Exact solution to peristaltic transport of power-law fluid in asymmetric channel with compliant walls

Effects of compliant wall properties on the peristaltic flow of a non-Newtonian fluid in an asymmetric channel are investigated.The rheological characteristics are characterized by the constitutive equations of a power-law fluid.Long wavelength and low Reynolds number approximations are adopted in the presentation of mathematical developments.Exact solutions are established for the stream function and velocity.The streamlines pattern and trapping are given due attention.Salient features of the key parameters entering into the present flow are displayed and important conclusions are pointed out.     

Similar solution for three‐dimensional flow in an Oldroyd‐B fluid over a stretching surface

The present investigation deals with the three‐dimensional flow of an Oldroyd‐B fluid over a stretching surface. The governing equations for the three‐dimensional flow are developed. Similarity transformations are invoked for the conversion of nonlinear partial differential equations into the coupled system of ordinary differential equations. Computations for the series solution are presented through implementation of homotopy analysis method. The salient features of the involved parameters have been pointed out. Copyright © 2011 John Wiley & Sons, Ltd.     

Peristaltic flow of Walter’s B fluid in endoscope

The peristaltic flow of a Walter’s B fluid in an endoscope is studied.The problem is modeled in a cylindrical coordinate system.The main theme of the present analysis is to study the endoscopic effects on the peristaltic flow of the Walter’s B fluid.To the best of the authors’ knowledge,no investigation has been made so far in the literatures to study the Walter’s B fluid in an endoscope.Analytical solutions are obtained using the regular perturbation method by taking δ as a perturbation parameter.The appro...     

Peristaltic motion of a third-order fluid in a planar channel

In order to determine the characteristics of the peristaltic transport of shear thinning non-Newtonian materials, the motion of a third-order fluid in a planar channel having walls that are transversely displaced by an infinite, harmonic traveling wave of large wavelength and negligibly small Reynolds number was analyzed using a perturbation expansion in terms of a variant of the Deborah number. Within the range of validity of this analysis, we found the pumping rate of a shear-thinning fluid is less than that for a Newtonian fluid having a shear viscosity the same as the lower-limiting viscosity of the nonNewtonian material. Also, the space of variables for which trapping of a bolus of fluid occurs is reduced for the shear-thinning fluid investigated here.     

Peristaltic motion of a power-law fluid in an asymmetric channel

The flow of a power-law fluid is investigated in an asymmetric channel caused by the movement of peristaltic waves with the same speed but with different amplitudes and phases on the flexible walls of the channel. The differential equation governing the flow is non-linear and can admit non-unique solutions. There exist two different physically meaningful solutions one satisfying the boundary conditions at the upper wall and the other at the lower wall. The effects of the power-law nature of the fluid on the pumping characteristics and axial velocity are studied in detail.     

Nonlinear bendings of symmetrically layered anisotropic rectangular plates

This paper has studied the nonlinear bending of symmetrically layered anisotropic rectangular plates under various supports. The uniformlys valid N-order asymptotic solutions of the deflection and stress function are derived by the singular perturbation methods offered in [1]. The analysis and calculations are given for simply and clamped supported, rectangular plates subjected to combined edge tensions and lateral loading in conjunction with the modified Galerkin procedure (a method of weighted residuals).     

Wall properties and heat transfer analysis of the peristaltic motion in a power‐law fluid

A mathematical analysis has been carried out to investigate the effect of elasticity of the flexible walls on the peristaltic flow of a power‐law fluid. The heat transfer analysis is further examined. Expressions of stream function, velocity, and temperature distributions are presented in closed form under long wavelength and low Reynolds number approximations. The effects of the various parameters entering into the mathematical analysis are sketched and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical and analytical simulation of peristaltic flows of generalized Oldroyd‐B fluids

In this paper, we study the peristaltic flows of generalized Oldroyd‐B fluids through the gap between concentric uniform tubes under the assumption of large wavelength and low Reynolds number approximations. The inner tube is rigid and the outer tube has a sinusoidal wave travelling down its wall. Homotopy perturbation and variational iteration methods are used for solution of the problem. The obtained solution is then used to discuss various interesting features of peristalsis. The effects of relaxation time, retardation time and radii of the tubes on pressure rise and friction forces (per wavelength on the inner and outer tubes) are discussed with illustrations. It is found that pressure rise diminishes with increase in relaxation time or the ratio of radii of inner and outer tubes. It increases with increasing retardation time. The effects of both time parameters on friction forces have the opposite behavior to that of pressure rise. Copyright © 2010 John Wiley & Sons, Ltd.     

k‐Version of finite element method in 2‐D polymer flows: Oldroyd‐B constitutive model

In this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process of two‐dimensional polymer flows utilizing Oldroyd‐B constitutive model. Alternate forms of the choices of dependent variables in the governing differential equations (GDEs) are considered and is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious over others. It is shown that: (a) since, the differential operator in the GDEs is non‐linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non‐symmetric and hence may have partial or completely complex basis and thus the resulting computational processes may be spurious. (b) Since the VC of the VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches always have possibility of spurious solutions. (c) Least squares process utilizing GDEs in u, v, p, τ (strong form of the GDEs) variables (as well as others) is variationally consistent. The coefficient matrices are always symmetric and positive definite and hence always have a real basis and thus naturally yield computational processes that are free of spurious solutions. (d) The theoretical solution of the GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solution are preserved, undoubtedly requires local approximations in higher order scalar product spaces . (e) LSP with local approximations in spaces provide an incomparable mathematical and computational framework in which it is possible to preserve desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of the Oldroyd‐B model or lack of it is examined for both model problems for increasing De. A mathematical idealization of the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when h_d→0. For both model problems shear rate is examined in the flow domain to establish validity of the Oldroyd‐B constitutive model. Copyright © 2006 John Wiley & Sons, Ltd.     

Peristaltic motion of third grade fluid in curved channel

＂ Analysis is performed to study the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel with wall properties. The resulting nonlinear partial differential equations are transformed to a single ordinary differential equation in a stream function by using the assumptions of long wavelength and low Reynolds number. This differential equation is solved numerically by employing the built-in routine for solving nonlinear boundary value problems （BVPs） through the software Mathematica. In addition, the analytic solutions for small Deborah number are computed with a regular perturbation technique. It is noticed that the symmetry of bolus is destroyed in a curved channel. An intensification in the slip effect results in a larger magnitude of axial velocity. Further, the size and circulation of the trapped boluses increase with an increase in the slip parameter. Different from the case of planar channel, the axial velocity profiles are tilted towards the lower part of the channel. A comparative study between analytic and numerical solutions shows excellent agreement.     

Peristaltic flow of a Jeffrey‐six constant fluid in a uniform inclined tube

In this work, we studied the peristaltic flow of a Jeffrey‐six constant fluid in a uniform tube. The governing equations of the Jeffrey‐six constant fluid were simplified by using the assumptions of long wave length and low Reynolds number approximation. The simplified form of equations were solved using the perturbation, homotopy analysis and finite difference methods. The comparison of the three solutions are shown graphically. The variation of pressure rise and frictional forces with the different parameters were also examined numerically. Results are presented at the end of the article. Copyright © 2011 John Wiley & Sons, Ltd.     

Exact and numerical simulation of peristaltic flow of a non‐Newtonian fluid with inclined magnetic field in an endoscope

In the present article, we have studied the effects of inclined magnetic field on the peristaltic flow of Jeffrey fluid through the gap between two coaxial inclined tubes. The inner tube is rigid, whereas the outer tube has sinusoidal wave traveling down its wall. The governing equations are simplified using long wave length and low Reynolds number approximations. Exact and numerical solutions have been derived for velocity profile. The expressions for pressure rise and friction force are calculated using numerical integration. Graphical results and trapping phenomenon is presented at the end of the article to see the physical behavior of different parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

Peristaltic flow of MHD Jeffrey fluid through finite length cylindrical tube

The present investigation studies the peristaltic flow of the Jeffrey fluid through a tube of finite length. The fluid is electrically conducting in the presence of an applied magnetic field. Analysis is carried out under the assumption of long wavelength and low Reynolds number approximations. Expressions of the pressure gradient, volume flow rate, average volume flow rate, and local wall shear stress are obtained. The effects of relaxation time, retardation time, Hartman number on pressure, local wall shear stress, and mechanical efficiency of peristaltic pump are studied. The reflux phenomenon is also investigated. The case of propagation of a non-integral number of waves along the tube walls, which are inherent characteristics of finite length vessels, is also examined.     

Spectral element method for viscoelastic flows in a planar contraction channel

A new algorithm, which combines the spectral element method with elastic viscous splitting stress (EVSS) method, has been developed for viscoelastic fluid flows in a planar contraction channel. The system of spectral element approximations to the velocity, pressure, extra stress and the rate of deformation variables is solved by a preconditioned conjugate gradient method based on the Uzawa iteration procedure. The numerical approach is implemented on a planar four‐to‐one contraction channel for a fluid governed by an Oldroyd‐B constitutive equation. The behaviour of the Oldroyd‐B fluids in the contraction channel is investigated with various Weissenberg numbers. It is shown that numerical solutions obtained here agree well with experimental measurements and other numerical predictions. Copyright © 2003 John Wiley & Sons, Ltd.     

Peristaltic flow of MHD Jeffrey fluid through finite length cylindrical tube

The present investigation studies the peristaltic flow of the Jeffrey fluid through a tube of finite length. The fluid is electrically conducting in the presence of an applied magnetic field. Analysis is carried out under the assumption of long wavelength and low Reynolds number approximations. Expressions of the pressure gradient, volume flow rate, average volume flow rate, and local wall shear stress are obtained. The effects of relaxation time, retardation time, Hartman number on pressure, local wall she...     

Peristaltic transport of a Herschel-Bulkley fluid in an inclined tube

Peristaltic flow of Herschel-Bulkley fluid in an inclined tube is analyzed. The velocity distribution, the stream function and the volume flow rate are obtained. Also, when the yield stress ratio τ→0, and when the inclination parameter α=0 and the fluid parameter n=1, the results agree with those of Jaffrin and Shapiro (Ann. Rev. Fluid Mech. 3 (1971) 13) for peristaltic transport of a Newtonian fluid in a horizontal tube. The effects of τ and n on the pressure drop and the mean flow are discussed through graphs. Furthermore, the results for the peristaltic transport of Bingham and power law fluids through a flexible tube are obtained and discussed. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of Herschel-Bulkley fluid on the flow characteristics.     

Time‐dependent finite element simulations of a shear‐thinning viscoelastic fluid with application to blood flow

A new finite element method is developed to simulate time‐dependent viscoelastic shear‐thinning flows characterized by the generalized Oldroyd‐B model. The focus of the algorithm is improved stability through a free‐energy dissipative scheme by using low‐order piecewise‐constant finite element approximations for stress. The algorithm is further modified by incorporating a pressure‐projection method, a DG‐upwinding scheme, a symmetric interior penalty DG method to solve the elliptic pressure‐update equation and a geometric multigrid preconditioner. The improved stability and cost to accuracy is compared when using higher order discontinuous bilinear approximation, where in addition, we consider the influence of a slope limiter for these elements. The algorithm is applied to the 2D start‐up‐driven cavity problem, and the stability of the free energy is illustrated and compared between element choices. An application of the model to modelling blood in small arterioles and channels is considered by simulating pulsatile blood flow through a stenotic arteriole. The individual influences of viscoelasticity and shear‐thinning within the generalized Oldroyd‐B model are investigated by comparing results to the Newtonian, generalized Newtonian and Oldroyd‐B models. Copyright © 2014 John Wiley & Sons, Ltd.     

Peristaltic flow of a Bingham fluid in a channel

We study the peristaltic transport of a Bingham fluid in a channel with small aspect ratio whose walls behave as a periodic traveling wave. The governing equations in the unyielded phase are obtained writing the integral formulation for the momentum balance. As shown in Fusi et al. (2015), this approach allows to overcome the so-called “lubrication paradox” which may arise in the thin film approximation. We consider the case in which the inlet flux is prescribed and the one in which the flow is driven by a given pressure drop. In both cases the solution of the problem is determined solving a nonlinear integral equation for the yield surface. We perform some numerical simulations to illustrate the behavior of the yield surface, assuming that the traveling wave describing the peristaltic motion has a sinusoidal shape.     

A variational analytical approach to time dependent flow of oldroyd B fluid in circular tube

In the present investigation the time dependent flow of an Oldroyd fluid B in a horizontal cylindrical pipe is stuided by the variational analytical approach developed by author. The time dependent problem is mathematically reduced to a partial differential equation of third order. Using the improved variational approach due to Kantorovich the partial differential equation can be reduced to a system of ordinary differential equations for different approximations. The ordinary differential equations are solved by the method of the Laplace transform which is led to an analytical form of the solutions. Project supported by TWAS and Chinese Academy of Sciences and the National Science Foundation of China     

On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

In this work we develop a mathematical model to predict the velocity profile for an unidirectional, incompressible and steady flow of an Oldroyd 6-constant fluid. The fluid is electrically conducting by a transverse magnetic field. The developed governing equation is non-linear. This equation is solved analytically to obtain the general solution. The governing non-linear equation is also solved numerically subject to appropriate boundary conditions (three cases of typical plane shearing flows) by an iterative technique with the finite-difference discretizations. A parametric study of the physical parameters involved in the problems such as the applied magnetic field and the material constants is conducted. The obtained results are illustrated graphically to show salient features of the solutions. Numerical results show that the applied magnetic field tends to reduce the flow velocity. Depending on the choice of the material parameters, the fluid exhibits shear-thickening or shear-thinning behaviours.