Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate is investigated. Exact solutions for the velocity field are established by means of the Fourier and Laplace transforms. The similar solutions for Maxwell and Newtonian fluids can be obtained as limiting cases of our results. In the absence of side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.     

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag–Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.      

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.     

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.     

Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate

The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained.     

Starting solutions for oscillating motions of an Oldroyd-B fluid over a plane wall

In this paper, we establish the starting solutions for oscillating motions of an Oldroyd-B fluid between two side walls perpendicular to a plane wall. The expressions for the velocity field and the associated tangential stress at the bottom wall are obtained, presented under integral and series form. These satisfy all imposed initial and boundary conditions. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. In the absence of side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Moreover, the obtained solutions can be specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motions.     

Unsteady MHD flow of a non-Newtonian fluid on a porous plate

This paper deals with the study of the MHD flow of non-Newtonian fluid on a porous plate. Two exact solutions for non-torsionally generated unsteady hydromagnetic flow of an electrically conducting second order incompressible fluid bounded by an infinite non-conducting porous plate subjected to a uniform suction or blowing have been analyzed. The governing partial differential equation for the flow has been established. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. The effect of presence of the material constants of the second order fluid on the velocity field is discussed.     

3D flow of a generalized Oldroyd-B fluid induced by a constant pressure gradient between two side walls perpendicular to a plate

This paper deals with the 3D flow of a generalized Oldroyd-B fluid due to a constant pressure gradient between two side walls perpendicular to a plate. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. Exact analytic solutions for the velocity and stress fields, in terms of the Fox H-function, are established by means of the finite Fourier sine transform and the Laplace transform. Solutions similar to those for ordinary Oldroyd-B fluid as well as those for Maxwell and second-grade fluids are also obtained as limiting cases of the results presented. Furthermore, 3D figures for velocity and shear stress fields are presented for the first time for certain values of the parameters, and the associated transport characteristics are analyzed and discussed.     

Some exact solutions of an elastico-viscous fluid

The exact solutions for the unsteady flow of an elastico-viscous fluid caused by general periodic oscillations are obtained. Further, the plate is assumed to be rigid as well as porous executing periodic rotary oscillations. The velocity field and shear stress are established.     

Analytic solution for MHD Transient rotating flow of a second grade fluid in a porous space

This article looks into the unsteady rotating magnetohydrodynamic (MHD) flow of an incompressible second grade fluid in a porous half space. The flow is induced by a suddenly moved plate in its own plane. Both the fluid and plate rotate in unison with the same angular velocity. Analytic solution of the governing flow problem is obtained by using Fourier sine transform. Based on the modified Darcy's law, expression for velocity is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. Several existing solutions of Newtonian fluid have been also deduced as limiting cases.     

Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection

The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect. An external uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate moves with a constant velocity and the two plates are kept at different but constant temperatures. Numerical solutions are obtained for the governing momentum and energy equations taking the Joule and viscous dissipations into consideration. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions are studied.     

Axial Couette flow of a generalized Oldroyd-B fluid due to a longitudinal time-dependent shear stress

Abstract

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder are determined by means of Hankel and Laplace transforms. The solutions that have been obtained, written in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are obtained as limiting cases of our general solutions.     

Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference

This paper concerns with studying the steady and unsteady MHD micropolar flow and mass transfers flow with constant heat source in a rotating frame of reference in the presence chemical reaction of the first-order, taking an oscillatory plate velocity and a constant suction velocity at the plate. The plate velocity is assumed to oscillate in time with a constant frequency; it is thus assumed that the solutions of the boundary layer are the same oscillatory type. The governing dimensionless equations are solved analytically after using small perturbation approximation. The effects of the various flow parameters and thermophysical properties on the velocity and temperature fields across the boundary layer are investigated. Numerical results of velocity profiles of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. The results show that there exists completely oscillating behavior in the velocity distribution.     

New exact solutions for magnetohydrodynamic flows of an Oldroyd-B fluid

This paper presents the new exact analytical solutions for magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The explicit expressions for the velocity field and the associated tangential stress are established by using the Laplace transform method. Three characteristic examples: (i) flow due to impulsive motion of plate, (ii) flow due to uniformly accelerated plate, and (iii) flow due to non-uniformly accelerated plate are considered. The solutions for the hydrodynamic flows are special cases of the presented solutions. Moreover, the similar solutions corresponding to Maxwell and Newtonian fluids in the presence as well as absence of a magnetic field appear as the limiting cases of our solutions. The influences of the exerted magnetic field on the flow are also graphically presented and discussed. In particular, graphical results for the Oldroyd-B fluid are compared with those of a Newtonian fluid.     

On the Rayleigh problem for a Sisko fluid in a rotating frame

The unsteady rotating flow of a Sisko fluid bounded by a suddenly moved infinite flat plate is investigated. The fluid is electrically conducting in the presence of a transverse applied time-dependent magnetic field. A highly non-linear differential equation resulting from the balance of momentum and mass, coupled with appropriate boundary and initial conditions is solved numerically. The numerical solutions for different values of the parameters are compared and discussed.     

Unsteady MHD two-phase Couette flow of fluid–particle suspension

The problem of two-phase unsteady MHD Couette flow between two parallel infinite plates has been studied taking the viscosity effect of the two phases into consideration. Unified closed form expressions are obtained for the velocities and the skin frictions for both cases of the applied magnetic field being fixed to either the fluid or the moving plate. The novelty of this study is that we have obtained the solution of the unsteady flow using the Laplace transform technique, D’Alemberts method and the Riemann-sum approximation method. The solution obtained is validated by assenting comparisons with the closed form solutions obtained for the steady states which have been derived separately and also by the implicit finite difference method. Graphical result for the velocity of both phases based on the semi-analytical solutions are presented and discussed. A parametric study of some of the physical parameters involved in the problem is conducted. The skin friction for both the fluid and the particle phases decreases with time on both plates until a steady state is reached, it is also observed to decrease with increase in the particle viscosity on the moving plate while an opposite behaviour has been noticed on the stationary plate.     

On the homotopy solution for Poiseuille flow of a fourth grade fluid

This paper develops a mathematical model with an aim to compute the analytic solution for the flow of a fourth grade fluid between two fixed porous walls. The flow is induced under the application of a constant pressure gradient. The arising nonlinear problem is treated analytically yielding a series solution by homotopy analysis method (HAM). Results of velocity and shear stresses at the walls are obtained. The impacts of several flow parameters are examined on the velocity and shear stresses.     

Rayleigh problem for a MHD Sisko fluid

The problem of unsteady unidirectional flow of an incompressible Sisko fluid bounded by a suddenly moved plate is studied. The fluid is magnetohydrodynamic (MHD) in the presence of a time-dependent magnetic field applied transversely to the flow. The non-linear flow problem arising from the laws of momentum, mass and suitable boundary and initial conditions is solved analytically using Lie symmetry method. The manner in which the various emerging parameters affect the structure of the velocity is delineated.     

A note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a time dependent shear stress

The unsteady flow of a generalized second-grade fluid through an infinite straight circular cylinder is considered. The flow of the fluid is due to the longitudinal time dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the governing equation corresponding to a second-grade fluid is introduced. The velocity field and the resulting shear stress are obtained by means of the finite Hankel and Laplace transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is used. The corresponding solutions for ordinary second-grade and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the material constants and of the fractional parameter on the velocity and shear stress variations is underlined by graphical illustrations.