Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.     

Analytic solution for flow of Sisko fluid through a porous medium

This paper presents the analytic solution for flow of a magnetohydrodynamic (MHD) Sisko fluid through a porous medium. The non-linear flow problem in a porous medium is formulated by introducing the modified Darcy’s law for Sisko fluid to discuss the flow in a porous medium. The analytic solutions are obtained using homotopy analysis method (HAM). The obtained analytic solutions are explicitly expressed by the recurrence relations and can give results for all the appropriate values of material parameters of the examined fluid. Moreover, the well-known solutions for a Newtonian fluid in non-porous and porous medium are the limiting cases of our solutions.     

Homotopy Solution for the Channel Flow of a Third Grade Fluid

The solution for the flow of a third grade fluid bounded by two parallel porous plates is given using homotopy analysis method (HAM). A comparison is made with the exact numerical solution for the various values of the physical parameters. It is found that a proper choice of the auxiliary parameter occurring in HAM solution gives very close results.     

Effect of heat transfer on a third grade fluid in a flat channel

The heat transfer analysis on the laminar flow of an incompressible third grade fluid through a porous flat channel is examined. The lower plate is assumed to be at a higher temperature than the upper plate. Analytical solution for temperature distribution is obtained for various values of the controlling parameters and discussed. The obtained analytical solution is also compared with the numerical solution. The comparison shows the fact that the accuracy is remarkable. Copyright © 2009 John Wiley & Sons, Ltd.     

MHD Peristaltic Transport of a Jeffery Fluid in a Channel With Compliant Walls and Porous Space

Here an attempt has been made to investigate the magnetohydrodynamic (MHD) flow of a non-Newtonian fluid filling the porous space in a channel with compliant walls. Constitutive equations of a Jeffery fluid are used in the mathematical modeling. The flow is created due to sinusoidal traveling waves on the channel walls. The resulting problem is solved analytically and series solution for a stream function is derived. The effects of pertinent flow parameters are discussed through graphs.     

Effects of partial slip on flow of a third grade fluid

This paper is an analytical study of the rotating flow of a third grade fluid past a porous plate with partial slip effects. It serves as a flow model for the study of polymers. The analytic solution has been determined using homotopy analysis method (HAM).The English text was polished by Yunming Chen.     

The effects of the side walls on the flow of a second grade fluid in ducts with suction and injection

The effects of the side walls on the flow in ducts with suction and injection are examined. Three illustrative examples are given. The first example considers the effect of the side walls on the flow over a porous plate. The second example considers the flow between two parallel porous plates and the third example is devoted to the investigation of the flow in a rectangular duct with two porous walls. Exact solution of the governing equation using the no-slip boundary condition and an additional condition are obtained. The expression of the velocity, the volume flux and the vorticity are given. It is found that for large values of the cross-Reynolds number near the suction region the flow for a Newtonian fluid does not satisfy the boundary condition, but it does not behave in the same way for a second grade fluid. Three examples considered show that there are pronounced effects of the side walls on the flows of a second grade fluid in ducts with suction and injection.     

New exact solutions of Stokes' second problem for an MHD second grade fluid in a porous space

We investigate a problem describing the oscillating flow of an incompressible magnetohydrodynamic (MHD) second grade fluid in a porous half space. Exact solutions for sine and cosine oscillations are developed by applying the Laplace transform method. The total obtained solution is a sum of steady and transient solutions. Particular attention is given to the effects of magnetic and porous medium parameters on the velocity. It is shown that previous results for a non-porous medium and hydrodynamic fluid are the limiting cases of the present problem. The results for velocity are plotted and discussed carefully.     

Hydromagnetic rotating flow of third grade fluid

This work investigates the flow of a third grade fluid in a rotating frame of reference. The fluid is incompressible and magnetohydrodynamic （MHD）. The flow is bounded between two porous plates, the lower of which is shrinking linearly. Mathematical modelling of the considered flow leads to a nonlinear problem. The solution of this nonlinear problem is computed by the homotopy analysis method （HAM）. Graphs are presented to demonstrate the effect of several emerging parameters, which clearly describe the flow characteristics.     

Heat and mass transfer on a MHD third grade fluid with partial slip flow past an infinite vertical insulated porous plate in a porous medium

The influence of third grade, partial slip and other thermophysical parameters on the steady flow, heat and mass transfer of viscoelastic third grade fluid past an infinite vertical insulated plate subject to suction across the boundary layer has been investigated. The space occupying the fluid is porous. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. An efficient numerical scheme of midpoint technique with Richardson’s extrapolation is employed to solve the governing system of coupled nonlinear equations of momentum, energy and concentration. Numerical calculations were carried out for different values of various interesting non-dimensional quantities in the slip flow regime with heat and mass transfer and were shown with the aid of figures. The values of the wall shear stress, the local rate of heat and mass transfers were obtained and tabulated. The analysis shows that as the fluid becomes more shear thickening, the momentum boundary layer decreases but the thermal boundary layer increases; the magnetic field strength is found to decrease with an increasing temperature distribution when the porous plate is insulated. The consequences of increasing the permeability parameter and Schmidt number decrease both the momentum and concentration boundary layer thicknesses respectively whereas an increase in the thermal Grashof number gives rise to the thermal boundary layer thickness.     

On the MHD flow of fractional generalized Burgers＇ fluid with modified Darcy＇s law

This work is concerned with applying the fractional calculus approach to the magnetohydrodynamic (MHD) pipe flow of a fractional generalized Burgers’ fluid in a porous space by using modified Darcy’s relationship. The fluid is electrically conducting in the presence of a constant applied magnetic field in the transverse direction. Exact solution for the velocity distribution is developed with the help of Fourier transform for fractional calculus. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the present analysis.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Propagation of torsional surface wave in anisotropic poroelastic medium under initial stress

The present work deals with the possibility of propagation of torsional surface wave in fluid saturated poroelastic layer lying over nonhomogeneous elastic half space. Both the media are assumed to be under compressive initial stress. The half space has two types of inhomogeneity, viz; hyperbolic and quadratic. The dispersion equation for torsional wave in porous layer has been derived and observed that the presence of fluid in pores increases the velocity of the torsional surface wave but the phase velocity diminishes due to the presence of compressive initial stress in the porous layer. It is also observed that the velocity of the torsional surface wave increases due to the increase of initial stress in inhomogeneous half space. The inhomogeneity factor due to quadratic and hyperbolic variations in rigidity, density and initial stress of the medium decreases the phase velocity as it increases.     

Flow of a third grade fluid due to an accelerated disk

The magnetohydrodynamic (MHD) flow induced by non‐coaxial rotation of porous disk and a third grade fluid at infinity is investigated. The disk is moving with uniform acceleration and rotating with a uniform angular velocity. Numerical solution of the governing nonlinear initial and boundary value problem is obtained. The effects of physical parameters on the velocity profiles are examined in detail. The present study shows that the constant acceleration part has a greater influence than the time part of the assumed variable velocity of the disk. Copyright © 2009 John Wiley & Sons, Ltd.     

A Local Model for the Energy Transfer in a Saturated Static Mixture

In the present work the transient energy transfer in a nonsaturated porous medium is studied, using a mixture theory viewpoint. The porous matrix is assumed homogeneous, rigid and isotropic, while the fluid is a Newtonian incompressible one and both are assumed static. Since the homogeneous matrix is not saturated, gradients of concentration are present. The porous medium and the fluid (a liquid) will be regarded as continuous constituents of a mixture that will have also a third constituent, an inert gas, assumed with zero mass density and thermal conductivity. The problem is described by a set of two partial differential equations which represent the energy balances for the fluid and the solid constituents. Isovalues for these two constituents are plotted, considering representative time instants and selected values for the energy equations coefficients and for the saturation.     

Flow due to non-coaxial rotation of a porous disk and a second grade fluid rotating at infinity

An exact solution for the three-dimensional flow due to non-coaxial rotation of a porous disk and a second grade fluid at infinity is obtained. It is shown that for uniform suction or uniform blowing at the disk, an asymptotic profile exists for the velocity distribution. The velocity depends on two parameters: one of them is the suction parameter or blowing parameter and the other is the visco-elastic parameter. Furthermore, it is found that when the value of the visco-elastic parameter is fixed, the velocity decreases with an increase in the value of the suction parameter and when the value of the suction parameter is fixed, the velocity increases with an increase in the value of the visco-elastic parameter.     

Theoretical and experimental aspects of vacuum impregnation of porous media using transparent etched networks

Vacuum impregnation is a process method in which air and native solution are removed from the porous space of a given porous material and replaced by an external solution. Vacuum impregnation is divided into two steps: Firstly, the porous material is immersed in a liquid solution and exposed to subatmospheric pressure for a given time to ensure that air trapped in the porous materials will be removed; secondly, atmospheric pressure is re-established and the external solution penetrates the pore structure of the porous material. The objective of this study was to describe the hydrodynamic mechanisms involved in vacuum impregnation of porous materials as a function of capillary number and viscosity ratio. To achieve the objectives proposed in the present study, a transparent glass micromodel 7.7 cm × 7.4 cm was first constructed using the photolithographic technique. In addition, a stainless steel vacuum tank was built. The tank top was covered with a transparent reinforced glass plate. The whole system was connected to a vacuum pump, and a conventional video camera was adapted to record the experiments. Liquid saturation was determined through the image analysis process. Capillary number and viscosity ratio were determined for the drainage and imbibition processes. For the systems studied, we conclude that transport mechanisms ranged between stable displacement and capillary fingering during the vacuum step (drainage) while transport mechanisms ranged between continuous capillary and discontinuous capillary domains during the atmospheric step (imbibition). Earlier work indicated that our proposed process should be even more efficient for realistically large systems.     

Transient response of fluid pressure in a poroelastic material under uniaxial cyclic loading

Poroelasticity is a theory that quantifies the time-dependent mechanical behavior of a fluid-saturated porous medium induced by the interaction between matrix deformation and interstitial fluid flow. Based on this theory, we present an analytical solution of interstitial fluid pressure in poroelastic materials under uniaxial cyclic loading. The solution contains transient and steady-state responses. Both responses depend on two dimensionless parameters: the dimensionless frequency Ω that stands for the ratio of the characteristic time of the fluid pressure relaxation to that of applied forces, and the dimensionless stress coefficient H governing the solid-fluid coupling behavior in poroelastic materials. When the phase shift between the applied cyclic loading and the corresponding fluid pressure evolution in steady-state is pronounced, the transient response is comparable in magnitude to the steady-state one and an increase in the rate of change of fluid pressure is observed immediately after loading. The transient response of fluid pressure may have a significant effect on the mechanical behavior of poroelastic materials in various fields.     

Exact solutions of starting flows for second grade fluid in a porous medium

This paper concentrates on the unsteady flows of a magnetohydrodynamic (MHD) second grade fluid filling a porous medium. The flow modeling involves modified Darcy's law. Three problems are considered. They are (i) starting flow due to an oscillating edge, (ii) starting flow in a duct of rectangular cross-section oscillating parallel to its length, and (iii) starting flow due to an oscillating pressure gradient. Analytical expressions of velocity field and corresponding tangential stresses are developed. These expressions are found to be significantly affected by the applied magnetic field, permeability of the porous medium and the material parameter of the fluid. Moreover, the influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn. Finally, a comparison is also made with the existing results in the literature.     

Seismic Wave Propagation in Composite Elastic Media

It has been known since the time of Biot–Gassman theory (Biot, J Acoust Soc Am 28:168–178, 1956, Gassmann, Naturf Ges Zurich 96:1–24, 1951) that additional seismic waves are predicted by a multicomponent theory. It is shown in this article that if the second or third phase is also an elastic medium then multiple p and s waves are predicted. Futhermore, since viscous dissipation no longer appears as an attenuation mechanism and the media are perfectly elastic, these waves propagate without attenuation. As well, these additional elastic waves contain information about the coupling of the elastic solids at the pore scale. Attempts to model such a medium as a single elastic solid causes this additional information to be misinterpreted. In the limit as the shear modulus of one of the solids tends to zero, it is shown that the equations of motion become identical to the equations of motion for a fluid filled porous medium when the viscosity of the fluid becomes zero. In this limit, an additional dilatational wave is predicted, which moves the fluid though the porous matrix much similar to a heart pumping blood through a body. This allows for a connection with studies which have been done on fluid-filled porous media (Spanos, 2002).