Based on a modified Darcy’s law for a viscoelastic fluid, Stokes’ first problem was extended to that for an Oldroyd-B fluid in a porous half space. By using Fourier sine transform, an exact solution was obtained. In contrast to the classical Stokes’ first problem for a clear fluid, there is a y-dependent steady state solution for an Oldroyd-B fluid in the porous half space, which is a damping exponential function with respect to the distance from the flat plate. The thickness of the boundary layer, which tends to be a limited value, is also different from that of a clear fluid. The effect of viscoelasticity on the unsteady flow in porous media is investigated. It was found if α>1∕4[(αt∕Re)+Re]2, oscillations in velocity occur obviously and the system exhibits viscoelastic behaviors, where α and αt are nondimensional relaxation and retardation times, respectively, Re is Reynold number in porous media. Some previous solutions of Stokes’ first problem corresponding to Maxwell fluid and Newtonian fluid in por...

Stokes first problem for an oldroyd-b fluid in a porous half space

In this paper, magnetohydrodynamic (MHD) Casson fluid flow in two lateral directions past a porous linear stretching sheet is investigated. Self-similar solutions are obtained and compared with the available data for special cases. It is found that the present results are in an excellent agreement with the available data. The dimensionless velocities and shear stresses are obtained in both directions. Pertinent results are presented graphically and discussed quantitatively with respect to variation in Casson flow parameter as well as other fluid flow parameters.

MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

Starting solutions for some unsteady unidirectional flows of a second grade fluid

We study the MHD flow and also heat transfer in a viscoelastic liquid over a stretching sheet in the presence of radiation. The stretching of the sheet is assumed to be proportional to the distance from the slit. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The resulting non-linear momentum differential equation is solved exactly. The energy equation in the presence of viscous dissipation (or frictional heating), internal heat generation or absorption, and radiation is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation using a new variable and using the Rosseland approximation for the radiation. The governing differential equations are solved analytically and the effects of various parameters on velocity profiles, skin friction coefficient, temperature profile and wall heat transfer are presented graphically. The results have possible technological applications in liquid-based systems involving stretchable materials.

Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet

We present two simple but elegant solutions for the flow of an Oldroyd-B fluid. First, we consider the flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in the case of blowing it admits no such solution. Second, we study the longitudinal and torsional oscillations of an infinitely long rod of finite radius. The solutions are found in terms of Bessel functions.

Exact solutions for some simple flows of an Oldroyd-B fluid

We consider the flow of an Oldroyd-B fluid occupying the space over an elastic sheet, due to the stretching of the sheet, in the presence of a constant free-stream velocity. By introducing a similarity transformation for the velocity field, as well as the three components of the stress tensor, the governing equations are reduced to a system of coupled non-linear ordinary differential equations. The resulting equations are solved numerically and by means of a perturbation in the Weissenberg number. A comparison of the two solutions shows exceedingly good agreement.

Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity

We have studied the non-radial flow of an Oldroyd-B fluid between intersecting planes. The effects of the Reynolds number Re and the elastic number σ are carefully delineated. Our results agree with the results of Strauss (Acta Mechanica, 21 (1975) 141) when σ = 0. In addition to studying the problem when σ ≠ 0, we carried out studies for values of Reynolds number which are two orders of magnitude larger than in previous studies. We find that there is an interesting structural change in the streamline patterns with changes in the Reynolds and Weissenberg numbers which was not observed in the earlier studies, and this has been documented for the first time.

Flow of an Oldroyd-B fluid between intersecting planes

The problem of two-dimensional steady laminar flow of an incompressible viscoelastic fluid through a porous cylindrical annulus with walls of different permeability is discussed. A quasilinearization method is used to solve the governing flow equations in polar coordinates for moderately large cross-flow through the porous walls. The numerical results are presented for large suction or blowing at the outer cylinder, for several values of the cross-flow Reynolds number and the non-Newtonian fluid parameters.

Application of quasilinearization to viscoelastic flow through a porous annulus

We study the flow of an Oldroyd-B fluid between two intersecting plates, one of which is fixed and the other moving along its plane. This problem was first considered by Strauss (1975) for the Maxwell fluid using a similarity transformation. We find that even in the case of a Maxwell fluid, which can be obtained by setting a specific parameter, say σ, in the Oldroyd-B model to zero, our results disagree with those of Strauss (1975). We find that circulating cells are present, adjacent to the stationary plate while Strauss (1975) finds them adjacent to the moving plate. We also delineate the effect of the coefficient σ, which is a measure of the elasticity of the flow, on the flow pattern. We find that an increase in the elastic parameter reduces the cellular structure.

R. K. Bhatnagar

Papers

Exact solutions for some simple flows of an Oldroyd-B fluid

Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity

Flow of an Oldroyd-B fluid between intersecting planes

Application of quasilinearization to viscoelastic flow through a porous annulus

Flow of a non-Newtonian fluid between intersecting planes of which one is moving