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Flow of a micropolar fluid through catheterized artery

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... This has been acknowledged by the engineering community and, thus, micropolar fluid flows have been extensively investigated in the last decade. In particular, one can find a vast amount of recent results concerning engineering applications in biomedicine, mostly in blood flow modeling (see, e.g., [2][3][4][5]). Rigorous, mathematical justification of various effective models describing flow of a micropolar fluid can be found in [6][7][8][9][10][11]. ...
... From the flux condition (3.4) 1 , we now have that Ω (2) ...
... with the boundary and initial conditions: The functionsf ( (2) (x, t) are given with: ...
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Motivated by the pipe network problems, in this paper, we consider the Leray’s problem for the nonstationary flow of a micropolar fluid. We prove that in an unbounded domain with cylindrical outlets to infinity, there exists a unique solution to the nonlinear micropolar equations which exponentially tends to the generalized nonstationary micropolar Poiseuille solution in each cylindrical outlet.
... A detailed literature and development of the model of micropolar fluid and their applications are given by Lukaszewicz [12] and Eringen [13]. Srinivasacharya and Srikanth [14] studied the steady flow of micropolar fluid through mild symmetric stenosis in the presence of the catheter. Mekheimer and El Kot [15] considered blood as micropolar fluid and discussed the effects of the asymmetry nature of the stenosis in their steady flow analysis. ...
... It is also observed that for few initial values of the increase in impedance is less and it increases exponentially for higher values of . Further the above results have been validated with the experimental of Back [4] and with the theoretical results for symmetric stenosis, as considered by Srinivasacharya and Srikanth [14]. Here, the results obtained by us are in similar lines with that of the experimental results, thus validating our approach. ...
... It is to be noted that the motion of micropolar fluid may get affected by the viscous action ( ) of the fluid elements, couple stresses ( ) and the direct coupling of Ratio of the diameter of the catheter to that of the vessel (in case of Back [4])/ratio of the radius of the catheter to that of the annular region (in the present study and as done by [14]) Impedance as obtained by Back [4] with 33% sugar-water solution with kinematic viscosity ] = 0.035 cm 2 s −1 Impedance in case of symmetric stenosis as in [14] the microstructure to the velocity field ( ). All these fluid parameters can have any value greater than or equal to zero. ...
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The blood flow through an overlapping clogged tapered artery in the presence of catheter is discussed. Since cholesterol deposition is resulting in the stenosis formation, velocity slip at the arterial wall is considered. The equations governing the fluid flow have been solved analytically under the assumption of the mild stenosis. The analysis with respect to various parameters arising out of fluid and geometry considered, on physiological parameters such as impedance and wall shear stress at the maximum height of the stenosis as well as across the entire length of the stenosis has been reported. A table summarizing the locations of extreme heights and the corresponding annular radii is provided. It is observed that the wall shear stress is the same at both the locations corresponding to the maximum height of the stenosis in case of nontapered artery while it varies in case of tapered artery. It is also observed that slip velocity and diverging tapered artery facilitate the fluid flow. Shear stress at the wall is increasing as micropolar parameter is decreasing and the trend is reversed in case of coupling number. The results obtained are validated by comparing them with the experimental and theoretical results.
... Srinivasacharya et al. [19] considered vanishing micro-rotation parameter in their analysis. However, it is understood that, in the neighborhood of the boundary, the rotation is only due to shear and therefore, as given by Ahmadi et al. [1] the gyration vector must be equal to the angular velocity. ...
... and the initial conditions (19) in case of micropolar fluid flow are ...
... Mishra and Nidhi [5] examined the effect of stenosis on non-Newtonian flow of blood in constricted blood vessel and concluded that the height of the stenosis has a direct relation with shear stress and impedance while reversal relationship with flow rate. Srinivasacharya and Srikanth [6] quoted that the flow patterns are strongly based on the geometry of artery and stenosis. ...
... As the flow is symmetric about its axes, all the variables are independent of θ. With the assumption that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow in an artery with mild stenosis, this implies that along the axial direction, the variation of all the flow characteristics are negligible except pressure given by Srinivasacharya and Srikanth [6]. Now the non-dimensional scheme is given by ...
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A mathematical model for blood flow through a bifurcated artery with mild stenosis in its parent artery is presented by taking blood as a copper suspended nanofluid with water as base fluid. The boundary of bifurcated artery is converted into a well-organized boundary using suitable transformations. The numerical solutions for temperature profiles, flow rate, impedance, and shear stress are found by using the finite difference method. The influence of admissible parameters on physical quantities on both sides of the apex is discussed graphically. It has been noticed that flow rate and impedance have been changed suddenly with the pertinent parameters on both sides of the apex. This occurs because of the backflow of the streaming blood at the onset of the lateral junction and secondary flow near the apex. All these observations on blood flow through bifurcated artery are of medical interest.
... The mathematical theory of equations of micropolar fluids and applications of these fluids to the theory of lubrication and to the theory of porous media is presented by Lukaszewicz [21]. Srinivasacharya and Srikanth [22] have studied the effects of micropolar flow through constricted annulus in the presence of catheter assuming the flow to be steady. Srinivasacharya and Srikanth [23] modelled the pulsatile flow of blood as micropolar fluid and studied the effects of catheter in the presence of simple stenosis. ...
... (27) in Eqs. (22) and (23), we get ...
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In this article pulsatile nature of blood flow through unsymmetric stenosed tapered artery in the presence of catheter has been modelled. Blood is represented by micropolar fluid. The analytical solutions for velocity and microrotation components are obtained in terms of Bessel functions of the first and second kind. Flow parameters such as the resistance to flow (impedance) in the artery and wall shear stress at the maximum height of the stenosis have been calculated and the effects of various parameters such as shape parameter (n), tapered parameter (ζ\zeta ), slip velocity (u1u_{1}, u2u_{2}), radius of the catheter (rcr_{c}), Reynolds number (Re), Strouhal number (σ\sigma ), micropolar parameter (m), coupling number (N) and height of the stenosis (ϵ\epsilon ) on impedance and wall shear stress are discussed. The locations of the maximum height of the stenosis and the annular radius which are dependent on both tapered parameter (ζ\zeta ) and shape parameter (n) are computed. It is observed that impedance is increasing while catheter radius, height of the stenosis, coupling number are increasing, while it is decreasing in case of shape parameter and micropolar parameter. Shape parameter has no effect on wall shear stress at the maximum height of the stenosis in case of non-tapered artery. However it is dependent on n in case of tapered artery. In particular wall shear stress decreases as stenosis is becoming more and more asymmetric in case of diverging tapered artery and the behaviour is exactly reverse in case of converging tapered artery. Also a comparison of the results for impedance of the present model with the experimental results of Back [4] have been carried out, it is observed that impedance increases significantly for higher values of the ratio of the radius of the catheter to that of the annular region is high.
... In this way, we obtain a coupled system of partial differential equations with four new viscosity coefficients. The model of micropolar fluids has been a subject of research in both the mathematical and engineering community: a comprehensive survey of the underlying mathematical theory can be found in the monograph by Lukaszezwicz [2], whereas recent results concerning engineering applications in biomedicine and blood flow modelling can be found in [3][4][5][6][7][8]. ...
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We consider the nonsteady flow of a micropolar fluid in a thin (or long) curved pipe via rigorous asymptotic analysis. Germano's reference system is employed to describe the pipe's geometry. After writing the governing equations in curvilinear coordinates, we construct the asymptotic expansion up to a second order. Obtained in the explicit form, the asymptotic approximation clearly demonstrates the effects of pipe's distortion, micropolarity and the time derivative. A detailed study of the boundary layers in space is provided as well as the construction of the divergence correction. Finally, a rigorous justification of the proposed effective model is given by proving the error estimates.
... Dash et al [17] studied the blood flow dynamics in a stenotic curved artery along with the effects of a catheter. Srinivasacharya and Srikanth [18] studied the oscillatory flow of micro-polar fluid through a mild contracted annulus in the stenotic artery and calculated wall shear stress and impedance. Reddy and Srikanth [19] studied a mathematical model of micro-polar fluid through overlapping tapered stenotic artery in the presence of the catheter with slip velocity. ...
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A theoretical model of blood-silver nanofluid flow through an ω\omega -shaped tapered stenotic artery in the presence of a catheter is understood in detail. The rheology of the blood is considered as that of a micro-polar fluid. Suspension of the silver nano-particles in the micro-polar fluid is proposed to investigate the temperature and concentration dispersion from the immersed temperature-sensitive drug-coated nano-particles. The effects of velocity discontinuity at the arterial wall in the stenotic and non-stenotic regions are considered. The outer surface of the catheter is layered with the temperature-sensitive, drug-coated nano-particles. The resulting mathematical formulation involving the coupled non-linear momentum, temperature and concentration equations is solved using the Homotopy Perturbation Method. The efficiency and convergence of the method to the modelled equations are discussed in detail. The consequent effects of the fluid and geometric parameters on pressure drop, flow rate, impedance and wall shear stress of the fluid flow are computed. It is noticed that the high volume fraction of the nano-particles in the blood results in high flow velocity, contributing to secondary flow regions, thus resulting in higher shear stress. Such high volume fraction of the nano-particle may lead to the pathological disorder called aneurysm. This physical model has an important application of drug delivery in biomedical and pharmaceutical industry to prevent obstructions in arteries. Further, the results obtained could be very useful in the manufacturing of related artificial devices.
... The constrictions (stenosis) may grow in series or may be of irregular shapes. To incorporate this in their model, several researchers (Srinivasacharya and Srikanth [26], Ramana Reddy et al. [15], Reddy and Srikanth [17]) considered the symmetric, asymmetric and ω-shaped nature of the stenosis. It is quite rational to consider composite nature of the stenosis by assuming that it is formed due to the aggregation of extra-vascular masses and blood cells which further results in porous nature of the stenosis. ...
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Investigation of couple stress fluid flow through ω\omega -shaped stenosed artery addresses many issues of blood flow in particular and circulatory system in general. In this model, physics of the flexible nature of the arterial tapered wall has been incorporated. The effects of pulsatile pressure gradient and stratification in both dynamic viscosity and couple stress viscosity are considered. To make the model more effective and realistic permeable stenosis, slip velocity and vanishing couple stresses at the arterial wall are also incorporated. The numerical solution of the transformed governing equations are obtained by using the finite difference method. The velocity profiles, the volumetric flow rate, the resistance to the flow and the wall shear stress are obtained numerically for various values of fluid and geometric parameters.
... Blood flow in small arteries and capillaries can exhibit non-Newtonian effects [31][32][33][34][35][36]. However, blood flow in a large arterial vessel and capillaries may be modeled as a Newtonian fluid [37][38][39][40]. ...
... Shukla et al. [4] studied the effect of stenosis using Casson and power law fluid models. Srinivasachary and Srikanth [5] analyzed the steady flow of micropolar fluid through the catheterized artery in the presence of stenosis. Several other studies were done considering the importance the non-Newtonian behaviour of the blood [6][7] . ...
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This paper explores the mathematical model for couple stress fluid flow through an annular region. The above model is used for studying the blood flow between the clogged (stenotic) artery and the catheter. The asymmetric nature of the stenosis is considered. The closed form expressions for the physiological parameters such as impedance and shear stress at the wall are obtained. The effects of various geometric parameters and the parameters arising out of the fluid considered are discussed by considering the slip velocity and tapering angle. The study of the above model is very significant as it has direct applications in the treatment of cardiovascular diseases.
... They found the exact solutions for the flow problem and shown the effects of various physical parameters. A mathematical model for the flow of micropolar fluid through catheterized artery has been analyzed by Srinivasacharya and Srikanth [21]. These three dimensional studies are considered in Cartesian coordinate system. ...
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In the present analysis, the unsteady peristaltic flow of an incompressible Carreau fluid is investigated in eccentric cylinders. The problem is measured in cylindrical coordinates. The governing equations are observed under the conditions of long wavelength and low Reynolds number approximations. The resulting highly nonlinear second order partial differential equations are solved by series solution technique. The relation for pressure rise is evaluated numerically by built-in technique with the help of mathematics software. As a special case, the present results are compared with the existing results given in the literature. The obtained results are then plotted to see the influence of different physical parameters on the velocity, pressure gradient and pressure rise expressions. The velocity profile is drawn for both two and three dimensions. The trapping boluses are also discussed through streamlines.
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MHD oscillatory blood flow in a channel as micropolar fluid in the presence of chemical reaction and a transverse magnetic field are studied. The partial differential equations governing the flow were formulated base on assumptions and already existing model. The partial differential equations were transformed to dimensionless equations with suitable variables. Analytical solution was obtained for the dimensionless equations. The pertinent parameters were investigated with graphs plotted and table generated using Matlab software. The study reveals that the parameters has significant influences on the flow.
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