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Unsteady axial viscoelastic pipe flows
Abstract
The main objective of this work is to examine in detail basic unsteady pipe flows and to investigate any new physical phenomena. We take the viscoelastic upper-convected Maxwell fluid as our non-Newtonian model and consider the flow of such a fluid in pipes of uniform circular cross-section in the following three cases: 1.(a) when the pressure gradient varies exponentially with time;2.(b) when the pressure gradient is pulsating;3.(c) a starting flow under a constant pressure gradient.In the first problem we looked separately at the pressure gradient rising exponentially with time and falling exponentially with time, i.e. the pressure gradient is proportional to e±α2τ. The behaviour of the flow field depends to a large extent on β where β2 = α2(1 ± Hα2) with H being the quotient of the Weissenberg and Reynolds numbers. In both cases for small |βη|, η being the radial distance from the axis, the velocity profiles are seen to be parabolic. However, for large |βη| the flows are vastly different. In the case of increasing pressure gradient the flow depicts boundary-layer characteristics while for decreasing pressure gradient the velocity depends on the wall distance.The case of a pulsating pressure gradient is investigated in the second problem. Here the pressure gradient is proportional to cos nτ. Again the flow depends to a large extent on a parameter β (β2 = in − n2H). For small values of |βη| the velocity profile is parabolic. However, it is found that, unlike Newtonian fluids, the velocity distribution for the upper-convected Maxwell fluid is not in phase with the exciting pressure distribution. In the case of large |βη| the solution displays a boundary-layer characteristic and the phase of the motion far from the wall is shifted by half a period.The final problem examines a flow that is initially at rest and then set in motion by a constant pressure gradient. A closed form solution has been obtained with the aid of a Fourier-Bessel series. The variation of the velocity across the pipe has been sketched and comparison made with the classical solution.
... The authors applied Laplace and finite Hankel transforms for sequential fractional derivatives to obtain a solution. In [11], the authors obtained a closedform solution for an upper-convected Maxwell fluid in a pipe by considering the pressure pulse as exponential, pulsating and constant. They derived results that were obtained with the aid of a Fourier-Bessel series. ...
... We assume that the exact solution of Eq. (10) with boundary conditions Eqs. (11) and (12) consists of the steady state solution u s , that satisfies the boundary conditions plus a transient solution u t , that is, ...
This study aimed to develop a mathematical model of an unsteady Burgers’ fluid in a circular cylinder with a trapezoidal pressure waveform described by an infinite Fourier series. An analytical solution was obtained for the governing equation using the Bessel transform method together with similarity arguments. The validity of the solution was verified using a numerical inversion method based on Stehfest’s method. Limiting cases were considered to examine the fluid flow performance of different fluids. Our results show that the Newtonian and Oldroyd-B fluids performed similar velocity time variation for the trapezoidal waveform of oscillating motion, whereas the velocity time variation was different for Maxwell and Burger’s fluids. Moreover, it is evident that the material constant of a Burgers’ fluid is another important factor that affects flow performance in an oscillating flow.
... The paper published by Boyko and Stone, 12 which considers a pressuredriven channel flow of an Oldroyd-B fluid, also provides a comprehensive review of pressure-driven viscoelastic flows. Other works which consider either steady-state or transient cases of pressure-driven viscoelastic flows are those of Rahaman and Ramiksoon, 13 Cruz et al., 14,15 Dressler and Edwards, 16 Letelier and Siginer 17,18 who treated the problem analytically or semianalytically, and those of Keshtiban et al. 19 and Yapici et al., 20 who treated the problem numerically, while Siline and Leonov 21 utilized both of the approaches. Instabilities present in various viscoelastic fluids is also a field of research, which has recently attracted considerable interest, and is comprehensively reviewed in a paper by Datta et al. 22 There has also been plenty of research concerning problems related to mathematical modeling of polymerization in various circumstances, with frontal polymerization receiving a lot of focus due to its presence in various phenomena and industrial processes, such as manufacturing of composite materials, 3D printing, stereolithography, etc. Work by Suslick et al. 23 provides a very broad as well as detailed overview of both theoretical and experimental aspects of both thermaland photo-frontal polymerization. ...
In this paper, we consider a pressure-driven flow of a viscoelastic fluid in a straight rectangular channel undergoing a solidification phase change due to polymerization. We treat the viscoelastic response of the fluid with a model based on the formalism of variable-order calculus; more specifically, we employ a model utilizing a variable-order Caputo-type differential operator. The order parameter present in the model is determined by the extent of polymerization induced by light irradiation. We model this physical quantity with a simple equation of kinetics, where the reaction rate is proportional to the amount of material available for polymerization and optical transmittance. We treat cases when the extent of polymerization is a function of either time alone or both position and time, and solve them using either analytical or semi-analytical methods. Results of our analysis indicate that in both cases, solutions evolve in time according to a variable-order decay law, with the solution in the first case having a hyperbolic cosine-like spatial dependence, while the spatial dependence in the second case conforms to a bell curve-like function. We infer that our treatment is physically sound and may be used to consider problems of more general viscoelastic flows during solidification, with the advantage of requiring fewer experimentally determined parameters.
... Their solution was further extended to non-Newtonian fluids using the second-grade fluid, Maxwell, and Oldroyd-B models in the following literature just to name a few [37,[40][41][42][43][44][45][46][47][48][49][50][51]. ...
In this study, we investigate the mobilization of mucus in a cylindrical tube of constant cross-section subjected to a Small Amplitude Oscillatory Shear (SAOS) assuming the viscoelastic behavior of mucus is described by the Oldroyd-B constitutive equation. The Laplace transform method was adopted to derive expressions for the velocity profile, average velocity, instantaneous flowrate and mean flowrate (i.e., mobilization) of the mucus within the tube. Additionally, a 2D finite element model (FEM) was developed using COMSOL Multiphysics software to verify the accuracy of the derived analytical solution considering both a Newtonian and an Oldroyd-B fluid. Furthermore, sensitivity studies were performed to evaluate the influence of the vibration frequency, vibration amplitude, mean relaxation time, and zero-shear viscosity on mucus mobilization. Results prove that the analytical and numerical results are within acceptable error tolerance and thus in excellent agreement. Besides, the parametric studies indicate a 48%, 57%, and 343% improvement in mucus mobilization when the mean relaxation time, vibration amplitude, and vibration frequency, respectively were increased by a factor of 6. Conversely, the mucus mobilization decreased by 25% when the zero-shear viscosity was increased by a factor of 6. In general, this study confirms that mucus mobilization in a tube can be improved by increasing the magnitude of vibration amplitude and vibration frequency. Similarly, the larger the magnitude of mucus mean relaxation time the better the mucus mobilization when the tube is subjected to boundary vibrations. Finally, mucus mobilization decreases as the magnitude of mucus viscosity increases.
... He studied the problem by considering time dependent shear stress on cylinders and impact of physical parameters on the velocity and shear stress. Rahaman and Ramkissoon [4] (1995) examined in detail the basic unsteady pipe flows to investigate the new physical phenomena. He considered the viscoelastic upper-convected Maxwell fluid as non-Newtonian model in pipes of uniform circular cross-section and studied the following three cases to give the results in close form by using Bessel equations: (a) when the pressure gradient varies exponentially with time; (b) when the pressure gradient is pulsating; (c) a starting flow under a constant pressure gradient. ...
Keeping in view of the complex fluid mechanics in bio-medicine and engineering, the Burgers' fluid with a fractional derivatives model analyzed through a rotating annulus. The governing partial differential equation solved for velocity field and shear stress by using integral transformation method and using Bessel equations. The transformed equation inverted numerically by using Gaver-Stehfest's algorithm. The approximate analytical solution for rotational velocity, and shear stress are presented. The influence of various parameters like fractional parameters, relaxation and retardation time parameters material constants, time and viscosity parameters are drawn numerically. It is found that the relaxation time and time helps the flow pattern, on the other hand other material constants resist the fluid rotation. Fractional parameters effect on fluid flow is opposite to each other. Finally, to check the validity of the solution there are comparisons for velocity field and shear stress for obtained results with an other numerical algorithm named Tzou's algorithm.
... Later, Womersley [3,4] published a paper focusing on flows in arteries, in which it is stressed the importance of the dimensionless parameter used by Sexl and Lambossy, and that became known as Womersley number. The oscillatory flow of non-Newtonian fluids has been studied theoretically by Pipkin [5], Etter and Schowalter [6], and Rahaman and Ramkissoon [7] for viscoelastic fluids, among others. Uchida [8] studied the flow of a Newtonian fluid due to pulsatile pressure gradient (i.e., superposition of oscillatory and constant pressure gradients). ...
The flow between two parallel plates driven by a pulsatile pressure gradient was studied analytically with a second-order velocity expansion. The resulting velocity distribution was compared with a numerical solution of the momentum equation to validate the analytical solution, with excellent agreement between the two approaches. From the velocity distribution, the analytical computation of the discharge, wall shear stress, discharge, and dispersion enhancements were also computed. The influence on the solution of the dimensionless governing parameters and of the value of the rheological index was discussed.
... The flow between rotating cylinders, start form rest, has applications in the food industry and is one of the most important problem of motion near rotating bodies. Unsteady pressuredriven flow of a classical Maxwell fluid in a pipe is studied by Rahaman and Ramkissoon [1]. Hayat et al. [2] obtained the velocity fields for some simple flows of Oldroyd-B fluids by using Fourier transform, also they studied the unsteady flows due to non-coaxial rotations of a disc fluid at infinity in [3][4][5][6]. ...
... Actually, the transient response of the viscoelastic fluid to external excitations will provide a new way to implement reliable microfluidic circuit components with dynamic modulation functions. The response of viscoelastic fluid has been previously studied for a start-up flow under an impulse or constant pressure gradient at rest [26][27][28][29] . The start-up flow for non-Newtonian fluid are particularly for the verification of the numerical methods, which are used for calculating transient flow responses [30][31][32] . ...
Transient flow responses of viscoelastic fluids to different external body forces are studied. As a non-Newtonian fluid, the viscoelastic fluid exhibits significant elastic response which does not raise in Newtonian fluid. Here, we investigate the transient response of a viscoelastic Poiseuille flow in a two-dimensional channel driven by external body forces in different forms. The velocity response is derived using the Oldroyd-B constitutive model in OpenFOAM. Responses in various forms like damped harmonic oscillation and periodic oscillation are induced and modulated depending on the fluid intrinsic properties like the viscosity and the elasticity. The external body forces like constant force, step force and square wave force are applied at the inlet of the channel. Through both time domain and frequency domain analysis on the fluid velocity response, it is revealed that the oscillation damping originates from the fluid viscosity while the oscillation frequency is dependent on the fluid elasticity. The velocity response of the applied square waves with different periods shows more flexible modulation signal types than constant force and step force. An innovative way is also developed to characterize the relaxation time of the viscoelastic fluid by modulating the frequency of the square wave force.
... An excellent computational study of pulsatile flow dwelling on nonlinear flow aspects was presented by Hung [43]. Rahaman and Ramkissoon [44] have studied the unsteady axial viscoelastic pipe flows under the influence of periodic pressure gradient. Sharp [45] studied the effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes. ...
In this paper, the problem of magnetohydrodynamic (MHD) human blood flow between two concentric cylinders filled with porous medium is described using Giesekus model. A uniform circular magnetic field is applied normal to the blood flow which is driven by an unsteady pulsating pressure gradient. The momentum equation and the Giesekus equation of state that characterize the fluid are solved considering the modified Darcy's law to account the resistance offered by the porous medium. Exact solutions for the axial velocity and volume flow rate are presented. The effects of various parameters on the blood pulsatile flow characteristics such as geometry, viscoelasticity, magnetic and permeability parameters are presented through graphical illustrations. The main results of the present study is that, the blood velocity distribution increases with an increase of permeability parameter of the porous medium, while it decreases as the magnetic and frequency parameters are increase. Also it is found that the effect of the magnetic field is to decrease the blood flow rate. The decrease in the flow rate is due to an increase in the apparent viscosity of blood due to the magnetic field. The results are presented graphically and the effects of the various parameters on the human blood flow rate are discussed and compared with reported literature values.
... Waters and King [3] discussed the Oldroyd-B fluid in a circular tube by taking Poiseuille flow into account. The investigation of basic unsteady pipe flow and viscoelastic upper-convected Maxwell fluid in uniform circular cross section, is available in [4]. They used the Fourier Bessel series to obtain the closed form solutions. ...
The solutions for velocity and stress are derived by using the methods of Laplace transformation and Modified Bessel’s equation for the rotational flow of Burgers’ fluid flowing through an unbounded round channel. Initially, supposed that the fluid is not moving with t = 0 and afterward fluid flow is because of the circular motion of the around channel with velocity ΩRtp with time positively grater than zero. At the point of complicated expressions of results, the inverse Laplace transform is alternately calculated by “Stehfest’s algorithm” and “MATHCAD” numerically. The numerically obtained solutions in the terms of the Modified Bessel’s equations of first and second kind, are satisfying all the imposed conditions of given mathematical model. The impact of the various physical and fractional parameters are also indeed and so presented by graphical demonstrations.
... Srivastava [12] was the first who studied the motions of Maxwell fluids through a circular cylinder and obtained analytical solutions. Other exact solutions for motions of Maxwell fluids in cylindrical domains have been obtained by Rahaman and Ramkisson [13], Fetecau and Corina Fetecau [14], Vieru et al. [15], Jamil and Fetecau [16], Jamil et al. [17], Zeb et al. [18], and Corina Fetecau et al. [19]. Recently, Nehad et al. [20] provided the first general solutions for rotational motions of rate type fluids between circular cylinders. ...
Abstract The rotational motion of fractional Maxwell fluids in an infinite circular cylinder that applies a time-dependent but not oscillating couple stress to the fluid is investigated using the integral transform technique. Such a flow model was not analyzed in the past both for ordinary and fractional rate type fluids. This is due to their constitutive equations which contain differential expressions acting on the shear stresses. The obtained solutions fulfill all the enforced initial and boundary conditions and are easily reduced to the solutions of Newtonian or ordinary Maxwell fluids having similar motion. At the end, the influence of pertinent parameters on velocity and shear stress variations is graphically underlined and discussed.
... The fractional constraints for Burgers' fluid (BFFD) can be established by replacing the total time derivative in Eqs. (3) and (4) with fractional derivatives, by the Caputo fractional differential operator [14,16] ...
In this research articale, we found analytical results for the flow of a Burgers' fluid within a round pipe having infinite length. The fractional derivatives approach is used by means of integral transform technique. Fluid motion has been induced by the circular motion of the pipe about its axis. Series form results are accomplished in the form of generalized G-function. Limiting results are also obtained from general solutions. Numerical computations and graphical discussion were made to observe influence of non-integer order fractional variables on the fluid flow.
... The fractional constraints for Burgers' fluid (BFFD) can be established by replacing the total time derivative in Eqs. (3) and (4) with fractional derivatives, by the Caputo fractional differential operator [14,16] ...
In this research articale, we found analytical results for the flow of a Burgers' fluid within a round pipe having infinite length. The fractional derivatives approach is used by means of integral transform technique. Fluid motion has been induced by the circular motion of the pipe about its axis. Series form results are accomplished in the form of generalized G-function. Limiting results are also obtained from general solutions. Numerical computations and graphical discussion were made to observe influence of non-integer order fractional variables on the fluid flow.
... The fractional constraints for Burgers' fluid (BFFD) can be established by replacing the total time derivative in Eqs. (3) and (4) with fractional derivatives, by the Caputo fractional differential operator [14,16] ...
In this research articale, we found analytical results for the flow of a Burgers' fluid within a round pipe having infinite length. The fractional derivatives approach is used by means of integral transform technique. Fluid motion has been induced by the circular motion of the pipe about its axis. Series form results are accomplished in the form of generalized G-function. Limiting results are also obtained from general solutions. Numerical computations and graphical discussion were made to observe influence of non-integer order fractional variables on the fluid flow.
... The results of Poiseuille flow in a rotating pipe for Oldroyd-B fluid are given in [2]. Rahaman et al. [3] obtained solutions for viscoelastic upper-convected Maxwell fluid in a rotational pipe-like domain using the Fourier-Bessel series. ...
Exact solutions for velocity field and tangential stress for rotational flow of a generalized Burgers’ fluid within an infinite circular pipe are derived by using the methods of Laplace and finite Hankel transformations. Firstly we take the position of fluid at rest and then the fluid flow due to the rotation of the pipe around the axis of flow having time dependant angular velocity. The exact solutions are presented in terms of the generalized Ga,b,c(.,t)-functions. The corresponding results can be freely specified for the same results of Burgers’, Oldroyd B, Maxwell, second grade and Newtonian fluids (performing the same motion) as particular cases of the results obtained earlier. The impact of the different parameters, individually and in comparison, are represented by graphical demonstrations. Secondly the numerical solutions for velocity and stress are also obtained with the help of Laplace transformation, Gaver Stehfest's algorithm and MATHCAD. Finally a comparison of both methods for the same problem is done and shows the consistency of results.
... By discussing about the Poiseuille flow in circular pipe, Waters and King [2] find results for Oldroyd-B fluid. Rahaman et al. [3] investigated about the viscoelastic upper-convected Maxwell fluid within round surface. For this purpose the Fourier Bessel series was used. ...
In this paper, we study the unsteady flow of fractional second grade fluid between an infinite annulus. Initially both, fluid and cylinders are at rest and motion is due to the rotation of the inner cylinder about its axis. In the governing equations we used the modified fractional derivative recently given by Caputo and Fabrizio, which is more suitable for viscoelastic as compare to usual fractional derivative. Our goal is to determine the solutions by usning integral transform techniques. The obtained solution is represented in terms of generalize G function which satisfy all the applied initial and boundary conditions. Furthermore, substituting favorable limits of different parameters, we get similar solutions for ordinary second grade fluid and recover well known solution for Newtonian fluid. For the validity of the result we made a good comparison between existing and our analytical solutions with the help of tables and graphs, which shows that both the solutions are equivalent. In the end the effect of the physical parameters on the fluid flow are illustrated graphically.
... The constitutive equations of non-Newtonian and rate type fluids are studied by Rajagopal [5] and differential type fluids have been studied by Dunn and Rajagopal [6]. Unsteady pressure flow of a classical Maxwell fluid in a cylinder is discussed by Rahaman and Ramkissoon [7]. In particular, the predictions of a fractional Maxwell model with the linear viscoelastic date in glass transition and a relaxation zeros is given by Palade et al. [8]. ...
In this paper, the velocity field and the time dependent shear stress of Maxwell fluid with Caputo fractional derivatives in an infinite long circular cylinder of radius R is discussed. The motion in the fluid is produced by the circular cylinder. The fluid is initially at rest and at time t=0+, cylinder begins to oscillate with the velocity fsinωt, due to time dependent shear stress acting on the cylinder tangentially. The hybrid technique used in this paper for the solution of the problem has less computational efforts and time cost as compared to other commonly used methods. The obtained solutions are in transformed domain, which are expressed in terms of modified Bessel functions I0(·) and I1(·). The inverse Laplace transformation has been calculated numerically by using MATLAB package. The semi analytical solutions for Maxwell fluid with fractional derivatives are reduced to the similar solutions for Newtonian and ordinary Maxwell fluids as limiting cases. In the end, numerical simulations have been performed to analyze the behavior of fractional parameter α, kinematic viscosity ν, relaxation time λ, radius of the circular cylinder R and dynamic viscosity μ on our obtained solutions of velocity field and shear stress.
... The computational study presented by Hung [15] is considered one of the excellent study in pulsatile nonlinear flow. Rahaman and Ramkissoon [16] studied the unsteady axial viscoelastic pipe flows under the influence of periodic pressure gradient. Bestman and Njoku [17] constructed a solution for hydrodynamic channel flow of viscous fluid induced by tooth pulses. ...
In this paper the unsteady magnetohydrodynamic (MHD) flow of Oldroyd–B fluid through a tube filled with porous medium was investigated. A uniform circular magnetic field was applied perpendicular to the flow direction. The fluid was driven by an axial time dependent pressure gradient which increased or decreased or pulsated with time. The governing equations were accurately solved taking in consideration the modified Darcy's law to count the resistance presented by the porous medium. The obtained velocity profiles had Bessel and modified Bessel functions of zero order representations. The effects of the various parameters on the flow characteristics such as non–Newtonian, magnetic and permeability parameters had been studied and discussed graphically. Figures were provided to compare the velocity profile of Newtonian fluid and Maxwell fluid with Oldroyd fluid. It was found that the velocity distribution increased with the increase of permeability parameter of the porous medium, but it decreased with the increased magnetic and frequency parameters. The flow was parabolic for small frequencies but it had boundary layer characters at large frequencies. Solutions of Maxwell and Newto-nian fluids were appeared as limited cases for the present analysis.
... The influence of thermal radiation and Joule heating on MHD flow of Maxwell fluid in the presence of thermophoresis has been considered by Hayat and Qasim (2010). Unsteady pipe flows of Maxwell fluid has been investigated by Rahaman et al. (1995) for different types of pressure gradients. Unsteady flow of a Maxwell Fluid over a Stretching Surface in the Presence of a Heat Source/Sink has been investigated by Mukhopadhyay (2012) and obtained a numerical solution for the flow field equations by the shooting method. ...
This article presents an unsteady incompressible Upper Convected Maxwell (UCM) fluid flow with temperature dependent thermal conductivity between parallel porous plates which are maintained at different temperatures varying periodically with time. Assume that there is a periodic suction and injection at the upper and lower plates respectively. The governing partial differential equations are reduced to non linear ordinary differential equations by using similarity transformations and the solution is obtained using differential transform method. The effects of various fluid and geometric parameters on the velocity components, temperature distribution and skin friction are discussed in detail through graphs.
... 2. The Maxwell and Oldroyd fluids as well as other models of incompressible viscoelastic fluids have been studied by numerous authors [35][36][37][38][39][40] (see also [22,[41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]). The operators K and L corresponding to the Maxwell fluid equations [22,[36][37][38][39][40][41][42][43][44] (creeping flows) and linearized hyperbolic Navier-Stokes equations [57][58][59] are defined in row 5, where τ is the relaxation time. ...
The present paper provides a systematic treatment of various decomposition methods for linear (and some model nonlinear) systems of coupled three-dimensional partial differential equations of a fairly general form. Special cases of the systems considered are commonly used in applied mathematics, continuum mechanics, and physics. The methods in question are based on the decomposition (splitting) of a system of equations into a few simpler subsystems or independent equations. We show that in the absence of mass forces the solution of the system of four three-dimensional stationary and nonstationary equations considered can be expressed via solutions of three independent equations (two of which having a similar form) in a number of ways. The notion of decomposition order is introduced. Various decomposition methods of the first, second, and higher orders are described. To illustrate the capabilities of the methods, more than fifteen distinct systems of coupled 3D equations are discussed which describe viscoelastic incompressible fluids, compressible barotropic fluids, thermoelasticity, thermoviscoelasticity, electromagnetic fields, etc. The results obtained may be useful when constructing exact and numerical solutions of linear problems in continuum mechanics and physics as well as when testing numerical and approximate methods for linear and some nonlinear problems.
... 2. The Maxwell and Oldroyd fluids as well as other models of incompressible viscoelastic fluids have been studied by numerous authors [35][36][37][38][39][40] (see also [22,[41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]). The operators K and L corresponding to the Maxwell fluid equations [22,[36][37][38][39][40][41][42][43][44] (creeping flows) and linearized hyperbolic Navier-Stokes equations [57][58][59] are defined in row 5, where τ is the relaxation time. ...
Во второй части статьи предлагаются общие методы декомпозиции, обобщающие классические представления. Получают развитие декомпозиции систем линейных и модельных нелинейных дифференциальных уравнений с частными производными, возникающих в механике сплошных сред, в частности в теории упругости, термоупругости, пороупругости и вязкоупругости, систематический подход к декомпозиции уравнений механики сплошных сред. Описаны несимметричный и симметричный методы декомпозиции различных классов трехмерных линейных (и модельных нелинейных) систем уравнений, которые используются в теории упругости, термоупругости и пороупругости, в механике вязких и вязкоупругих несжимаемых жидкостей и сжимаемых баротропных газов. Эти методы основаны на расщеплении систем связанных уравнений на несколько более простых независимых уравнений и использовании двух функций тока. Показано, что при отсутствии массовых сил любое решение рассматриваемых стационарных и нестационарных трехмерных систем выражается через решения двух независимых уравнений. Предложены методы прямой декомпозиции, не требующие разложения правой части системы уравнений на составляющие. Предложены обобщения рассмотренных методов декомпозиции на системы высоких порядков, а также на специальные классы модельных нелинейных уравнений. Даны примеры декомпозиции конкретных систем. Формулы и отдельные независимые уравнения, приведенные в работе, существенно упрощают качественное исследование и интерпретацию наиболее важных физических свойств широкого класса систем связанных уравнений механики сплошной среды и позволяют изучать их волновые и диссипативные свойства. Приведенные результаты можно использовать для точного интегрирования линейных систем механики, а также для тестирования численных методов, применяемых для решения нелинейных уравнений механики сплошных сред.
[In the second part of work presents the general decomposition methods for systems of linear partial differential equations that arise in continuum mechanics, in particular, in the theory of elasticity and thermoelasticity and poroelasticity. A systematic approach to the decomposition of the equations of continuum mechanics is proposed. Asymmetrical and symmetrical decomposition methods for various classes of three-dimensional linear (and model nonlinear) systems of equations arising in the theory of elasticity, thermoelasticity, and thermoviscoelasticity, the mechanics of viscous and viscoelastic incompressible and compressible barotropic gas are described. These methods are based on the decomposition of systems of coupled equations into several simpler independent equations and the use of two stream functions. It is shown that in the absence of body forces any solution of considered steady and unsteady three-dimensional systems is expressed in terms of solutions of two independent equations. The methods of direct decomposition that do not require expansion of the right hand side of the equations into the components are proposed. A generalization of the considered methods to the decomposition of higher orders systems of equations, as well as to special classes of model nonlinear equations are obtained. The examples of the decomposition of specific systems are given. Formulas and split equations given in the work significantly simplify the qualitative study and the interpretation of the most important physical properties of a wide class of coupled systems of equations for continuum mechanics and allow studying their wave and dissipative properties. These results can be used for the exact integration of linear systems of mechanics, as well as for testing of numerical methods for nonlinear equations of continuum mechanics.]
... In the meantime a lot of papers regarding such motions have been published. The interested readers can see for instance the papers7891011121314 and their related references. However, it is worthy pointing out that all above mentioned works dealt with problems in which the velocity is given on the boundary. ...
The velocity and the shear stress, corresponding to the unsteady flow of an Oldroyd-B fluid in an infinite circular cylinder subject to a time-dependent couple, are established by means of the Hankel transform. The similar solutions for Maxwell, second grade and Newtonian fluids can be obtained as limiting cases of general solutions. Finally, the influence of the material parameters on the velocity profile and the shear stress is spot-lighted by means of graphical illustrations.
... and this can be easily solved by separation of variables, expressing the solution with Bessel functions. 24 In particular, the Newtonian case is recovered when, in addition, λ 1 = 0. In such a case, the system (4.10) ...
The unsteady flow of a viscoelastic fluid in a straight, long, rigid pipe, driven by a suddenly imposed pressure gradient is studied. The used model is the Oldroyd-B fluid modified with the use of a nonconstant viscosity, which includes the effect of the shear-thinning of many fluids. The main application considered is in blood flow.
Two coupled nonlinear equations are solved by a spectral collocation method in space and the implicit trapezoidal finite difference method in time. The presented results show the role of the non-Newtonian terms in unsteady phenomena.
... Finite speeds of propagation are also provided by models of incompressible viscoelastic fluids [13][14][15][16][17][18][19]. A number of exact solutions and various problems for viscoelastic and other Non-Newtonian fluids have been considered in many studies (see, for example, [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]). ...
... For the past few years, attention has been given to study oscillatory flow of viscoelastic fluid in pipes. [1], [4], [9], [15], [17]. [12] Pointed that in the investigation of the behaviour of polymer melts, the main problem is to get the correct nonlinear behaviour in both shear and extension. ...
The flow of viscoelastic fluids in industries poses a number of challenges, not least from a
modelling point of view. Research is needed to further understand and be able to predict the flow behaviour of such fluids and to investigate ways of improving their processing. The
properties of viscoelastic fluids cannot be captured by the momentum and continuity
equations alone. As results, additional constitutive equation is needed for the characterisation of these fluids. Here, the study focuses on the momentum and continuity equation together with additional constitutive equation. These equations collectively form the model called the Oldroyd-B Model. We have derived a forward finite difference scheme for a creeping Oldroyd fluid flow. The derived scheme is then tested for convergence and error stability and thereafter, the scheme is used to simulate the flow in a circular curved pipe. In this simulation, we need to determine the pressure loss of the fluid when flowing in a curved pipe so as to determine the appropriate size of pump needed to keep the fluid flowing.
... During recent years quite a number of papers on this type of flow have been published. Unsteady, pressure-driven flow of a classical Maxwell fluid in a pipe was studied by Rahaman and Ramkissoon [5]. Exact solutions were obtained as infinite Fourier-Bessel series. ...
In this research article, the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative model through an infinite circular cylinder is studied by means of the finite Hankel and Laplace transforms. The motion is produced by the cylinder, that after time t=0+, begins to rotate about its axis with an angular velocity Ωt
p
. The solutions that have been obtained, presented under series form in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The corresponding solutions that have been obtained can be easily particularized to give the similar solutions for Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions.
The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of Second grade fluid with fractional derivatives as a limiting case of our general solutions corresponding to the Oldroyd-B fluid with fractional derivatives, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G-functions.
Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.
This paper reviews analytical solutions for the accelerated flow of an incompressible Newtonian fluid in a pipeline. This problem can be solved in one of two ways according to the (1) imposed pressure gradient or (2) flow rate. Laminar accelerated flow solutions presented in a number of publications concern cases where the two driving mechanisms are described by simple mathematical functions: (a) impulsive change; (b) constant change; (c) ramp change, etc. The adoption of a more complex and realistic description of the pressure gradient or flow rate will be associated with a profound mathematical complexity of the final solution. This is particularly visible with the help of the universal formula derived by several researchers over the years and discussed in this paper. In addition to the solutions strictly defined for laminar flow, an interesting extension of this theory is the theory of underlying laminar flow for the analysis of turbulent accelerated pipe flows (TULF model developed by García García and Alvariño). The TULF model extends the Pai model developed more than 60 years ago, which has been previously used for steady flows only. The discussed solutions extend the theory of analytical solutions of simplified two-dimensional Navier–Stokes equations and can be used not only to study the behavior of liquids during accelerating pipe flow but they can also be used to test the accuracy of commercial CFD codes.
In this study, we investigate the mobilization of mucus in a cylindrical tube of constant cross-section subjected to a Small Amplitude Oscillatory Shear (SAOS) assuming the viscoelastic behavior of mucus is described by the Oldroyd-B constitutive equation. The Laplace transform method was adopted to derive expressions for the velocity profile, average velocity, instantaneous flowrate and mean flowrate (i.e., mobilization) of the mucus within the tube. Additionally, a 2D finite element model (FEM) was developed using COMSOL Multiphysics software to verify the accuracy of the derived analytical solution considering both a Newtonian and an Oldroyd-B fluid.
In this paper, the unsteady two layer model of contaminant Oldroyd-B fluid through a circular pipe with porous medium are investigated. The governing equation solved by analytically using Bessel function. The effects of time varying pressure gradient on various parameters are discussed with through graphs.
In this note, the exact solutions of velocity field and associated shear stress corresponding to the flow of second-grade fluid in a cylindrical pipe, subject to a sinusoidal shear stress, are determined by means of Laplace and finite Hankel transform. These solutions are written as sum of steady-state and transient solutions, and they satisfy governing equations and all imposed initial and boundary conditions. The corresponding solutions for the Newtonian fluid, performing the same motion, can be obtained from our general solutions. At the end of this note, the effects of different parameters are presented and discussed by showing flow profiles graphically.
In previous study on the flow of fractional viscoelastic fluid between two coaxial cylinders, due to the complexity of computation, researchers commonly assume that the flow area is a rectangular pipe. This paper studies two-dimensional magnetohydrodynamic (MHD) flow and heat transfer of fractional Oldroyd-B nanofluid between two coaxial cylinders with variable pressure. The boundary layer fractional cylindrical governing equations are firstly formulated and derived. Meanwhile, the numerical solutions are obtained by using the newly developed finite difference method combined with L1-algorithm, whose validity is confirmed by the comparison with the exact solution constructed. Results show that the fractional derivative parameters α1, β and exponent parameter δ related to the wall temperature have significant effects on velocity and temperature fields. Specifically, the velocity decreases with the augment of α1, but increases as β rises up. The temperature profiles intersect each other for different δ. Meanwhile, increasing δ leads to an increase in temperature near the wall of inner tube, but a decline near the outer cylinder, which demonstrates increasing exponent parameter can enhance heat transfer.
In this paper, helices of a generalized Oldroyd-B fluid have been analyzed through a horizontal circular pipe. The circular pipe is taken in the form of a circular cylinder. The analytical solutions are determined for velocities and shear stresses due to the unsteady helical flow of a generalized Oldroyd-B fluid. The general solutions are derived by using finite Hankel and discrete Laplace transforms to satisfy the imposed conditions and the governing equations. The special cases of our general solutions are also perused performing the same motion for fractional and ordinary Maxwell fluid, fractional and ordinary second-grade fluid and fractional and ordinary viscous fluid as well. The graphical illustration is depicted in order to explore how the two velocities and shear stresses profiles are impacted by different rheological parameters, for instance, fractional parameter, relaxation time, retardation time, material non-zero constant, dynamic
viscosity and few others. Finally, ordinary and fractional operators have various similarities and differences on a circular pipe for helicoidal behavior of fluid flow.
The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.
This chapter considers the simple fluid of multiple integral-type models with fading memory. Secondary flows are determined in the case of laminar longitudinal flows in approximately triangular and square conduits, when the flow is driven by small-amplitude oscillatory pressure gradients. The chapter presents the mathematical background and determining the velocity field of laminar Newtonian unsteady flow in non-circular pipes. This is followed by an analysis of the pulsating flow in circular pipes of a viscoelastic fading memory fluid of the multiple integral type. A mathematical expression for the axial velocity is developed for flow in non-circular pipes driven by a pressure gradient oscillating around a non-zero mean. The analytical steps that lead to the determination of the transversal velocity field are developed. Plots that depict the main features of axial and secondary flow fields are also presented. Fully developed Newtonian flows in pipes are rectilinear when they are driven by a longitudinal pressure gradient, properties are kept constant, no wall-suction is considered, and the pipe is a straight cylinder of arbitrary cross-section. No secondary flows are possible under these conditions. The linearity of the equations of motion makes it possible to use a wide variety of analytical methods that yield either exact closed-form, or approximate solutions.
This work is to examine theoretically unsteady flow of viscoelastic fluid in a circular cylinder and to investigate the existence of steady solutions. We consider the flow is solely due to pressure gradient in following three forms: (i) sinusoidal; (ii) square; (iii) constant. Based on the pressure gradient described, three flow situations are solved and the exact analytical solutions are given in terms of Bessel functions. In all the three problems, it is found that the variation of the unsteady and steady solution mainly depends for small time, while for large time the two solutions are identical.
Oscillating motion of Oldroyd-B flow in a capillary tube at a triangular sine and cosine pressure waveform is considered. An analytical solution of velocity field is obtained for an oscillating laminar flow, which can be used to determine the existence of trapezoidal waveform effect on the flow. When both triangular sine and cosine components are dominate, the oscillating flow can result in significant trapezoidal waveform performance. The dimensionless oscillating frequency, amplitude, and fluid material parameters, are primary factors affecting the flow performance of an oscillating flow in a capillary tube.
By solving the momentum and energy equations of the oscillating flow with the fractional Maxwell model, which is driven by a sinusoidally varying pressure gradient, the general analytical solutions of the velocity profile, the temperature profile and the enhanced thermal diffusivity were obtained. Based upon the analysis of the enhanced thermal diffusivity, it is found that the parameters influenced the enhanced heat transfer for the problem are the Womersley number Wo, the Deborah number De, the dimensionless amplitude Δx/R and the Prandtl number of the fluid Pr. The heat transfer with fractional Maxwell model exhibits resonance phenomena similar to the flow of the viscoelastic fluid. And the number of the resonance peaks increases with the decrease of De, the frequency of the first peak decreases with the decrease of De. In addition, the frequencies where the resonance occurs are not concerned with the Pr and Δx/R.
By combining the momentum equation and the constitutive equation, a mathematical model for the pulsating laminar flow of the upper-convected Maxwell viscoelastic fluid under periodic pressure gradient in concentric annuli was set up. The velocity profile was obtained by using the intrinsic function method and the factors influencing the velocity profile were discussed. The results show that the velocity in annuli changes periodically with time due to the periodic pressure gradient. Fluid elasticity is responsible for the difference of velocity profile for Newtonian fluid and non-Newtonian fluid.
The velocity field and the shear stress corresponding to motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders are es-tablished by means of the Hankel transforms. The flow of the fluid is produced due to the time dependent axial shear stress applied on the boundary of the inner cylinder. The exact solutions, presented under a series form, can easily be specialized to give similar solutions for the Maxwell, second grade and Newtonian fluids performing the same mo-tion. Finally, some characteristics of the motion as well as the influence of the material constants on the behavior of the fluid are underlined by graphical illustrations.
We study the unsteady blood flow in a straight, long, rigid pipe, driven by an oscillatory pressure gradient. Three different non-Newtonian models for blood are considered and compared. One of them turns out to offer the best fit of experimental data, when the rheological parameters are suitably fixed. Numerical results are obtained by a spectral collocation method in space, and by an implicit trapezoidal finite difference method in time.
In order to solve the velocity profile and pressure gradient of the unsteady unidirectional slip flow of Voigt fluid,
Laplace transform method is adopted in this research. Between the parallel microgap plates, the flow motion is
induced by a prescribed arbitrary inlet volume flow rate which varies with time. The velocity slip condition on the
wall and the flow conditions are known. In this paper, two basic flow situations are solved, which are a suddenly
started and a constant acceleration flow respectively. Based on the above solutions, linear acceleration and oscillatory
flow are also considered.
The velocity profile and pressure gradient of an unsteady state unidirectional
MHD flow of Voigt fluids moving between two parallel surfaces under magnetic
field effects are solved by the Laplace transform method. The flow motion
between parallel surfaces is induced by a prescribed inlet volume flow rate that
varies with time. Four cases of different inlet volume flow rates are considered
in this study including (1) constant acceleration piston motion, (2) suddenly
started flow, (3) linear acceleration piston motion, and (4) oscillatory piston
motion. The solution for each case is elaborately derived, and the results of
associated velocity profile and pressure gradients are presented in analytical
forms.
This paper presents an analysis for the unsteady flow of an incompressible Maxwell fluid in an oscillating rectangular cross section. By using the Fourier and Laplace transforms as mathematical tools, the solutions are presented as a sum of the steady-state and transient solutions. For large time, when the transients disappear, the solution is represented by the steady-state solution. The solutions for the Newtonian fluids appear as limiting cases of the solutions obtained here. In the absence of the frequency of oscillations, we obtain the problem for the flow of the Maxwell fluid in a duct of a rectangular cross-section moving parallel to its length. Finally, the required time to reach the steady-state for sine oscillations of the rectangular duct is obtained by graphical illustrations for different parameters. Moreover, the graphs are sketched for the velocity.
The velocity field and the associated shear stresses corresponding to the unsteady flow of generalized Maxwell fluid on oscillating rectangular duct have been determined by means of double finite Fourier sine and Laplace transforms. These solutions are also presented as a sum of the steady-state and transient solutions. The solutions corresponding to Maxwell fluids, performing the same motion, appear as limiting cases of the solutions obtained here. In the absence of w, namely the frequency, and making α → 1, all solutions that have been determined reduce to those corresponding to the Rayleigh Stokes problem on oscillating rectangular duct for Maxwell fluids. Finally, some graphical representations confirm the above assertions.
Helical flows for Oldroyd-B fluid are studied in concentric cylinders and a circular cylinder. The fractional calculus approach in the constitutive relationship model Oldroyd-B fluid is introduced and a generalized Jeffreys model with the fractional calculus has been built. Exact solutions of some unsteady helical flows of Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and Laplace transform for fractional calculus. For α=β=1α=β=1, λ1=0λ1=0, λ2=0λ2=0 or λ1=λ2=0λ1=λ2=0 they reduce to the similar solutions for a Maxwell fluid, a second grade or Newtonian fluid, respectively. Numerical results for helical flow in pipe are presented.
In this paper we extend some of our previous works on continua with stress threshold. In particular here we propose a mathematical model for a continuum which behaves as a non-linear upper convected Maxwell fluid if the stress is above a certain threshold and as a Oldroyd-B type fluid if the stress is below such a threshold. We derive the constitutive equations for each phase exploiting the theory of natural configurations (introduced by Rajagopal and co-workers) and the criterion of the maximization of the rate of dissipation. We state the mathematical problem for a one-dimensional flow driven by a constant pressure gradient and study two peculiar cases in which the velocity of the inner part of the fluid is spatially homogeneous.
The wall friction induced by the oscillating flow of the fractional derivative Maxwell viscoelastic fluid in the pipe is investigated. The velocity and the shear stress solutions of the flow are solved. The friction is derived from the shear stress expression and analysed by numerical simulation. From analysis, it is found that the friction amplitude exhibits resonance-like phenomena. Moreover, the number of the resonance-like peaks, the enhancement magnitude, and the resonance-like frequency of the same order vary with the pipe radius and rheological parameters of fluids. When the radius overtakes the critical value, the friction curve monotonously decreases, and when the radius is big enough, the enhancement disappears.
Flow of a power-law fluid with a suddenly applied arbitrary time-dependent pressure gradient in a horizontal circular tube is presented. The momentum equations are solved numerically using a implicit finite-difference technique for the time-dependent velocity profiles for both start up and oscillating pressure gradients. Dimensionless curves are presented showing velocity profile development and phase angle variation for a variety of power-law index values.
In this paper, the uniqueness of solution for internal bounded unsteady flows of a shortmemory fluid is first established. Closed-form solutions are then obtained for the equations characterizing flows of such fluids in circular and rectangular tubes of uniform cross-section under an arbitrary pressure gradient. Special cases including the oscillatory flow between two parallel plates are discussed.
Solutions to the equation of motion have been obtained for a viscoelastic fluid which obeys an Oldroyd three‐constant constitutive equation. The solutions apply to a fluid at rest which at zero time is subjected to (1) a step change in pressure gradient or (2) a sinusoidally varying pressure gradient. From these results one can predict the response of the fluid to an arbitrary time‐dependent pressure gradient. Curves are presented which show how the flow of a viscoelastic fluid, when subjected to a step change in pressure, can overshoot the final steady‐state flow. Amplitude and phase relationships between different variables are presented for a sinusoidally varying pressure gradient. Comparisons are made with available experimental data.
The transient response of a non-Newtonian power-law fluid to several assumed forms of pressure pulse in a circular tube is analysed by the semi-direct variational method of Kanntovorich. Velocity profiles are shown for several power-law indices, and by comparing the results for the Newtonian case with the exact solution given by Szymanski, it is observed that the results are good to 5%. More accurate solutions have been found for the case involving Newtonian fluid flow. New results are reported concerning the effect of a triangular pressure pulse on the development and transient response of the flow field of a non-Newtonian fluid.
An exact solution of pulsating laminar flow superposed on the steady motion in a circular pipe is presented under the assumption of parallel flow to the axis of pipe. Total mass of flow on time average is found to be identified with that given byHagen-Poiseuille's low calculated on the steady component of pressure gradient. The phase lag of velocity variation from that of pressure gradient increases from zero in the steady motion to 90 in the pulsation of infinite frequency. Integration of work for changing kinetic energy of fluid through one period is vanished, while that of dissipation of energy by internal friction remains finite and excess amount caused by the components of periodic motion is added to the components of steady flow.It is found that the given rate of mass flow is attained in pulsating motion by giving the same amount of average gradient of pressure as in steady flow, but that excess works to the steady case are necessary for maintenance of this motion.Eine exakte Lsung der pulsierenden laminaren Strmung in einem Kreisrohr wird angegeben mit der Annahme, dass die Richtung dem Geschwindigkeitsvektor der Rohrachse parallel ist. Die Durchflussmenge stimmt berein mit der aus der stationren Druckgefllekomponente gerechneten Menge. Fr die Erhaltung der Bewegung dagegen ist die der Dissipation entsprechende Extraarbeit notwendig. Die Quantitt dieser Arbeit hngt ab von den Frequenzen der Stromschwingungen.
Unter dem im Titel genannten Effekt ist folgendes zu verstehen: E. G. Richardson hat bei Untersuchungen der Amplituden von Luftschwingungen in Helmholtzschen Resonatoren gefunden, da entgegen allen Erwartungen die Amplituden ihr Maximum nicht im zentralen Teil erreichen, sondern in einer der Wand nahen Schicht, um dann gegen die Wand zu auf Null herabzusinken. Ferner wurden empirische Gesetzmigkeiten fr dieses Maximum aufgestellt. Es wird nun gezeigt, da alle diese Gesetzmigkeiten qualitativ und quantitativ aus den Stokes-Navierschen Grundgleichungen der Hydrodynamik, als fr eine oszillierende Laminarstrmung in einem Kreisrohr charakteristisch, hergeleitet werden knnen.
The transient thermal response of a power law type non-Newtonian, laminar boundary layer flow past a flat plate is investigated. Consideration is given to the case of a step change in surface temperature. The transient heat flux and details of the temperature field are obtained and have been illustrated graphically. The range of Prandtl numbers investigated was 5–1000 while the viscosity index was allowed to vary 0.1–5.0.