effects on the thermal features [29]. They soon extended their studies to the case of a mixed electroosmotic and pressuredriven flow [31]. ...

There has been a growing interest in the development of microchannel heat sinks deploying electrically modulated fluid flow in recent years. The efficient design of such devices requires heat transfer models that can account for complex distributions of heat generation in microelectronics. In this paper, expressions are obtained for temperature distribution and Nusselt number of thermally developing mixed electroosmotic and pressure-driven flow through circular/slit microchannels of axially non-uniform wall heat flux. The heating section is considered to be of a finite length in order to simulate a physically more realistic situation. Both the Joule heating and axial conduction effects are considered in the model. By comparing the results for linear, sinusoidal, and exponential distributions of the wall heat flux with the predictions of full numerical simulations, it is shown that the analytical solutions presented are accurate up to a Péclet number of 10. This threshold is demonstrated to be larger than the maximum Péclet number encountered in practical applications involving electroosmotic pumping mechanisms. After justification of the model, a parametric analysis is executed, revealing that the average Nusselt number is a decreasing function of the EDL thickness and pressure-driven velocity, irrespective of the wall heat flux distribution. Moreover, whereas a higher Joule heating rate is accompanied by a smaller value of the average Nusselt number for pure electroosmotic and pressure-assisted flows, the opposite is true in the presence of a significant back pressure.

... Bianchi et al. [2] (2000), studied electroosmotically-driven micro flows in T-junctions using a finite element formulation based on the Gouy-Chapman approximation. Maynes and Webb [3](2003)analytically studied fully developed electroosmotically generated convective transport for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Keisuke H. and Prashanta D. [4] (2004) solved electroosmosis flow by simulation of Navier Stoke flow, poission-Boltzmann and heat transfer equations in two dimensions, They studied the thermal effected parameters (temperature distribution, heat coefficient and Nussult number) at steady state and pure electroosmosis flow. ...

... Electric Field density : (3) Poisson equation : ...

Electrokinetic micropumps received more attention due to its applications in pumping of biological and chemical fluids, such as blood, DNA, and saline PBSs. In this paper the electroosmotic flow in microchannel has been numerically investigated by developing a model from the basic governing equations (continuity, momentum, energy, Laplace and poison- boltzmann equations) through a square channel. These equations were numerically solved by CFD program for two fluids (water and PBS). The study covers a selected wide range of external applied electric field and zeta potential to show their effects on electroosmotic flow. Thermal characteristics of electroosmotic flow have been also studied by calculating the temperature distribution through electroosmosis micropump region. The results obtained show a considerable effect of the applied electric field, concentration of electrolyte fluid and zeta potential on the velocity and flow rate. The PBS gave higher velocity and flow rate compared with water, and there is a slight increase in temperature due to small effect of Joule heating.

... Hydrodynamically developing flow between two parallel plates for electroosmotically generated flow has been numerically analyzed by Yang et al. [5]. Maynes and Webb [6] analytically studied fully developed electroosmotically generated convective transport for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Yang et al. [7] investigated forced convection in rectangular ducts with electrokinetic effects for both hydrodynamically and thermally fully developed flow. ...

... (a)result of [19] (b)result of present model Figure [ 6] velocity profile in center line along channel length as a comparisin between present model and [ 19] In contrast, Figure 7 depicts the velocity contours as comparison between present model and [19]. ...

The growing interest of researchers in the phenomenon of electrokinetic can be seen as a direct and natural response to the importance and significance of its applications in our life. Transporting of fluids under the influence of electroosmosis or electrophoresis has significance implication nowadays in the fabrication of microfluidic devices and thus in biomedicine and micro-electromechanical systems. Obvious examples are separation and lap-on-a-chip devices. The overall aim of the present paper is to investigate how electroosmotic flow could effectively contribute to the intracellular biomolecules flow. The significance of the current study arises from being one of the early attempts to study electroosmosis in Iraq. Utilizing the numerical analysis, two mathematical models have been built in order to shapes the electroosmosis-based microflow of the negatively charged messenger proteins to the negatively charged nucleus. To shed more light on the electroosmotic flow behavior through microchannels with contraction, the well-known computer software COSMOL has been used to develop the simulations. The potential impacts of several design parameters on temperature distribution and velocity profiles for a variety of contraction shapes have been analyzed and presented. Such design parameters include wall zeta potential, external electric field, and channel wall material.

... For instance, the analysis of electro-osmotic flows in microchannels can be drastically altered by imposing non-isothermal conditions. In this direction, some analytical results for electro-osmotic flows in microchannels with different geometries can be found in Maynes and Webb [2], Su et al. [3] and Horiuchi and Dutta [4]. In parallel, Tang et al. [5] and Tang et al. [6] developed numerical studies under steady and transient conditions, determining the influence of the Joule effect on the electroosmotic flow. ...

... Replacing the above equations into Eqs. (2) and (3), we obtain that, ...

In this work, we develop a new thermal analysis for an electro-osmotic flow in a rectangular microchannel. The central idea is very simple: the Debye length that defines the length of the electrical double-layer depends on temperature T. Therefore, if exists any reason to include variable temperature effects, the above length should be utilized with caution because it appears in any electro-osmotic mathematical model. For instance, the presence of the Joule effect is a source that can generate important longitudinal temperature gradients along the microchannel and the isothermal hypothesis is no longer valid. In this manner, the Debye length is altered and as a consequence, new longitudinal temperature gradient terms appear into the resulting governing equations. These terms are enough to change the electric potential and the flow field. Taking into account the above comments, in the present study the momentum equations together with the energy and Poisson conservation equations are solved by using a regular perturbation technique. For this purpose, we introduce a dimensionless parameter α that measures the temperature deviations of a reference temperature. For practical cases, this parameter is small compared with unity and the theoretical predictions show; however, that for the used values of this parameter, the volumetric flow rate decreases in comparison with the isothermal case.

... Studies on heat transfer characteristics of electrokinetic flows have assumed great importance primarily because of the inherent importance of thermal transport in micro-electro-mechanically actuated flows in modern age industrial applications. Maynes and Webb [16,17] investigated thermally fully developed flow situation in microchannels for pure electroosmotic and combined pressuredriven and electroosmotic flows. Horiuchi and Dutta [18] provided analytical solutions for temperature distribution and Nusselt number for thermally developing electroosmotic flows through straight microchannels. ...

... Substituting Eq. (16) in Eq. (17), and after nondimensionlization, the electroneutrality condition may be expressed as follows: ...

In this article, we investigate the combined consequences of magnetohydrodynamic forces and interfacial slip on the heat transfer characteristics of streaming potential mediated flow in narrow fluidic confinements by following a semianalytical formalism. Going beyond the celebrated Debye–Hückel linearization, we obtain a closed form analytical expression for velocity and induced streaming potential through the consistent description of finite conductance of the immobilized Stern layer. We report an augmentation in the streaming potential field as attributable to the wall slip activated enhanced electromagnetohydrodynamic transport of the ionic species within the EDL. In particular, we demonstrate the key role of induced streaming potential in altering thermal transport and Nusselt number variation considering the concurrent interplay of hydrodynamic slip lengths, magnetic effects, viscous dissipation, and Joule heating. We also show the implications of Stern layer conductivity and magnetohydrodynamic influence on system irreversibility through entropy generation analysis due to fluid friction and heat transfer. Finally, our results have significant scientific and technological consequences in the novel design of future generation energy efficient devices and could be useful in further advancement of theory, simulation, and experimental work.

... Fluid momentum from thin EDL is then transferred to bulk fluid by viscous stresses. EOF has shown positive effect on performance enhancement [13][14][15]. Using Poisson-Boltzmann equation and the Navier-Stokes equations, Morini et al. [16] numerically studies EOF in MCHS having rectangular and trapezoidal cross sections. ...

With continuous miniaturization of modern electronic components, the need of better cooling devices also keeps on increasing. The improper thermal management of these devices not only hampers the efficiency but can also cause permanent damage. Among various techniques, microchannel heat sink has shown most favourable performance. To further enhance the performance, two techniques i.e., active and passive are used. In passive technique, no external power source is required like heat sink design alteration and working fluid modification. External power source is necessary for heat transfer augmentation in the microchannel heat sink when using the active approach. Due to compact size of microchannel, active techniques are not used more often. However, the present work highlights the different active technique used in microchannel i.e., Electrostatic forces, flow pulsation, magnetic field, acoustic effects, and vibration active techniques. Above mentioned techniques have been analysed in detail.

... Electroosmotic flow (EOF), a representative electrokinetic phenomenon, can be found under the action of an external electric field (Pretorius et al. 1974;Masliyah and Bhattacharjee 2006;Maynes and Webb 2003). Compared to conventional mechanical pumping flow, electroosmotic flow has numerous advantages. ...

The electroosmotic flow of fractional Maxwell fluid in microchannels with isosceles right-triangular cross-sections is presented in this paper. The finite difference method is applied to build the numerical model based upon the Navier–Stokes equations, fractional Maxwell constitutive equation, and Poisson–Boltzmann equation. Under the action of an applied AC electric field, numerical solutions are derived for different cases considering different high-wall zeta-potential conditions. Finally, a detailed discussion of the effects of some dimensionless parameters on the velocity profile and volumetric flow rate is given, with numerical and graphical interpretations. The results show that the sign of wall zeta-potential affects the shape of the velocity profile. The required time to reach a stable periodic oscillation depends on the assumed value of the fractional parameter, relaxation time, and angular Reynolds number. And the amplitude of the volumetric flow rate will be affected by all these parameters.

... After that, many researchers concentrated on the heat transfer characteristics in microtubes. Maynes and Webb [327] investigated the electroosmotically generated convective transport in a parallel plate and circular microtube subject to maintaining a constant heat flux and wall temperature. Najjaran et al. [328] investigated the induced charge electrokinetic phenomenon to intensify microchannels' convective heat transfer and mixing process. ...

This paper aims to develop a review of the electrokinetic flow in microchannels. Thermal characteristics of electrokinetic phenomena in microchannels based on the Poisson–Boltzmann equation are presented rigorously by considering the Debye–Hückel approximation at a low zeta potential. Several researchers developed new mathematical models for high electrical potential with the electrical double layer (EDL). A literature survey was conducted to determine the velocity, temperature, Nusselt number, and volumetric flow rate by several analytical, numerical, and combinations along with different parameters. The momentum and energy equations govern these parameters with the influences of electric, magnetic, or both fields at various preconditions. The primary focus of this study is to summarize the literature rigorously on outcomes of electrokinetically driven flow in microchannels from the beginning to the present. The possible future scope of work highlights developing new mathematical analyses. This study also discusses the heat transport behavior of the electroosmotically driven flow in microchannels in view of no-slip, first-order slip, and second-order slip at the boundaries for the velocity distribution and no-jump, first-order thermal-slip, and second-order thermal-slip for the thermal response under maintaining a uniform wall-heat flux. Appropriate conditions are conferred elaborately to determine the velocity, temperature, and heat transport in the microchannel flow with the imposition of the pressure, electric, and magnetic forces. The effects of heat transfer on viscous dissipation, Joule heating, and thermal radiation envisage an advanced study for the fluid flow in microchannels. Finally, analytical steps highlighting different design aspects would help better understand the microchannel flow’s essential fundamentals in a single document. They enhance the knowledge of forthcoming developmental issues to promote the needed study area.

... Heat transfer analysis is important in microfluidics at microscale, e.g., electronic device cooling [24] and microchannel heat sink [25] . The first study on fully developed EOF thermal properties in microchannels was carried out by Maynes and Webb [26] . They obtained a closed-form expression for the temperature distribution by considering the Debye-Hückel approximation. ...

The heat transfer of the combined magnetohydrodynamic (MHD) and electroosmotic flow (EOF) of non-Newtonian fluid in a rotating microchannel is analyzed. A couple stress fluid model is scrutinized to simulate the rheological characteristics of the fluid. The exact solution for the energy transport equation is achieved. Subsequently, this solution is utilized to obtain the flow velocity and volume flow rates within the flow domain under appropriate boundary conditions. The obtained analytical solution results are compared with the previous data in the literature, and good agreement is obtained. A detailed parametric study of the effects of several factors, e.g., the rotational Reynolds number, the Joule heating parameter, the couple stress parameter, the Hartmann number, and the buoyancy parameter, on the flow velocities and temperature is explored. It is unveiled that the elevation in a couple stress parameter enhances the EOF velocity in the axial direction.

... The effects of the EDL near the solid-liquid interface and the flow-induced electrokinetic field on the pressure-driven flow and heat transfer through a rectangular microchannel were reported by Yang et al. [24]. Thermally fully-developed, electroosmotically generated convective transport has been analyzed by Maynes and Webb [25] for some shapes of microtube. They presented analytic solutions for dimensionless temperature and Nusselt number for some channels. ...

Thermal characteristics of time-periodic electroosmotic flow are analyzed in a micro-annulus under the influence of various alternating electric fields. Representative hydrodynamic and thermal quantities, i.e. volumetric flow rate and Nusselt number, demonstrate oscillatory behaviors approaching a quasi-steady state if few periods of time elapse. An important parameter named dimensionless frequency, which normally affects the diffusion mechanism, is responsible for the penetration depth of momentum and energy into the fluid in the radial direction. The higher the dimensionless frequency, the smaller the transport phenomena diffuse radially into the bulk fluid; while the advection mechanism is intensified and consequently Nusselt number is increased. The cooling-mode mean Nusselt number is slightly smaller than the heating- mode one. A variety of waveforms is examined in the present research; their performances are then compared together using proper measures. A strength index is introduced to evaluate the relative ability of each individual excitation in a half period of time. A key parameter named thermal/frictional index is also utilized to assess the total effectiveness of the system in different circumstances especially when the basic excitation functions are applied. Unlike the vigorous square waveform, the sawtooth one is the most efficient.

... The circular geometry was one of the earliest to be investigated, as per the work of Rice and Whitehead, [19], who dealt with non-negligible thickness of the EDL compared to the classical treatment by Smoluchowski and predicted a maximum in the apparent viscosity of the flow (the so-called electro-viscous effect). Maynes and Webb studied the convective heat transfer for purely electro-osmotic [33] and both pressure and electro-osmotically driven flows, [34] considering the relative extension of the Debye length to the channel half-width, indicated with Z. Their analysis was limited to circular and parallel-plate ducts and to fully-developed velocity profiles and revealed how the temperature profile and Nusselt number strongly depend on the values of Z, of the non-dimensional Joule heating and, where applicable, on the relative magnitude of the external electric field to pressure gradient imposed. For simple geometries, Moghadam ([35] obtained the exact solution for the entrance region of a circular capillary considering the contribution of viscous dissipation, which was compared to Joule heating. ...

Microchannel heat sinks are able to provide high cooling capabilities in terms of heat flux rates. This makes them particularly interesting for the thermal management of electronic components such as CPUs, which have high power density and small dimensions. Pressure drop of the coolant across the microchannels may, however, be significant and give rise to viscous heating, thereby preventing the practical use of these devices. When the coolant is a polar fluid and the channel walls possess a net electric charge, an alternative means of moving the fluid is through an applied external electric field. The flow which originates is called electro-osmotic (EOF). EOF does not require moving parts, is free of vibrations and does not need lubrication, but is subject to Joule heating of the fluid and has flow and heat transfer characteristics which differ from those of pressure-drive flows. In spite of several previous investigation on EOF, no attention has been paid to the changes in velocity and temperature distributions caused by modifying the base cross-section of the channels which may be circular, rectangular or polygonal, thanks to the current capabilities of microfabrication. This work investigates numerically the influence of smoothing the corners of the cross-section at fixed hydraulic diameter on the values of the Poiseuille and Nusselt numbers for the laminar, steady and fully developed, electro-osmotic flow in a rectangular channel subject to uniform heat flux and Joule heating. Several aspect ratios are considered, as are different values of the ratio of Joule heating to heat flux through the walls. The results highlight a very slight increase of the Poiseuille number with the radius of curvature, whereas the Nusselt number experiences a significant improvement. Correlations are obtained for both the Poiseuille and Nusselt number as a function of the radius of curvature, aspect ratio and Joule heating-to-heat flux ratio.

... Recently, due to the proved effectiveness of electroosmosis-based microchannel heat sinks [ 5 , 21 ], the thermal features of electroosmotic flow have attracted researchers' attention. Maynes and coworkers pioneered the studies on the thermal transport characteristics of electroosmotic flow in circular and slit microchannels both in the absence [22] and in the presence [23] of viscous dissipation assuming constant heat flux/temperature boundary conditions. Their works were followed by a long line of research works on fully-developed thermal aspects of electroosmotic flow [24][25][26][27][28][29][30][31] . ...

Asymmetries in boundary condition are inevitable in practice in microfluidic channels, despite being rarely addressed from theoretical perspectives. Here, by arriving at closed form analytical solutions, we bring out a unique coupling between asymmetries in surface charge and heat transfer in electroosmotically driven microchannel flows. For illustration, we assume that the channel is laterally composed of two parts, each having specified values of the zeta potential and the wall heat flux. Considering low zeta potentials, we obtain analytical solutions in terms of infinite series for the dimensionless forms of the electric potential, the velocity, and the temperature distributions. We demonstrate that, by carefully adjusting the governing parameters, a variety of flow patterns may be achieved, a property that is crucial in applications such as liquid-phase transportation and mixing. Moreover, we show that the average velocity is a linear function of both the zeta potential ratio and the coverage factor. We further show that the average Nusselt number increases when part of the channel having the larger heat flux enlarges and the zeta potential of the part having the smaller surface charge increases. Hence, the maximum heat transfer rates are achieved when the boundary conditions are symmetrical.

... Considering the aforementioned intricacies in coupling thermal, electrical and hydro-dynamical effects in micro-confinements, research efforts towards addressing various aspects of thermo-solutal convection of electrolyte solutions have turned out to be relatively inadequate, despite having widespread applications in processes like water treatment, charge separation, zeta-potential determination, waste heat recovery, and energy conversion (Barragán & Kjelstrup 2017;Dietzel & Hardt 2016Jokinen et al. 2016;Li & Wang 2018;Sandbakk et al. 2013;Würger 2008Würger , 2010. As such, the research focus in this domain has been directed primarily towards incorporating non-isothermal effects as a secondary force in the alteration of hydrodynamics of simple fluids (Chakraborty 2006;Garai & Chakraborty 2009;Huang & Yang 2006;Maynes & Webb 2003;Sadeghi et al. 2011;Sánchez et al. 2018;Tang et al. 2003;Xuan et al. 2004;Xuan 2008). Therefore, except for some limited physical scenarios, however, such an exclusive effect has not been utilized to a significant practical benefit (Dietzel & Hardt 2016Zhang et al. 2019). ...

Enhancing solute dispersion in electrically actuated flows has always been a challenging proposition, as attributed to the inherent uniformity of the flow field in the absence of surface patterns. Over the years, researchers have focused their attention towards circumventing this limitation, by employing several fluidic and geometric modulations. However, the corresponding improvements in solute dispersion often turn out to be inconsequential. Here we reveal that by exploiting the interplay between an externally imposed temperature gradient, subsequent electrical charge redistribution and ionic motion, coupled with the rheological complexities of the fluid, one can achieve enhancement of up to one order of magnitude of solute dispersion in a pressure-driven flow of an electrolyte solution. Our results demonstrate that the complex coupling between thermal, electrical, hydrodynamic and rheological parameters over small scales, responsible for such exclusive phenomenon, can be utilized in designing novel thermally actuated microfluidic and bio-microfluidic devices with favourable solute separation and dispersion characteristics.

... However, it was recently shown that Joule heating is relatively small as compared to the overall heat removed from the devices to be cooled, promising a bright future for electro-osmotic cooling [22,23]. The research studies on the thermal aspects of electro-osmotic flow were pioneered by Maynes and coworkers who explored the fully developed heat transfer characteristics for low and high zeta potentials with and without considering the viscous dissipation effects [24][25][26]. More recently, analytical solutions were developed to study the heat transfer features of pure electro-osmotic flow in microchannels of different geometries under the H1 thermal boundary condition, which refers to a constant axial wall heat flux while having a constant wall temperature within each cross section [27][28][29]. ...

The heat generated by microprocessors has an extremely non-uniform spatial distribution with hotspots that have heat fluxes several times larger than the background flux. Hence, for an accurate design of microchannel heat sinks used for cooling of microelectronic devices, models are required that can take such a non-uniform distribution of wall heat flux into account. In this study, analytical solutions are obtained for hydrodynamically fully-developed but thermally developing mixed electroosmotic and pressure-driven flow in a rectangular microchannel with a peripherally uniform but axially non-uniform distribution of the wall heat flux. It is assumed that the heat flux is applied over a finite length, to mimic a physically more realistic situation, and the Péclet number is small so that lateral temperature variations are negligible as compared to the axial variations of temperature. By comparing the results with those of full numerical simulations for exponential, sinusoidal, and stepwise distributions of wall heat flux, it is demonstrated that the solutions obtained are accurate up to a Péclet number of 10. Fortunately, this value is larger than the maximum Péclet number of electroosmotic microflows. Furthermore, it is shown that smoother distributions of wall heat flux give rise to higher heat transfer rates. The model developed in this study can pave the way for modeling of hotspots in more complicated microfluidic devices.

... Considering the aforementioned intricacies in coupling thermal, electrical and hydro-dynamical effects in micro-confinements, research efforts towards addressing various aspects of thermo-solutal convection of electrolyte solutions have turned out to be relatively inadequate, despite having widespread applications in processes like water treatment, charge separation, zeta-potential determination, waste heat recovery, and energy conversion (Barragán & Kjelstrup 2017;Dietzel & Hardt 2016Jokinen et al. 2016;Li & Wang 2018;Sandbakk et al. 2013;Würger 2008Würger , 2010Xie et al. 2018). As such, the research focus in this domain has been directed primarily towards incorporating non-isothermal effects as a secondary force in the alteration of hydrodynamics of simple fluids (Chakraborty 2006;Garai & Chakraborty 2009;Huang & Yang 2006;Keramati et al. 2016;Maynes & Webb 2003;Sadeghi et al. 2011;Sánchez et al. 2018;Tang et al. 2003;Xuan et al. 2004a;Xuan 2008;Xuan et al. 2004b;Yavari et al. 2012). Therefore, except for some limited physical scenarios, however, such an exclusive effect has not been utilized to a significant practical benefit (Dietzel & Hardt 2016Zhang et al. 2019). ...

Enhancing solute dispersion in electrically actuated flows has always been a challenging proposition, as attributed to the inherent uniformity of the flow field in absence of surface patterns. Over the years, researchers have focused their attention towards circumventing this limitation, by employing several fluidic and geometric modulations. However, the corresponding improvements in solute dispersion often turn out to be inconsequential. Here we unveil that by exploiting the interplay between an externally imposed temperature gradient, subsequent electrical charge redistribution and ionic motion, coupled with the rheological complexities of the fluid, one can achieve up to one order of magnitude enhancement of solute dispersion in a pressure-driven flow of an electrolyte solution. Our results demonstrate that the complex coupling between thermal, electrical, hydro-dynamic and rheological parameters over small scales, responsible for such exclusive phenomenon, can be utilitarian in designing novel thermally-actuated micro and bio-microfluidic devices with favorable solute separation and dispersion characteristics.

... Compared to other methods of improving thermal conductivity like increased concentration, external magnetic field, etc. [2,5,10], the present method yields larger enhancements, as well as transient control over the thermal conductivity, which helps to modulate the coolant as per requirement. The findings may find potential implications in enhanced thermal management of micro-nanoscale devices [21], or in microfluidics based bioanalysis using temperature gradient focussing [22,23], and so on. ...

Electrophoresis has been shown as a novel methodology to enhance heat conduction capabilities of nanocolloidal dispersions. A thoroughly designed experimental system has been envisaged to solely probe heat conduction across nanofluids by specifically eliminating the buoyancy driven convective component. Electric field is applied across the test specimen in order to induce electrophoresis in conjunction with the existing thermal gradient. It is observed that the electrophoretic drift of the nanoparticles acts as an additional thermal transport drift mechanism over and above the already existent Brownian diffusion and thermophoresis dominated thermal conduction. A scaling analysis based on the thermophoretic and electrophoretic velocities from classical Huckel-Smoluchowski formalism is able to mathematically predict the thermal performance enhancement due to electrophoresis. It is also inferred that the dielectric characteristics of the particle material is the major determining component of the electrophoretic amplification of heat transfer. Influence of surfactants has also been probed into and it is observed that enhancing the stability via interfacial charge modulation can in fact enhance the electrophoretic drift, thereby enhancing heat transfer calibre. Also, surfactants ensure colloidal stability as well as chemical gradient induced recirculation, thus ensuring colloidal phase equilibrium and low hysteresis in spite of the directional drift in presence of electric field forcing. The findings may have potential implications in enhanced and tunable thermal management of micro-nanoscale devices and in thermo-bioanalysis within lab-on-a-chip devices.

... Dutta et al. [6] have presented a new algorithm to investigate the mixed electroosmotic-and pressure-driven microflows in the microgeometries. Maynes and Webb [7] have analyzed the electroosmotically generated transport in a parallel microchannel and microtube. Tang et al. [8] have given numerical results for the electrokinetic transport of a viscous fluid in a microchannel using a Crank-Nicolson finite difference scheme. ...

A theoretical study is conducted for magnetohydrodynamic pumping of electroosmotic non-Newtonian physiological nanoliquids through a two-dimensional microfluidic channel. The Sutterby rheological nanofluid model is utilized to characterize the liquid. The normalized two-dimensional conservation equations for mass, longitudinal and transverse momentum, energy and solutal concentration are reduced with lubrication approximations (long wavelength and low Reynolds number assumptions). A coordinate transformation is employed to map the unsteady problem from the wave laboratory frame to a steady problem in the wave frame. Slip and convective conditions are imposed at the channel walls. The emerging boundary value problem is solved numerically using MATLAB software. The flow is effectively controlled by many geometric parameters, viz., electroosmosis, Hartmann and Sutterby fluid parameters. It is observed from the analysis that the rise in magnetic and electroosmosis effects leads to a reduction in the axial velocity field. The radiation parameter decreases the temperature for the positive value of Joule heating parameter and the trend is revered for the negative Joule heating parameter. This study is encouraged by exploring the nanofluid dynamics in peristaltic transport as symbolized by heat transport in biological flows, novel pharmacodynamics pumps and gastrointestinal motility enhancement. The study is also relevant to MHD biomimetic blood pumps.

... The experimental studies 9,10 can also be seen in the direction of electrokinetic microflows. Maynes and Webb 11 have analyzed the electro-osmotically generated convective transport in a parallel microchannel and in a circular microtube by focusing at the channel/tube size effect on the subject of the Debye length for different Joule heating values. Horiuchi et al 12 have obtained the heat transfer characteristics on the mixed electro-osmotic and pressure-driven flows in a microchannel. ...

In the present article, the theoretical investigation is presented for the mixed electrokinetic and pressure‐driven transport of couple stress nanoliquids in a microchannel with the effect of magnetic field and porous medium. This topic has gained a remarkable scope in nanoscale electro‐osmotic devices. The formulation of the present mathematical problem is simplified using the Debye‐Hückel linearization assumption. The merging model has important features such as the thermal Grashof number, solutal Grashof number, Joule heating, Helmholtz‐Smoluchowski velocity. The analytical solutions are presented for the axial velocity, temperature, and solute concentration. The expressions for the heat transfer rate, solute mass transfer rate, and surface shear stress function at the walls are also presented. The results display that, the velocity of the couple stress nanofluid is less in the case of pure electro‐osmotic flow as compared to that of combined electro‐osmotic and pressure‐driven flow. When the Joule heating parameter vanishes, the temperature and solute concentration profiles are linear, otherwise nonlinear. The shear stress function is larger in the case of pure electro‐osmotic flow and it is smaller for the combined effects of electro‐osmotic and pressure gradient. The present analysis places a significant observation that the various zeta potential plays an influential role in heartening fluid velocity. The analysis is relevant to electrokinetic hemodynamics and microfluidics.

... Si consideri un fluido Newtoniano monofase incomprimibile, in moto laminare, completamente sviluppato, in regime stazionario, con proprietà fisiche costanti, all'interno di un microcondotto rettilineo a sezione circolare, in assenza di variazione di pressione lungo l'asse del condotto. In letteratura sono numerose le soluzioni proposte per geometrie rettangolari e trapezoidali, mentre solo pochi autori hanno analizzato flussi elettrosmotici in condotti a sezione circolare [24][25][26][27]. ...

In questo lavoro viene determinata la distribuzione di velocità e temperatura per un liquido polare, in moto completamente
sviluppato in un microcondotti a sezione circolare, con moto imposto da un campo elettrico situato tra le sezioni di ingresso e
di uscita del condotto. Il problema riveste notevole importanza nel settore delle pompe elettrosmotiche, utilizzate per
sospingere liquidi in microcondotti in assenza di gradienti di pressione, utilizzando solo gli effetti elettrodinamici generati nella
electric double layer. Si considera un condotto circolare, in cui un fluido Newtoniano, monofase, incomprimibile riceve calore
in condizioni al contorno H, in regime stazionario, nella regione di moto completamente sviluppato. Il problema è dapprima
risolto analiticamente, introducendo l’ipotesi di Debye Hückel, valida per modesti valori del campo elettrico applicato, per
determinare la distribuzione spaziale del potenziale elettrico. I profili di velocità e temperatura sono individuati dalla soluzione
delle equazioni di Navier Stokes e di bilancio energetico, considerando la generazione di potenza nel liquido per effetto Joule.
Infine il problema generale è risolto numericamente, ricorrendo al software FlexPDE per la risoluzione di un sistema di tre
equazioni differenziali alle derivate parziali. I risultati sono analizzati parametricamente al variare delle principali grandezze
fisiche e geometriche.

... Uno studio analitico dei fenomeni di scambio termico convettivo in microtubi sottoposti a condizioni di flusso termico costante o di temperatura costante alla parete è stato recentemente condotto da Maynes e Webb [8,9]. ...

SOMMARIO In questo lavoro viene presentato il modello di un micro-dissipatore in silicio a canali rettangolari percorso da un flusso laminare liquido elaborato mediante una pompa ad effetto elettro-osmotico. Il modello permette di simulare il funzionamento di una micro-pompa ad elettro-osmosi (EOP) fornendo al progettista informazioni di dettaglio, quali l'andamento del potenziale elettrico all'interno del microcanale, la distribuzione di velocità impressa nella generica sezione, la relazione esistente tra differenza di potenziale imposta agli elettrodi e portata di fluido elaborata e la distribuzione di temperatura nel canale. Il calcolo dell'andamento del potenziale elettrico all'interno del microcanale viene effettuato attraverso la soluzione dell'equazione di Poisson-Boltzmann dell'equilibrio elettrostatico. La distribuzione di velocità indotta all'interno del canale dal campo elettrico esterno è individuata andando a risolvere le equazioni di Navier-Stokes e la distribuzione di temperatura all'interno del canale è determinata andando a risolvere l'equazione di bilancio dell'energia nel caso di fluido sottoposto a condizione termica al contorno di tipo H1 nell'ipotesi di moto completamente sviluppato. Vengono infine presentati i risultati di alcune simulazioni ottenuti risolvendo numericamente le equazioni del modello con il metodo degli elementi finiti. INTRODUZIONE I fenomeni di trasporto all'interno di microcanali aventi dimensioni caratteristiche inferiori al millimetro sono oggetto di intenso studio teorico e sperimentale ormai da una decina di anni. In questi ultimi tempi, grazie agli enormi progressi effettuati nello sviluppo delle tecniche di microlavorazione delle superfici, si sono moltiplicate le applicazioni tecnologiche in cui viene sfruttato l'elevato potere di drenaggio termico di flussi laminari all'interno di microcanali. Al fine di generare flussi convettivi all'interno di condotti aventi diametri idraulici inferiori ai 100 µm le convenzionali tecniche basate sulla creazione di un gradiente di pressione tra le sezioni estreme del canale possono entrare in crisi a causa dei significativi salti di pressione richiesti. Le pompe miniaturizzate che sono state sviluppate fino ad ora hanno palesato evidenti limiti di efficienza ed affidabilità. Un metodo alternativo è quello di "trascinare" il fluido applicando un campo elettrico esterno così da sfruttare il fenomeno della "elettro-osmosi" (pompe EOP). Tale fenomeno è noto da almeno due secoli (Reuss [1]). Gli effetti dovuti ai fenomeni di accoppiamento elettrostatico tra fluido e parete all'interno di microcanali sono stati indagati da Mala et al. [2,3], da Yang e Li [4], Yang et al. [5] e Li [6]. In questi lavori viene proposta una soluzione analitica dell'equazione di Navier-Stokes per flussi interni laminari in presenza di forze di massa di tipo elettrostatico in sezioni rettangolari. E' stato evidenziato il ruolo delle interazioni elettrostatiche tra fluido e parete sul fattore di attrito [2,3,4,5,6] e sul numero di Nusselt [3,5]. In particolare, è stato previsto un aumento del fattore d'attrito in funzione della concentrazione ionica presente nel liquido e una dipendenza del numero di Nusselt medio dal numero di Reynolds anche in regime di moto laminare. Santiago [7] analizza il comportamento dinamico di un liquido trascinato per effetto elettro-osmotico (Electro Osmotic Flow) in un canale delimitato da due lastre piane parallele. Uno studio analitico dei fenomeni di scambio termico convettivo in microtubi sottoposti a condizioni di flusso termico costante o di temperatura costante alla parete è stato recentemente condotto da Maynes e Webb [8,9]. Tang et al. [10] hanno analizzato l'entità dei fenomeni di riscaldamento dovuti all'effetto Joule nei liquidi elaborati da pompe EOP; è stato evidenziato come la dipendenza dalla temperatura dei coefficienti di diffusione ionica e la generazione di calore in seno al liquido dovuto all'effetto Joule possono influenzare notevolmente il profilo di velocità e lo scambio termico all'interno di un canale. Arulanandam e Li [11] hanno condotto uno studio numerico volto alla caratterizzazione delle pompe EOP; il loro modello permette di investigare la dipendenza della portata volumetrica elaborata dalle caratteristiche geometriche del microcanale, dal valore della concentrazione ionica nel liquido e dal potenziale elettrico applicato agli elettrodi. In questo lavoro verrà studiato un microdissipatore in silicio dotato di canali rettangolari percorso da un liquido mosso per effetto di una pompa EOP. La simulazione del funzionamento della micropompa EOP fornisce al progettista informazioni di dettaglio, quali l'andamento del potenziale elettrico all'interno del microcanale, la distribuzione di velocità impressa nella generica sezione, la relazione esistente tra differenza di potenziale imposta agli elettrodi, la portata di fluido elaborata e la distribuzione di temperatura nel canale. Vengono infine presentati i risultati di alcune simulazioni effettuate andando a risolvere numericamente le equazioni del modello proposto mediante il metodo degli elementi finiti (FEM).

... The first research works in this field were performed by Maynes and coworkers [19e21]. Their studies were dealing with flow in slit and circular microchannels at low zeta potentials [19], flow through circular microcapillaries at high zeta potentials [20], and viscous heating effects on the thermal transport characteristics for both slits and microtubes [21], all considering the thermally fully developed conditions. They considered the classical boundary condition of constant wall heat flux and showed that the solution of the case with a constant wall temperature boundary condition is also a special case of the general solution. ...

We consider the forced convection heat transfer associated with a mixed electroosmotically and pressure-driven flow through a rectangular microchannel. The thermal boundary condition is considered to be the constant wall heat flux of second type, H2, and the flow is assumed to be both hydrodynamically and thermally fully developed. Series solutions are obtained for the electrical potential, velocity, and temperature fields as well as the Nusselt number. The results show that the Nusselt number is generally an increasing function of the channel width to height ratio with the exception being the pressure-assisted flow with surface cooling for which a weak decreasing dependence on the aspect ratio is observed. It is also found that an increase in the Joule heating parameter is accompanied by a lower Nusselt number, unless for pressure-opposed flows for which the trend is reversed at higher aspect ratios. The Joule heating effects, however, are not much important when the minimum channel dimension to Debye length ratio is above 200. Furthermore, increasing either of the velocity scale ratio or electric-double-layer thickness leads to smaller Nusselt numbers. Nevertheless, the impact of these parameters diminishes when either the velocity scale ratio takes very large values or electric-double-layer gets very thin.

... In an another study, they (Bandopadhyay and Chakraborty, 2012a, 2012b, 2012b have discussed analytical solutions of velocity distribution and streaming potential with a consideration of dynamical interplay between interfacial electrokinetics and a combined dissipative and elastic behaviour of flow through narrow confinements. A few studies have been dedicated to understand the thermal transport phenomena in electroosmotically driven flow in narrow confinements (Burgreen and Nakache, 1964;Levine et al., 1975;Patankar and Hu, 1998;Yang et al., 1998;Dutta et al., 2002;Maynes and Webb, 2003;Horiuchi and Dutta, 2004;Chakraborty, 2006;Rawool and Mitra, 2006;Zade et al., 2007;Chen et al., 2013;Duwairi and Abdullah, 2007) with and without considering the effects of axial pressure gradients. Such studies are important in many practical engineering applications dealing with the internal heat generation arising out of the temperature rise of the narrow fluidic passage. ...

This paper presents a theoretical analysis of non-Newtonian (power-law obeying) fluid in a narrow confinement subjected to the combined consequences of interfacial electrokinetics, rheology, and superimposed magnetic field. We devote special attention on the exploitation of magnetic field and power-law exponent, in the development of induced streaming potential and thermofluidic energy transfer characteristics over small scales. In an effort to do so, going beyond the Debye-Hückel limit, we first derive an expression for streaming potential by invoking the consequences of strong EDL (electrical double layer) interactions in the narrow fluidic passage and finite conductance of the Stern layer. In particular, we solve thermal energy transport equation with an illustrative case of classical uniform wall heat flux boundary and considering the volumetric heat generation effects due to viscous dissipation as well as Joule heating. Our results demonstrate that the applied magnetic field imparts a retarding influence on the induced streaming potential development, whereas, it results in enhancement of heat transfer rate. Moreover, additional influences of power law index show reduction in heat transfer as well as the streaming potential magnitude. We unveil the optimal combinations of power law index and the magnetic field which lead to the minimization of the global total entropy generation in the system. We believe that theoretical results presented in this research will be useful in the development of novel narrow fluidic energy efficient devices under electrokinetic modulation.

... Hydro dynamically developing flow between two parallel plates for electroosmotically generated flow has been numerically analyzed by Yang et al. [5]. Maynes and Webb [6] analytically studied fully developed electroosmotically generated convective transport for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Yang et al. [7] investigated forced convection in rectangular ducts with electrokinetic effects for both hydrodynamically and [9] studied the electro-osmotic flow of power-law fluid and heat transfer in a micro-channel with effects of Joule heating and thermal radiation. ...

Electrokinetic micropumps received extra attention due to its applications in pumping of
biological and chemical fluids, such as blood, DNA, and saline PBSs. In this paper the
electroosmotic flow in microchannel has been numerically investigated by developing a
model from the basic governing equations (continuity, momentum, energy, and Laplace and
Poison- Boltzmann equations) through a square channel with contraction of different
geometries. Utilizing the numerical analysis, a mathematical model has been built in order to
study the electroosmosis-based microflow of the negatively charged messenger proteins to the
negatively charged nucleus to shed more light on the electroosmotic flow behavior through
microchannels with contraction. The results obtained show a considerable effect of the zeta
potential on the velocity and many parameters studied such as shapes, depth and number of
contractions and channel wall material. Adding materials with different thermal conductivity
to channel wall, affect the heat distribution throughout channel length. In particular, five wall
materials typically used in Microelectromechanical systems (MEMS). It shows that a material
with low thermal conductivity works as isolation against dissipating the heat generated from
Joule heating effect.

... Electroosmotic flow is characterized by Joule Heating which arises due the fluid electrical resistivity and the applied electric field. Maynes and Webb [3] were the first to consider the thermal aspects of electroosmotic flow. They analytically studied the convective transport for a parallel plate microchannel and circular microtube under constant heat flux and constant wall temperature condition for fully-developed region. ...

Thermal transport characteristics of combined electroosmotic and pressur-driven flow of non-Newtonian (power-law) nanofluids through a microchannel have been studied. Non-Newtonian fluids can influence the thermal behaviour of the flow by affecting the rate of heat convection and viscous dissipation. Electroosmotic phenomenon causes resistance heating of the fluid, called Joule heating. Joule heating becomes a noteworthy phenomenon in microscale flow dynamics as the thickness of the ionic charged layer or electric double layer (EDL) is only about one-tenth to one-hundredth of the channel height. Nanofluids are known to have better heat transfer characteristics than conventional fluids at microscale. A complete parametric study has been carried out to investigate the effect of different flow and electrolytic parameters on the thermal behaviour of the flow. The governing equations have been solved semi-analytically under constant wall heat flux condition taking into account the effects of viscous dissipation and Joule heating. Power-law fluids of both shear-thinning and shear-thickening nature have been considered. The governing equations have been solved only for hydrodynamically and thermally fully-developed flow. Three nanofluidic parameters have been taken into consideration, namely: viscosity, electrical permittivity and electrical resistivity. These parameters have been introduced as ratios with reference to the corresponding properties of a conventional fluid.

... However, Joule heating can be positively used under some circumstances, such as to control the thermal environments in microfluidic devices. Enhancement of heat transfer by Joule heating induced thermal convection was analytically examined using idealized thermal boundary conditions [70][71][72]. A normalized source term is introduced to represent the ratio of Joule heating to surface heat flux by Horiuchi et al. [73,74], in their thermal analysis of pure electroosmotic flows and mixed electroosmosis/pressure-driven flows. ...

Microfluidics has been undergoing fast development in the past two decades due to its promising applications in biotechnology, medicine, and chemistry. Towards these applications, enhancing concentration sensitivity and detection resolution are indispensable to meet the detection limits because of the dilute sample concentrations, ultra-small sample volumes and short detection lengths in microfluidic devices. A variety of microfluidic techniques for concentrating analytes have been developed. This article presents an overview of analyte concentration techniques in microfluidics. We focus on discussing the physical mechanism of each concentration technique with its representative advancements and applications. Finally, the article is concluded by highlighting and discussing advantages and disadvantages of the reviewed techniques.

... (The correct citation is given below.) Themost common form of citation, which appears in over 30 publications despite apparently originating as recently as 2003, [18] is disturbingly erroneous,i ncluding af abricated article title,t ranslated journal title,a nd incorrect volume number and pagination. ...

Electro-osmosis and electrophoresis were discovered by F. F. Reuss in Moscow in 1807. Or so the story goes. This essay critically examines the contributions of three scientists to the discovery of electrokinetic phenomena. The evidence suggests that Reuss did indeed discover electro-osmosis, which takes its name (indirectly) from the work of Porrett. Contrary to current consensus, Gautherot made the earliest known observation of electrophoresis.

... There is lots of phenomenon in nano/micro scales microfluidic which have become a research hot spot in different areas especially in scientific and engineering fields [1]. In most micro/nano fluidic system, we deal with electrolyte solution, glass plate surface and micro/nano pumps [2][3] Electrokentic flows have become one of the most important methods for investigation of micro-nano fluidic [4][5]. One of the most important methods for micro/nano fluid pumping is Electroosmotic flow. ...

In this paper, we present our results about simulation of 2D-EOF in Nano/Micro scales porous media using lattice Boltzmann method (LBM) in micro-channel for EOF. The high efficient numerical code use strongly high nonlinear Poisson Boltzmann equation to predicate behavior of EOF in complex geometry. The results are developed with precisely investigation of several effective parameters on permeability of EOF, such as geometry (channel height and number and location of charge), external electric field, thickness of Debye length (ionic concentration), and zeta potential. Our results are in excellent agreement with available analytical results. Our results show that for certain external electric field, zeta potential and porosity, there is an optimal Kh parameter (ionic concentration and channel height in this study) for velocity profiles. Based on the current study, homogenous zeta potential distribution on solid porous media, zeta potential and thickness of Debye length (Kh parameter) can dramatically affect on EOF permeability linearly or non-linearly, depend on amount of quantities. Thus, different arrangements are also considered. We show that prediction of EOF behavior in complex geometry with regarding role of effective parameters is completely possible for various applicable conditions.
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The heat transfer characteristics and entropy generation of an electroosmotic flow in curved rectangular nanochannels under high zeta potential conditions are investigated numerically with consideration of the steric effects. The numerical solutions of the Nusselt number (Nu) and the local entropy generation rate (SG) are derived under constant wall heat flux assumption. The variation trends of Nu and SG with corresponding parameters, including the curvature ratio of the curved nanochannel (δ), the effective ionic size (a0) and the wall heat flux (qw) are investigated. Results reveal that, for small values of qw (qw < 10), the consideration of the steric effects (corresponding to a non zero assumption for a0) leads to a lower SG and an enhanced Nu simultaneously. In this case, to further reduce entropy generation, a curved nanochannel with a large curvature ratio δ is adopted. However, for relatively large values of qw, the steric effects will have a negative effect on entropy generation minimization, especially for a curved nanochannel with large curvature ratio. Our results may be useful in designing the efficient thermal nano-equipment.

At high values of interfacial electrokinetic potentials, the effects of finite ionic sizes are consequential and must be taken in the purview. This paper addresses the non-trivial implications of steric effects on thermal transport of an electroosmotic-pressure driven flow within a microchannel under an imposed heat flux. The ion size is introduced into the modified Poisson-Boltzmann equation by the steric factor, which allows considering the ions as point charges or finite sizes. The non-linear, non-dimensional form of the electrokinetic potential, momentum, and energy equations are numerically solved in a spatial scheme discretized into the non-equally spaced elements. Our main point here is that the intricate physical interplay among zeta potential, pressure gradient, and steric effects has a considerable impact on the temperature distribution. Results indicated that by decreasing the surface cooling, the effect of ion finite size is significantly intensified while other factors become of peripheral importance. Also, unfavourable pressure gradients tend to mask the effect of additional heat generated by increasing the channel aspect ratio. Finally, adverse pressure gradients generated higher velocity gradients at the surface, results in a higher convective cooling and consequently an improved thermal dissipation.

Alternating current electroosmotic flow and associated heat transfer under constant surface heat flux conditions are numerically examined in a micro-annular channel. The channel surfaces are arbitrarily heated and/or cooled, while the resultant heating is normally transferred downstream by the advection mechanism. An important feature of time-periodic electrokinetic flow is that there is not any preferential axial direction. If the excitation frequency is sufficiently small and/or the liquid kinematic viscosity is sufficiently large, momentum diffuses far into the bulk fluid and thus the advection mechanism is intensified. The reverse is true for relatively large frequency and small kinematic viscosity values. Mean Nusselt number fluctuates over a period of time; its time-averaged magnitude is dependent on the electrokinetic diameter and the dimensionless frequency. This quantity approaches a specific value regardless of the thermal scale and the wall heat flux ratios. The system may attain optimal effectiveness for particular magnitudes of the frequency and electrokinetic diameter. An interesting case may exist in the cooling mode where the axial variation of the mean fluid temperature vanishes temporarily while the state of constant surface temperature is instantly satisfied; at the particular points in time, the advection mechanism does not contribute to the development of the liquid temperature field.

In recent years, electromagnetohydrodynamically modulated control and hydroelectric energy conversion through narrow fluidic devices have emerged as promising means for controlling and manipulating liquid flows in diverse applications. Such processes include development of smart sensors, micrototal analysis systems (μTAS), capillary electrophoresis, electrochromatography, mixing, flow cytometry, DNA hybridization and analysis, cell manipulation, cell patterning, immunoassay, enzymatic reactions, and molecular detection, etc. Accordingly, in the present chapter, we discuss the fundamental theories and elucidate the semi analytical and numerical approaches for analysis of the electromagnetohydrodynamic forces and their effect on thermofluidic control and energy transfer characteristics in narrow fluidic confinements. The consequences of electromagnetohydrodynamic forces and interfacial slip on the streaming potential development has been discussed in detail for pressure driven flows in a narrow fluidic confinement. It has been inferred that wall slip activated electro-magnetohydrodynamic transport can enhance the induced streaming potential and intensifies the convective heat transfer rate. Furthermore, we have also shown an analysis for combined electroosmotic and pressure-driven flows through narrow confinements, subjected to spatially varying non-uniform magnetic field. It is revealed that one can augment heat transfer rate for such a situation by judiciously choosing the spatially varying magnetic field strength. Next, we highlight the collective interaction of the fluid rheology, kinematics, volumetric effects of ionic species (steric effect), and the electrodynamics leading to giant augmentations in the energy conversion efficiency. For all the studies reported above, exergy analysis can indicate the route for optimal designs of process and the reduction in the thermodynamic irreversibility. Finally, the specific points emerging out from the research are concluded and relevant application areas have been discussed.

This study aims to investigate thermal transport characteristics and analyze entropy generation of thermally fully-developed electroosmotic flow (EOF) in a rotating rectangular microchannel subject to constant wall heat flux. In the two-dimensional rectangular domain, the solutions for electric potential and flow velocity are obtained in terms of eigenfunction series by means of separation of variables technique, and the temperature field, which takes into consideration the effects of heat convection, Joule heating and viscous dissipation, is numerically solved. Results are generated to show the influences due to various factors, including rotational frequency, depth/width ratio of the rectangular channel, distribution of zeta potentials, strength of Joule heating and magnitude of wall heat flux, on the thermal transport characteristics. In particular, how the Nusselt number and the entropy generation vary with these factors are explored in details. To pursue optimization in energy utilization, an optimal aspect ratio is found at which the minimum entropy generation can be achieved. We believe that the findings in the present study will be useful in the design of energy efficient micro-systems which utilize the dual electrokinetic and centrifugal pumping effects.

We develop an electrokinetic technique that continuously manipulates colloidal particles to concentrate into patterned particulate groups in an energy efficient way, by exclusive harnessing of the intrinsic Joule heating effects. Our technique exploits the alternating current electrothermal flow phenomenon which is generated due to the interaction between non-uniform electric and thermal fields. Highly non-uniform electric field generates sharp temperature gradients by generating spatially-varying Joule heat that varies along the radial direction from a concentrated point hotspot. Sharp temperature gradients induce a local variation in electric properties which, in turn, generate a strong electrothermal vortex. The imposed fluid flow brings the colloidal particles at the centre of the hotspot and enables particle aggregation. Furthermore, maneuvering structures of the Joule heating spots, different patterns of particle clustering may be formed in a low power budget, thus opening up a new realm of on-chip particle manipulation process without necessitating a highly focused laser beam which is much complicated and demands higher power budget. This technique can find its use in Lab-on-a-chip devices to manipulate particle groups, including biological cells.

In the present study, the electroosmotic and pressure driven flow of nanofluid in a microchannel with homogeneous surface potential is investigated by using the Poisson-Boltzmann equation. The flow filed is assumed to be two-dimensional, laminar, incompressible, and steady. Distribution of nanoparticles in the base fluid is assumed to be homogeneous; therefore the nanofluid flow is modeled as single phase. The thermal conductivity of the nanofluid is modeled by using the Patel model to account for temperature dependency. In order to validate the numerical solution, the results are compared with available analytical solutions and the comparison shows a good agreement with the results. Then, the effects of different parameters such as ion molar percentage, volume fraction, and nanoparticles’ diameter on the flow filed and heat transfer is examined. The results show that by fixing the electric field and increasing the pressure gradient, the local Nusselt number will decrease, and by fixing the pressure gradient and enhancing the electric field, the Nusselt number increases. Increasing the diameter of nanoparticle in a fixed volume fraction will decrease the average Nusselt number. The average Nusselt number increase about 45%, 35% and 25% while nanoparticles’ diameters are 100, 110 and 120. When Γ=0.05, the average Nusselt number increase 10% while ion concentration changes 〖10〗^(-4) to 〖10〗^(-2). Furthermore, the direction and magnitude of velocity and concavity of velocity profile can be controlled by choosing a suitable phase angle between electrical and pressure driven flow parameters.

The first part of the present research is to estimating the relationship between the overall
friction power and rotating speed of a 4-stroke single cylinder diesel engine practically by using
an electrical motor which was directly connected to the engine with a suitable coupling, and the
input power was controlled by using an inverter,(current and volts) were measured by using a
digital clamp-meter for different rotating speeds of (600,1200,1800, 2400, 3000 and 3600) rpm.
From the data achieved an equation (polynomial third order) was estimated which represents
the friction losses as a function of engine speed. The estimated equation is used in simulation
program (QBASIC ), for predicting the effects of friction on some parameters (power
,efficiency and fuel consumption ) . The second goal is to developing the estimated equation
and using it many cylinders and many displacement volume diesel engines by multiplying it by
displacement ratio and number of cylinders. Finally the results showed good agreement.

We consider the forced convection heat transfer associated with a mixed electroosmotically and pressure-driven flow through a rectangular microchannel. The thermal boundary condition is considered to be the constant wall heat flux of second type, H2, and the flow is assumed to be both hydrodynamically and thermally fully developed. Series solutions are obtained for the electrical potential, velocity, and temperature fields as well as the Nusselt number. The results show that the Nusselt number is generally an increasing function of the channel width to height ratio with the exception being the pressure-assisted flow with surface cooling for which a weak decreasing dependence on the aspect ratio is observed. It is also found that an increase in the Joule heating parameter is accompanied by a lower Nusselt number, unless for pressure-opposed flows for which the trend is reversed at higher aspect ratios. The Joule heating effects, however, are not much important when the minimum channel dimension to Debye length ratio is above 200. Furthermore, increasing either of the velocity scale ratio or electric-double-layer thickness leads to smaller Nusselt numbers. Nevertheless, the impact of these parameters diminishes when either the velocity scale ratio takes very large values or electric-double-layer gets very thin.

This article presents a mathematical model for studying peristaltic mechanism of combine pressure and electro-osmotically driven flow of ionic liquids through a micro-channel having electrokinetic effects. The velocity slip and the thermal slip conditions at the channel wall are taken into account for investigating the thermomechanical interactions. The micro-channel is assumed to have porous structure. The governing equations for fluid flow and heat transfer in the electrical double layer (EDL) together with the Poisson-Boltzmann equation are considered. The analytical solutions have been obtained under low Reynolds number and long wave length assumptions. It is also assumed that the channel height is much greater than the thickness of the electrical double layer (EDL). The essential features of electro-osmotically driven flow and associated heat transfer characteristics in a micro-channel are clearly demonstrated by varying dimensionless parameters for velocity profile, temperature profile, pressure distribution, stream function, wall shear stress and the Nusselt number. The pressure drop exhibits a linear dependence on the flow rate. The study reveals that the electro-osmotic parameter has an enhancing effect on the size of the trapping bolus while the reducing effect on porous permeability of the channel. The temperature distribution is significantly influenced by Joule heating parameter and Brinkman number. The study bears the potential applications in biomedical engineering for the development of microfluidic devices in particular microfluidic pump to transport small volume of ionic liquids by maintaining temperature distribution.

Several lab-on-chip applications deal with the flow of fluids through microchannels. Some applications require recycle flows in such systems. In this work we show how this can be achieved in microfluidic systems using electroosmosis. The system consists of flow through a main channel and a side channel with a provision for recycle. The main flow is pressure-driven, and the recycle flow is achieved by imposing an electric field across the side channel. When the applied electric field is greater than a critical value, the flow in the side channel is reversed and recycle is induced. The critical applied electric field is investigated as a function of zeta potential. The effect of the applied electric field on pressure and flow rate in the main and side channels is analyzed. The influence of radius and position of the side channel on flow behavior has also been studied.

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter.

Elektroosmose und Elektrophorese wurden 1807 von F. F. Reuss in Moskau entdeckt – so oder ahnlich lautet die Geschichte. Dieser Essay beleuchtet die Beitrage von drei Wissenschaftlern bei der Entdeckung der elektrokinetischen Phanomene. Ziemlich sicher ist, dass Reuss tatsachlich die Elektroosmose entdeckt hat, wobei deren Name (indirekt) aus den Arbeiten von Porrett stammt. Entgegen dem derzeitigen Konsens stammt die fruheste Beobachtung der Elektrophorese von Gautherot.

This article is aimed to present an analytic study of electro-osmotic (EO) pumping of a micro-duct with inserted fin vanes. Finned structures are known to be an efficient and very important tool in conducting heat generated in fluid transport system, such as Joule’s heating in micro-fluidic devices. The present semi-analytical analysis is performed under the Debye–Hückel approximation (DHA), enabling us to explicitly investigate the combined effects of various parameters for optimizing the EO pumping rate while retaining substantial fin vanes for heat removal. A mathematical model based on the solutions of two fundamental EO flows is introduced to explain how the choice of the fin vane width may optimize the EO pumping rate in the general case. Moreover, we present the optimized EO flow rates in diagrams plotted on the plane of zeta potentials, which may serve as an easily used reference for engineering design and applications.

In this work, convection heat transfer for combined electro-osmotic and pressure driven flow of power-law fluid through a microtube has been analyzed. Typical results for velocity and temperature distributions, friction coefficient, and Nusselt number are illustrated for various values of key parameters such as flow behavior index, length scale ratio (ratio of Debye length to tube radius), dimensionless pressure gradient, and dimensionless Joule heating parameter. The results reveal that friction coefficient decreases with increasing dimensionless pressure gradient, and classical Poiseuille solutions can be retrieved as the dimensionless pressure gradient approaches to infinite. To increase the length scale ratio has the effect to reduce Nusselt number, while the influence of this ratio on Nusselt number diminishes as the pressure gradient increases. With the same magnitude of dimensionless Joule heating parameter, Nusselt number can be increased by increasing both the flow behavior index and dimensionless pressure gradient for surface cooling, while the opposite behavior is observed for surface heating. Also, singularities occurs in the Nusselt number variations for surface cooling as the ratio of Joule heating to wall heat flux is sufficiently large with negative sign.

In the present study, the heat transfer characteristics of thermally developed nanofluid flow through a parallel plate microchannel are investigated under combined influences of pressure-driven and streaming potential effects. The analytical solution for electrokinetic flow in microchannel is obtained by employing the Debye–Hückel linearization. The classical boundary condition of uniform wall heat flux is considered in the analysis, and the effects of viscous dissipation as well as Joule heating are also taken into account. Furthermore, based upon the velocity field and temperature field, the Nusselt number variations are induced, and the variations of local and total entropy generation of nanofluids are also performed. Concisely, the results show the profiles of streaming potential decrease with the dimensionless EDL thickness, whereas the Nusselt number increases with the dimensionless EDL thickness. An enhanced heat transfer performance with increasing nanoparticle volume fraction can be witnessed. The local entropy generation gradually grows from the centerline toward the wall. Beside, the total entropy generation obviously grows with increasing Br.

In this work, we investigate the heat transfer characteristics of thermally developed nanofluid flow through a parallel plate microchannel under the combined influences of externally applied axial pressure gradient and transverse magnetic fields. The analytical solutions for electromagnetohydrodynamic (EMHD) flow in microchannels are obtained under the Debye–Hückel linearization. The classical boundary condition of uniform wall heat flux is considered in the analysis, and the effect of viscous dissipation as well as Joule heating is also taken into account. In addition, in virtue of the velocity field and temperature field, the Nusselt number variations are induced. The results for pertinent dimensionless parameters are presented graphically and discussed in briefly.

In this study, the perturbation method was implemented to analytically solve the governing equations relevant to both hydrodynamically and thermally fully developed power-law fluid and plug flows through parallel-plates and circular microchannels under constant isoflux thermal and slip boundary condition. The temperature-dependent properties, being viscosity and thermal conductivity, were considered along with non-linear slip condition in the analysis in addition to viscous dissipation. The velocity, temperature and constant property Nusselt number closed form expressions were derived. Then, Nusselt number corresponding to temperature-dependent thermophysical properties was numerically obtained due to their complexity nature. Numerical simulations were also performed for verifying the analytical results. The results indicated that the property variations and slip condition significantly affected thermo-fluid characteristics, and these parameters should be included to achieve a better thermal design for microdevices.

Electrokinetic forces are emerging as a powerful means to drive microfluidic systems with flow channel cross-sectional dimensions in the tens of micrometers and flow rates in the nanoliter per second range. These systems provide many advantages such as improved analysis speed, improved reproducibility, greatly reduced reagent consumption, and the ability to perform multiple operations in an integrated fashion. Planar microfabrication methods are used to make these analysis chips in materials such as glass or polymers. Many applications of this technology have been demonstrated, such as DNA separations, enzyme assays, immunoassays, and PCR amplification integrated with microfluidic assays. Further development of this technology is expected to yield higher levels of functionality of sample throughput on a single microfluidic analysis chip.

Electrokinetic (EK) micropumps have been fabricated and
demonstrated in which electroosmotic flow is used to transport fluids.
Deionized water and pure acetonitrile have been used as working fluids
to achieve low current density pumping conditions. These EK pumps have
no moving parts and can generate maximum pressures of more than 20 atm
at 2 kV applied voltage. Minimizing and controlling electrolytic gas
generation is a major concern. Gas generated at the downstream electrode
surfaces appears to be forced to dissolve into surrounding fluid at high
pressure (>7 atm) condition, and this permits a stable pump
operation. Measurements of flow rate have been used to estimate pump
structure parameters

Methodology and analysis software have been developed that produce high-spatial-resolution field measurements of depthwise uniform, linear shear, and parabolic components of the velocity from particle-image videos in which the depthwise direction is not resolved. These velocity components can be approximately identified with uniform electrokinetic flow, electrokinetic flow with a ζ-potential difference between the bottom and top surfaces, and pressure-driven flow, respectively. This software has extracted measurements of the uniform and parabolic components of mixed pressure-driven and electrokinetic particle flows over heterogeneous boundaries. This decomposition of flow components simplifies the interpretation of complicated microflows.

In the present study, the steady laminar forced convection problem of heat transfer in the fully developed constant property flow of liquids through a certain class of channels is analyzed by considering the contribution of heat due to viscosity. The effects of viscous dissipation on the heat transfer are emphasized. A class of sufficiently long straight channels with uniform cross-sectional area is chosen such that the solutions for velocity and temperature fields are deducible directly from the equations of the boundary curves. The wall temperature is allowed to vary linearly in the axial direction, and some heat-source distribution, other than that due to viscosity, is imagined to be present in the flow field. The general solution of the problem for the given class of channels is given directly by avoiding the details of the mathematical treatment of the governing equations. To illustrate the general mathematical derivations and to visualize the effects of viscous dissipation, some basic examples have been investigated and the graphical representation of several relevant results is given in a number of figures. In the last section of this study, various relevant results and figures have been discussed from the point of view of viscous dissipation phenomena.

This paper is concerned with the heat-transfer problem of laminar forced convection in noncircular pipes with arbitrary heat generation and prescribed heat flux at the wall. This class of boundary-value problems with Neumann conditions is approached by the method of conformal mapping. The solutions in terms of two analytic functions are established. This greatly enlarges the possibilities of analyses to many configurations which are otherwise not easily attainable. The example of an indented pipe of cardioid section is investigated in detail.

In a valuable contribution, Rice and Whitehead (16) studied theoretically electrokinetic flow in a narrow cylindrical capillary, subject only to the restriction that the zeta-potential be sufficiently low for the Debye-Huckel approximation to be acceptable (i.e., ζ ≲ 25 mV for a 1-1 electrolyte). This is a severe restriction in practice, where zeta-potentials as high as 100–200 mV are frequently encountered. The objective of this work is therefore to extend the 'Rice and Whitehead theory to higher surface potentials. The predictions obtained for streaming potential should permit an improved interpretation of experimental data taken during the course of zeta-potential determinations of fine capillaries and porous media. Of academic interest is the prediction of a maximum in the electroviscous retardation effect with respect not only to electrokinetic radius (as already reported by Rice and Whitehead) but also with respect to zeta-potential.

A recently developed imaging tool is employed to study scalar transport in microfabricated fluidic manifolds. Using the two-color fluorescence-based method to generate molecular fluid flow tracers, we investigate electrokinetic flow through channel geometries commonly encountered in micro-total analytical system research and development.

The characteristics of electroosmotic flow in a cylindrical microchannel with nonuniform zeta potential were investigated in this paper. The Poisson–Boltzmann equation and momentum equation were used to model the electrical double-layer field and the flow field. The numerical results show the distorted electroosmotic velocity profiles resulting from the axial variation of the zeta potential. Also, the influences of the unequal section size and the direction of the zeta potential change on the velocity profile, the induced pressure distribution, and the volumetric flow rate are discussed in this paper. The simulation results revealed possible effects of bioadhesion in microchannels on the electroosmotic flow in biochip devices.

This paper investigates the effects of the EDL at the solid-liquid interface on liquid flow and heat transfer through a microchannel between two parallel plates at constant and equal temperatures. A linear approximate solution of the Poisson-Boltzmann equation is used to describe the EDL field near the solid-liquid interface. The electrical body force resulting from the double layer field is considered in the equation of motion. The equation of motion is solved for the steady state flow. Effects of the EDL field and the channel size on the velocity distribution, streaming potential, apparent viscosity, temperature distribution and heat transfer coefficient are discussed in this paper.

Using the Graetz problem with axial conduction as an illustrative example a method for solution of an important class of linear partial differential equations is developed. The method is a combination of orthogonal collocation and matrix diagonalization. The reason for the very high accuracy, which is obtained by collocation, is discussed in terms of the eigenvalues of the collocation operator. These are found to increase much faster than the true eigenvalues for k > N 2 where N is the number of collocation points, and this permits a high accuracy also in the "penetration region" of the solution where Fourier Series are slowly convergent. Explicit formulas for the asymptotic Nu-number for large and small Pe-numbers are developed in an appendix. They are based on a perturbation of the eigenfunctions of the simplified model with either infinite or zero Pe-number. A number of variants of the Graetz problem, which can be solved by a repetition of the present computations, are proposed.

A dual pump and a buffer pump have been integrated on a silicon wafer for chemical analyzing systems. These pumps realize constant and rippleless liquid flow of a small volume. The controllable flow rate of the pumps was up to about 40 μl/min and the maximum pumping pressure was about 1 mH2O. A sample injector made up of two three-way valves has also been fabricated with micromachining.

Electroosmotic motion through charged, narrow-bore channels and capillaries is analyzed for the case where there are dominantly-axial gradients in the composition of the flowing electrolyte. The channel width is assumed to be large compared with the Debye screening length, and the electroosmotic slip velocity along the channel wall is taken to vary locally with the ionic strength, pH and electric field. Owing to the wall slip condition, the velocity distribution is nonlinearly coupled to the composition variations within the fluid. The prototype problem studied is one in which buffer ions and other solutes (e.g. analytes) are initially distributed in a sample zone that is sandwiched between uniform running buffer. For the situations considered, the conductivity of the sample zone differs significantly from that of the running buffer; such configurations are common to stacking and electroosmotic pumping protocols. In a frame of reference that moves with the mean velocity of the flow, the velocity field exhibits flow separation in the neighborhood of the conductivity variations and this gives rise to solutal mixing and dispersion in and about the sample zone.

This paper contains an analytical study of electrokinetic flow in very fine capillary channels of rectangular cross section. It is a natural extension of the general theory of electrokinetic flow which heretofore was limited to channels of large electrokinetic radius or to interfaces exhibiting low source potential. The practical implications of the results of the study are explored.

The entry flow induced by an applied electrical potential through microchannels between two parallel plates is analyzed in this work. A nonlinear, two-dimensional Poisson equation governing the applied electrical potential and the zeta potential of the solid–liquid boundary and the Nernst–Planck equation governing the ionic concentration distribution are numerically solved using a finite-difference method. The applied electrical potential and zeta potential are unified in the Poisson equation without using linear superposition. A body force caused by the interaction between the charge density and the applied electrical potential field is included in the full Navier–Stokes equations. The effects of the entrance region on the fluid velocity distribution, charge density boundary layer, entrance length, and shear stress are discussed. The entrance length of the electroosmotic flow is longer than that of classical pressure-driven flow. The thickness of the electrical double layer (EDL) in the entry region is thinner than that in the fully developed region. The change of velocity profile is apparent in the entrance region, and the axial velocity profile is no longer flat across the channel height when the Reynolds number is large.

The forced-convection problems in channels of fully developed laminar
flow with heat sources and constant walltemperature gradient are approached by
the method of complex variables. lt is shown that the stated problems are reduced
to the determination of some analytic functions which satisfy the boundary
conditions. Basic problems, including flows in equilateral triangular ducts and
in elliptical tubes, are studied. It is shown that the cases of circular tubes
and of parallel plates with a gap are directly deducible from that of elliptical
tubes. All these solutions are expressed in closed forms by polynomials. (auth)

The flow profile of electroosmosis in capillary electrophoresis was studied by using a dye and a rectangular capillary. The movement of the dye is observed with a microscope—video system, and then advances per unit time are measured from the recorded video tapes. The medium at the central portion moves like a plug flow, and the zone front at the edges are ahead of the central portion. The flow profile in a capillary column with a circular cross-section is proposed. The flow profiles of ionic solutes are also discussed.

An imaging system based on microscope optics and a charge-coupled device camera is used to form high-resolution images of the liquid core inside narrow fused-silica capillaries. This technique allows direct examination of the fluid motion inside the capillary under electrokinetic and hydrodynamic (Poiseuille) conditions. Two experiments are described. The first involves monitoring the front of the fluorescence-labeled solvent as it travels through the capillary. The second is an examination of the local velocities using submicron-sized particles as probes. The results are discussed in context of band broadening in capillary electrophoresis. 22 refs., 6 figs.

Heat transfer by convection is emphasized by the present treatment,
which includes mass transfer by convection primarily by analogy. After
establishing a sound theoretical base at the molecular level, giving
insight into the physical origin of transport properties, equations
describing convective transport on the continuum level are derived.
Examples are given of physical situations described by one-dimensional
formulations. On the basis of these basic phenomena and solution
techniques, such complex applications of the theory as laminar and
turbulent duct and boundary layer flows are treated. Attention is also
given to integral methods, natural convection, boiling, and
condensation.

This paper is a theoretical study of electrokinetic flow in narrow cylindrical capillaries. It is concerned with the dependence of the usual electrokinetic phenomena on the electrokinetic radius. The results obtained for this dependence must, however, be treated with caution for the higher values of the interface potential due to the use of the Debye-Hückel approximation. Of interest is the prediction of a maximum in the electroviscous effect.

An electroosmosis based fluid propulsion system is described. The electroosmotic pump, operating in a high electric field, is isolated from other components by a grounding joint that is electrically conductive and permits the pumped fluid to be hydrodynamically coupled to contents downstream without leakage. The pump was used in single- and double-line flow injection analysis (FIA) systems. The determinations of chloride and iron(III) are discussed as representative examples. The experimental results showed excellent reproducibilities (relative standard deviation 0.4-0.8%), reflecting the stability and the reliability of the pump, The system can also be operated in a hybrid manner, On-line preconcentration (based on the electrostacking effect as used in capillary zone electrophoresis) is performed first, followed by FIA. The electrosmotically pumped fluid can be hydrostatically coupled to propel other fluids that are electrically too conductive or too resistive to be directly pumped.

An overview is given of research activities in the field of fluid components or systems built with microfabrication technologies. This review focuses on the fluidic behaviour of the various devices, such as valves, pumps and flow sensors as well as the possibilities and pitfalls related to the modelling of these devices using simple flow theory. Finally, a number of microfluidic systems are described and comments on future trends are given.

Although proper buffering is essential for repeatable separations in capillary electrophoresis, electroosmotic flow can be developed in many solvents without the addition of buffer components. In this study we detected electroosmotic flow in fused silica capillary in the following solvents without the addition of ionic species: water, deuterium oxide, acetonitrile, acetone, 2-butanone, formamide, N-methylformamide, N,N-dimethylformamide, methanol, ethanol, 1-propanol, dimethyl sulfoxide, ethyl acetate, tetrahydrofuran, and morpholine. The electroosmotic flow was in the range of 3.4×10−9–1.8×10−m2/V s. About 100 times weaker electroosmosis was detected in glacial acetic acid. We also detected electroosmotic flow in N-methylacetamide, which is solid at room temperature (melting range 28–30°C). No electroosmotic flow was found in inert solvents such as n-hexane and chloroform. The zeta potential of the fused silica capillary was calculated for the solvents where electroosmotic flow was found. The zeta potential of the capillary wall was also studied in water–methanol mixtures, where it has a maximum at about 40% methanol concentration. ©1999 John Wiley & Sons, Inc. J Micro Sep 11: 199–208, 1999

Four Capillary Electroseparation methods are distinguished. All have an ultimate efficiency limited only by axial diffusion and are in principle capable of achieving 106 plates in <1 hour.
The main limitation to performance arises from Ohmic heating of the electrolyte. While forced convection at 10ms−1 is recommended to keep tubes cool, the parabolic temperature profile within the electrolyte limits the tube bore which can be used. A simple limiting expression is derived: (dc/μm)3 (E/kV m−1)3 (c/mol dm−3) <3.3×109.

A hybrid miniature chemical analysis system is realized using two
piezoelectrically driven silicon micro-pumps and a separate glass flow
through a cell with a potassium-sensitive ISFET. The measurement
protocol is such that the sample solution does enter the detector but
does not pass the sensitive pump valves, thus improving the practical
applicability of the system. During its operation the sensor is
continuously calibrated with a very low consumption of calibrating
solution. With a measurement rate of 4 samples per minute the
consumption of calibrant is only 2 ml/hour

The effects of the electrical double layer near the solid/liquid interface on liquid flow through a rectangular microchannel are analyzed in this work. Based on the Debye–Hückel approximation, a linear solution of a two-dimensional Poisson–Boltzmann equation governing the electrical potential distribution in the cross-section of rectangular channels is presented to describe the electrical double-layer field near the solid/liquid interface. An additional body force originating from the presence of the electrical double layer is considered to modify the conventional Navier–Stokes equation. By using the Green's function formula, an exact solution to this modified Navier–Stokes equation in rectangular microchannels is obtained. The effects of the electrical double-layer field, channel size and the electrical static charge on the fluid velocity distribution, streaming potential, volume flow rate, friction coefficient and apparent viscosity have been discussed. The results show that, for diluted liquid, the liquid flow in rectangular microchannels is influenced significantly by the electrical double layer and hence deviates from the flow characteristics described by classical fluid mechanics.

The effects of the electric double layer near the solid–liquid interface and the flow induced electrokinetic field on the pressure-driven flow and heat transfer through a rectangular microchannel are analyzed in this work. The electric double layer field in the cross-section of rectangular microchannels is determined by solving a non-linear, two-dimensional Poisson–Boltzmann equation. A body force caused by the electric double field and the flow-induced electrokinetic field is considered in the equation of motion. For steady-state, fully-developed laminar flows, both the velocity and the temperature fields in a rectangular microchannel are determined for various conditions. The flow and heat transfer characteristics with⧹without consideration of the electrokinetic effects are evaluated. The results clearly show that, for aqueous solutions of low ionic concentrations and a solid surface of high zeta potential, the liquid flow and heat transfer in rectangular microchannels are significantly influenced by the presence of the electric double layer field and the induced electrokinetic flow.

Thermal band broadening is known to be caused by the temperature dependence of ionic mobility. This dependence also strongly influences the temperature of the capillary by providing positive feedback between the temperature and power density. Previous thermal models of capillary electrophoresis have not fully considered this “autothermal effect”. We show that it always causes a capillary to run hotter than is predicted by a constant conductivity model; temperature excursions two times greater are typical.We propose that the thermally induced parabolic distortion of the migration velocity can be countered with an opposing Poiseuille (pressure-driven) flow. Dispersion calculations indicate that it may be possible to obtain plate numbers in excess of 106 m-1 even in very large bore (400 μm) capillaries.

The characteristics of electroosmotic flow in rectangular microchannels were investigated in this paper. A 2D Poisson–Boltzmann equation and the 2D momentum equation were used to model the electric double layer field and the flow field. The numerical solutions show significant influences of the channel cross-section geometry (i.e. the aspect ratio) on the velocity field and the volumetric flow rate. Also, the numerical simulation of the electroosmotic flow reveals how the velocity field and the volumetric flow rate depend on the ionic concentration, zeta potential, channel size and the applied electrical field strength.

A new tool for imaging both scalar transport and velocity fields in liquid flows through microscale structures is described. The technique employs an ultraviolet laser pulse to write a pattern into the flow by uncaging a fluorescent dye. This is followed, at selected time delays, by flood illumination with a pulse of visible light which excites the uncaged dye. The resulting fluorescence image is collected onto a sensitive CCD camera. The instrument is designed as an oil immersion microscope to minimize beam steering effects. The caged fluorescent dye is seeded in trace quantities throughout the active fluid, thus images with high contrast and minimal distortion due to any molecular diffusion history can be obtained at any point within the microchannel by selectively activating the dye in the immediate region of interest. We report images of pressure- and electrokinetically driven steady flow within round cross section capillaries having micrometer scale inner diameters. We also demonstrate the ability to recover the velocity profile from a time sequence of these scalar images by direct inversion of the conserved scalar advection-convection equation.

Independent control of electroosmosis is important for separation science techniques such as capillary zone electrophoresis and for the movement of fluids on microdevices. A capillary electrophoresis microdevice is demonstrated which provides more efficient control of electroosmosis with an applied external voltage field. The device is fabricated in a glass substrate where a 5.0 cm separation channel (30 microm wide) is paralleled with two embedded electrodes positioned 50 microm away in the substrate. With this structure, greatly reduced applied external potentials (< or = 120 V compared to tens of kilovolts) still effectively altered electroosmosis. The efficiency for the control of electroosmosis by the applied external field is improved by approximately 40 times over published values.

This contribution addresses the problem of solute dispersion in a free convection electrophoretic cell for the batch mode of operation, caused by the Joule heating generation. The problem is analyzed by using the two-problem approach originally proposed by Bosse and Arce (Electrophoresis 2000, 21, 1018-1025). The approach identifies the carrier fluid problem and the solute problem. This contribution is focused on the latter. The strategy uses a sequential coupling between the energy, momentum and mass conservation equations and, based on geometrical and physical assumptions for the system, leads to the derivation of analytical temperature and velocity profiles inside the cell. These results are subsequently used in the derivation of the effective dispersion coefficient for the cell by using the method of area averaging. The result shows the first design equation that relates the Joule heating effect directly to the solute dispersion in the cell. Some illustrative results are presented and discussed and their implication to the operation and design of the device is addressed. Due to the assumptions made, the equation may be viewed as an upper boundary for applications such as free flow electrophoresis.

The theory behind and operation of an electroosmotically induced hydraulic pump for microfluidic devices is reported. This microchip functional element consists of a tee intersection with one inlet channel and two outlet channels. The inlet channel is maintained at high voltage while one outlet channel is kept at ground and the other channel has no electric potential applied. A pressure-induced flow of buffer is created in both outlet channels of the tee by reducing electroosmosis in the ground channel relative to that of the inlet channel. Spatially selective reduction of electroosmosis is accomplished by coating the walls of the ground channel with a viscous polymer. The pump is shown to differentially transport ions down the two outlet channels. This ion discrimination ability of the pump is examined as a function of an analyte's electrophoretic velocity. In addition, we demonstrate that an anion can be rejected from the ground channel and made to flow only into the field-free channel if the electrophoretic velocity of the anion is greater than the pressure-generated flow in the ground channel. The velocity threshold at which anion rejection occurs can be selectively tuned by changing the flow resistance in the field-free channel relative to the ground channel.

We have characterized electroosmotic flow in plastic microchannels using video imaging of caged fluorescent dye after it has been uncaged with a laser pulse. We studied flow in microchannels composed of a single material, poly(methyl methacrylate) (acrylic) or poly(dimethylsiloxane) (PDMS), as well as in hybrid microchannels composed of both materials. Plastic microchannels used in this study were fabricated by imprinting or molding using a micromachined silicon template as the stamping tool. We examined the dispersion of the uncaged dye in the plastic microchannels and compared it with results obtained in a fused-silica capillary. For PDMS microchannels, it was possible to achieve dispersion similar to that found in fused silica. For the acrylic and hybrid microchannels, we found increased dispersion due to the nonuniformity of surface charge density at the walls of the channels. In all cases, however, electroosmotic flow resulted in significantly less sample dispersion than pressure-driven flow at a similar velocity.

The hydrodynamic problem of electroosmotic flow in a cylindrical capillary with random zeta potential is solved in the limit of small Deybe length and low Reynolds number. Averages are defined over multiple experiments and the mean axial velocity is found to be a plug flow. The variance of the velocity exhibits parabolic-like variation across the capillary. Average concentrations of samples transported by the flow are approximated by defining an effective diffusivity coefficient. Theoretical formulas for the average concentration are supported by numerical experiments.

The transient aspects of electroosmotic flow in a slit microchannel are studied. Exact solutions for the electrical potential profile and the transient electroosmotic flow field are obtained by solving the complete Poisson-Boltzmann equation and the Navier-Stokes equation under an analytical approximation for the hyperbolic sine function. The characteristics of the transient electroosmotic flow are discussed under influences of the electric double layer and the geometric size of the microchannel.

Microfluidics–A review Micropump and sample injector for integrated chemical analysis systems

- P Gravensen
- J Branebjerg
- O S Jensen

P. Gravensen, J. Branebjerg, O.S. Jensen, Microfluidics–A review, J. Micromech. Microeng. 3 (1993) 168–182. [2] S. Shoji, S. Nakagawa, M. Esashi, Micropump and sample injector for integrated chemical analysis systems, Sens. Actuators, A 21 (1990) 189–1192.

Fluorescence-based visualization of electro-osmotic flow in microfabricated systems, Pro-[31] The second fundamental problem in heat transfer of laminar forced convection

- D W Arnold
- P H L N Paul
- Tao

D.W. Arnold, P.H. Paul, Fluorescence-based visualization of electro-osmotic flow in microfabricated systems, Pro-[31] L.N. Tao, The second fundamental problem in heat transfer of laminar forced convection, J. Appl. Mech. 415–420 (1962).

Liquid transport in rect-angular microchannels by electro-osmotic pumping, Col-loids Surf

- S Arulanandam
- L Dongqing

S. Arulanandam, L. Dongqing, Liquid transport in rect-angular microchannels by electro-osmotic pumping, Col-loids Surf. A 161 (2000) 89–102.

Physicochemical Hydrodynamics

- R.F Probstein
- R.F Probstein

Viscous dissipation effects on thermal entrance heat transfer in laminar and turbulent pipe flows with uniform wall temperature, ASME Paper No. 74-HT-50

- J W Ou
- K C Cheng

J.W. Ou, K.C. Cheng, Viscous dissipation effects on thermal entrance heat transfer in laminar and turbulent pipe flows with uniform wall temperature, ASME Paper No. 74-HT-50, ASME, New York, 1974.

Microfluidics–A review

- P Gravensen
- J Branebjerg
- O S Jensen

P. Gravensen, J. Branebjerg, O.S. Jensen, Microfluidics–A
review, J. Micromech. Microeng. 3 (1993) 168–182.

Microfluidics–A review

- Gravensen