# New Exact Solutions for an Oldroyd-B Fluid in a Porous Medium

Authors: I. Khan; M. Imran; K. Fakhar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

New exact solutions for unsteady magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid have been derived. The Oldroyd-B fluid saturates the porous space. Two different flow cases have been considered. The analytical expressions for velocity and shear stress fields have been obtained by using Laplace transform technique. The corresponding solutions for hydrodynamic Oldroyd-B fluid in a nonporous space appeared as the limiting cases of the obtained solutions. Similar solutions for MHD Newtonian fluid passing through a porous space are also recovered. Graphs are sketched for the pertinent parameters. It is found that the MHD and porosity parameters have strong influence on velocity and shear stress fields.


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## Body

## 1. Introduction

The behavior of viscoelastic materials in particularly the response of many polymeric liquids are best described by Oldroyd-B fluid model [1]. Further, these, fluids are also quite useful in chemical and process industry due to the fact that they encounter both the memory and elastic effects exhibited by most polymers and biological liquids. In early 1970, Waters and King [2, 3] have obtained first time the exact solutions to these fluids using Laplace transform technique. Later, several authors [4–20]have discussed the different motions of Oldroyd-B fluids in different geometries using various approaches for the solution.Recently, the concept of studing the fluid flows through porous medium has gained much attention largely due to their several technological and industrial applications such as geothermal energy extrusion, oil recovery, food processing, ground water flow, irrigation problems, and the biophysical sciences where the human lungs, for example, are modeled as a porous layer. The literature survey revealed that very few attempts are available in which the flows of electrically conducting Oldroyd-B fluid through a porous medium are studied [21–26]. Therefore, the basic objective of the present paper is to study the motion of Oldroyd-B fluid taking into account both magnetic and porosity effects.To the best of authors knowledge so far no study has been reported in which the MHD flow of an Oldroyd-B fluid passing through a porous medium has been considered for the following two cases: (i) flow due to impulsive motion of the plate and (ii) flow due to uniform motion of the plate. Therefore the main objective of this paper is to make such an attempt. The rest of the paper is arranged as follows. The mathematical formulation of the problem is given in Section2. Section 3 comprises the solution of the problem. The graphical results are displayed and discussed in the last section.


## 2. Problem Formulation

For the MHD flow through a porous medium, the continuity and momentum equations are given by (Tan and Masuoka [22])(2.1)div⁡V=0,(2.2)ρ(dVdt)=div⁡T+J×B-μϕk(1+λr∂∂t)V,
where V=(u,v,w) denotes the velocity vector, ρis the fluid density, Jis the current density, Bis the total magnetic field, T is the Cauchy stress tensor, d/dt is the material time derivative, ϕ(0<ϕ<1) is the porosity, and k>0is the permeability of the porous medium.The Cauchy stress tensorT for an incompressible Oldroyd-B fluid is given by [1](2.3)T=-pI+S,
in which the extra stress tensor S satisfies(2.4)S+λ[∂S∂t+(V⋅∇)S-LS-SLT]=μ[A1+λr(∂A1∂t+(V⋅∇)A1-LA1-A1LT)],
where -pI is the spherical part of the stress due to the constraint of incompressibility, μ is the dynamic viscosity, L is the velocity gradient, A1=L+LT is the first Rivlin-Ericksen tensor, and λ and λr(<λ) are, respectively, the relaxation and retardation times.The second term on the right side of (2.2) is called the Lorentz force and can be written as(2.5)J×B=(-σB02u,0,0),
in which B0 is the magnitude of a uniform magnetic field B0 applied in a direction normal to the fluid motion and σ is the electrical conductivity of the fluid.By taking the velocity field of the form(2.6)V=(u(y,t),0,0),
the continuity equation is automatically satisfied and the momentum equation in the absence of external pressure gradient, together with (2.3)–(2.5) having in mind the initial condition(2.7)S(y,0)=0,
finally gives(2.8)ρ(1+λ∂∂t)∂u(y,t)∂t=(1+λ∂∂t)∂T(y,t)∂t-(1+λ∂∂t)σB02u-μϕk(1+λr∂∂t)u+(1+λ∂∂t)T(y,t)=μ(1+λr∂∂t)∂u(y,t)∂t,
where T(y,t)=Sxy(y,t) is the tangential stress.


## 3. Solution of the Problem

### 3.1. Flow due to Impulsive Motion of the Plate

Consider the unsteady flow of an incompressible Oldroyd-B fluid occupying the upper porous half-space of(x,y) plane.The fluid is bounded by a rigid plate at y=0 such that the positive Y-axis is taken normal to the plate and X-axis is taken parallel to the plate. The Oldroyd-B fluid is assumed to be electrically conducting under the influence of a uniform magnetic field B0 applied in a direction normal to the flow. Initially, both the fluid and the plate are at rest. At time t=0+, the plate is suddenly jerked and the motion in the fluid is induced in the direction parallel to X-axis. Under these assumptions the flow is governed by (2.8) along with the following initial and boundary conditions:(3.1)u(y,0)=0,T(y,0)=0,y>0,(3.2)u(0,t)=U0,t>0.
Furthermore, the natural conditions(3.3)u(y,t),T(y,t)⟶0,asy⟶∞,
have to be also satisfied.Using the following dimensionless variables [26]:(3.4)τ=tλ,ξ=ycλ,U=uU0,S=TρcU0,
the governing (2.8) take the following form(3.5)∂U∂τ=∂S∂ξ-M2U-1K(1+α∂∂τ)U,(1+∂∂τ)S=(1+α∂∂τ)∂U∂ξ.
The corresponding initial and boundary conditions are(3.6)U(ξ,0)=0,S(ξ,0)=0,(3.7)U(0,τ)=1,τ>0,(3.8)U(ξ,τ),S(ξ,τ)⟶0,asξ⟶∞,
where(3.9)M2=σB02λρ,1K=μϕλρk,α=λrλ,c=μρλ.
Applying Laplace transform to (3.5), using the initial conditions (3.6), we attain the following system in the transformed q-plane:(3.10)∂S̅(ξ,q)∂ξ=(M2+1K)U̅(ξ,q)+(αK+1)qU̅(ξ,q),(1+q)S̅(ξ,q)=(1+αq)∂U̅(ξ,q)∂ξ,
where(3.11)U̅(0,q)=1q;U̅(ξ,q),S̅(ξ,q)⟶0,asξ⟶0.
Eliminating S̅(ξ,q) between (3.10) and solving the resulting equation for U̅(ξ,q) using conditions (3.11), we get(3.12)U̅(ξ,q)=1qexp⁡(-ξa3αq2+a4q+a5q+α0),
or(3.13)U̅(ξ,q)=1qexp⁡(-ξa3w(q)α)wherew(q)=q2+a4q+a5q+α0.
From (3.10),it is easy to write(3.14)S̅(ξ,q)=-a3αq2+a4q+a5q2+qexp⁡⁡(-(ξa3/α)(q2+a4q+a5)/(q+α0))(q2+a4q+a5)/(q+α0)
or equivalently(3.15)S̅(ξ,q)=-a3αq2+a4q+a5q2+qexp⁡(-(ξa3/α)w(q))w(q).
Now in order to determine U(ξ,τ)=ℒ-1{U̅(ξ,q)}, we are writing (3.13) as follows:(3.16)U̅(ξ,q)=u̅1(q)u̅2(ξ,q),
where(3.17)u̅1(q)=1q,u̅2(ξ,q)=exp⁡(-ξa3αw(q)).
If we denote u1(τ)=ℒ-1{u̅1(q)}and u2(ξ,τ)=ℒ-1{u̅2(ξ,q)}, then it is easy to write(3.18)u1(τ)=1,(3.19)u2(ξ,τ)=ξa32απ∫0∞δ(τ-u)uuexp⁡(-ξ2a34αu-a6u)du-ξa3a72απ∫0τ∫0∞δ(τ-s-u)usexp⁡(-ξ2a34αu-α0s-a6u)×J1(2a7us)duds.
Using the convolution formula(3.20)U(ξ,τ)=∫0τu1(τ-s)u2(ξ,s)ds.
Equations (3.18) and (3.19) give(3.21)U(ξ,τ)=ξa32απ∫0τ1ssexp⁡(-a3ξ24αs-a6s)ds-ξa3a72απ∫0τ∫0w1(s-w)wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7(s-w)w)dwds.
Similarly the Laplace inverse of (3.15) is derived as(3.22)S(ξ,τ)=-a3απτexp⁡(-a3ξ24ατ-a6τ)+a3a7απ∫0τ1sexp⁡(-a3ξ24α(τ-s)-α0s-a6(τ-s))×J1(2a7s(τ-s))ds-a5a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s)ds+a5a7a3απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7w(s-w))dwds-a9a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s-(τ-s))ds+a9a7a3απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w)-(τ-s))×J1(2a7w(s-w))dwds,
where(3.23)a1=M2+1K,a2=M2+1+1K+αK,a3=αK+1,a4=a2a3,a5=a1a3,a6=a4-α0,a7=a5-α0(a4-α0),a8=a4-1,a9=a8-a5.


### 3.2. Flow due to Uniform Motion of the Plate

Here we consider the flow situation in which motion in the fluid is induced by the uniform acceleration of the plate inx-direction. The governing equations, initial conditions, and a part of boundary conditions are same. The only boundary condition (3.2) is replaced by (3.24)u(0,t)=At.
Using a similar method of solution as in previous section, the derived expressions for velocity, and shear stress fields are given as follows:(3.25)U(ξ,τ)=ξa32απ∫0τ(τ-s)ssexp⁡(-a3ξ24αs-a6s)ds-ξa3a72απ∫0τ∫0w(τ-s)(s-w)wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7(s-w)w)dwds,S(ξ,τ)=-a11a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s)ds+a11a3a7απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7w(s-w))dwds-a5a3απ∫0τ(τ-s)sexp⁡(-a3ξ24αs-a6s)ds+a5a7a3απ∫0τ∫0w(τ-s)wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7w(s-w))dwds-a10a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s-(τ-s))ds+a10a7a3απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w)-(τ-s))×J1(2a7w(s-w))dwds,
where U=u/λA, S=T/ρcλA, a10=a5+1-a4 and a11=a4-a5.


## 3.1. Flow due to Impulsive Motion of the Plate

Consider the unsteady flow of an incompressible Oldroyd-B fluid occupying the upper porous half-space of(x,y) plane.The fluid is bounded by a rigid plate at y=0 such that the positive Y-axis is taken normal to the plate and X-axis is taken parallel to the plate. The Oldroyd-B fluid is assumed to be electrically conducting under the influence of a uniform magnetic field B0 applied in a direction normal to the flow. Initially, both the fluid and the plate are at rest. At time t=0+, the plate is suddenly jerked and the motion in the fluid is induced in the direction parallel to X-axis. Under these assumptions the flow is governed by (2.8) along with the following initial and boundary conditions:(3.1)u(y,0)=0,T(y,0)=0,y>0,(3.2)u(0,t)=U0,t>0.
Furthermore, the natural conditions(3.3)u(y,t),T(y,t)⟶0,asy⟶∞,
have to be also satisfied.Using the following dimensionless variables [26]:(3.4)τ=tλ,ξ=ycλ,U=uU0,S=TρcU0,
the governing (2.8) take the following form(3.5)∂U∂τ=∂S∂ξ-M2U-1K(1+α∂∂τ)U,(1+∂∂τ)S=(1+α∂∂τ)∂U∂ξ.
The corresponding initial and boundary conditions are(3.6)U(ξ,0)=0,S(ξ,0)=0,(3.7)U(0,τ)=1,τ>0,(3.8)U(ξ,τ),S(ξ,τ)⟶0,asξ⟶∞,
where(3.9)M2=σB02λρ,1K=μϕλρk,α=λrλ,c=μρλ.
Applying Laplace transform to (3.5), using the initial conditions (3.6), we attain the following system in the transformed q-plane:(3.10)∂S̅(ξ,q)∂ξ=(M2+1K)U̅(ξ,q)+(αK+1)qU̅(ξ,q),(1+q)S̅(ξ,q)=(1+αq)∂U̅(ξ,q)∂ξ,
where(3.11)U̅(0,q)=1q;U̅(ξ,q),S̅(ξ,q)⟶0,asξ⟶0.
Eliminating S̅(ξ,q) between (3.10) and solving the resulting equation for U̅(ξ,q) using conditions (3.11), we get(3.12)U̅(ξ,q)=1qexp⁡(-ξa3αq2+a4q+a5q+α0),
or(3.13)U̅(ξ,q)=1qexp⁡(-ξa3w(q)α)wherew(q)=q2+a4q+a5q+α0.
From (3.10),it is easy to write(3.14)S̅(ξ,q)=-a3αq2+a4q+a5q2+qexp⁡⁡(-(ξa3/α)(q2+a4q+a5)/(q+α0))(q2+a4q+a5)/(q+α0)
or equivalently(3.15)S̅(ξ,q)=-a3αq2+a4q+a5q2+qexp⁡(-(ξa3/α)w(q))w(q).
Now in order to determine U(ξ,τ)=ℒ-1{U̅(ξ,q)}, we are writing (3.13) as follows:(3.16)U̅(ξ,q)=u̅1(q)u̅2(ξ,q),
where(3.17)u̅1(q)=1q,u̅2(ξ,q)=exp⁡(-ξa3αw(q)).
If we denote u1(τ)=ℒ-1{u̅1(q)}and u2(ξ,τ)=ℒ-1{u̅2(ξ,q)}, then it is easy to write(3.18)u1(τ)=1,(3.19)u2(ξ,τ)=ξa32απ∫0∞δ(τ-u)uuexp⁡(-ξ2a34αu-a6u)du-ξa3a72απ∫0τ∫0∞δ(τ-s-u)usexp⁡(-ξ2a34αu-α0s-a6u)×J1(2a7us)duds.
Using the convolution formula(3.20)U(ξ,τ)=∫0τu1(τ-s)u2(ξ,s)ds.
Equations (3.18) and (3.19) give(3.21)U(ξ,τ)=ξa32απ∫0τ1ssexp⁡(-a3ξ24αs-a6s)ds-ξa3a72απ∫0τ∫0w1(s-w)wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7(s-w)w)dwds.
Similarly the Laplace inverse of (3.15) is derived as(3.22)S(ξ,τ)=-a3απτexp⁡(-a3ξ24ατ-a6τ)+a3a7απ∫0τ1sexp⁡(-a3ξ24α(τ-s)-α0s-a6(τ-s))×J1(2a7s(τ-s))ds-a5a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s)ds+a5a7a3απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7w(s-w))dwds-a9a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s-(τ-s))ds+a9a7a3απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w)-(τ-s))×J1(2a7w(s-w))dwds,
where(3.23)a1=M2+1K,a2=M2+1+1K+αK,a3=αK+1,a4=a2a3,a5=a1a3,a6=a4-α0,a7=a5-α0(a4-α0),a8=a4-1,a9=a8-a5.


## 3.2. Flow due to Uniform Motion of the Plate

Here we consider the flow situation in which motion in the fluid is induced by the uniform acceleration of the plate inx-direction. The governing equations, initial conditions, and a part of boundary conditions are same. The only boundary condition (3.2) is replaced by (3.24)u(0,t)=At.
Using a similar method of solution as in previous section, the derived expressions for velocity, and shear stress fields are given as follows:(3.25)U(ξ,τ)=ξa32απ∫0τ(τ-s)ssexp⁡(-a3ξ24αs-a6s)ds-ξa3a72απ∫0τ∫0w(τ-s)(s-w)wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7(s-w)w)dwds,S(ξ,τ)=-a11a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s)ds+a11a3a7απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7w(s-w))dwds-a5a3απ∫0τ(τ-s)sexp⁡(-a3ξ24αs-a6s)ds+a5a7a3απ∫0τ∫0w(τ-s)wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w))×J1(2a7w(s-w))dwds-a10a3απ∫0τ1sexp⁡(-a3ξ24αs-a6s-(τ-s))ds+a10a7a3απ∫0τ∫0w1wexp⁡(-a3ξ24α(s-w)-α0w-a6(s-w)-(τ-s))×J1(2a7w(s-w))dwds,
where U=u/λA, S=T/ρcλA, a10=a5+1-a4 and a11=a4-a5.


## 4. Limiting Cases

In this section we want to reduce the solutions obtained in the previous section to their limiting cases.(1) The solutions (3.21) and (3.22) (impulsive motion of the plate), and (3.25) (uniform motion of the plate) for hydrodynamic Oldroyd-B fluid (M=0)in a nonporous space (1/K=0) reduce to the following equations:
(4.1)U(ξ,τ)=ξ2απ∫0τ1ssexp⁡(-ξ24αs-(1-α0)s)ds-ξα0(α0-1)2απ∫0τ∫0w1(s-w)wexp⁡(-ξ24α(s-w)-α0w-(1-α0)(s-w))×J1(2α0(α0-1)(s-w)w)dwds,(4.2)S(ξ,τ)=-απτexp⁡(-ξ24ατ-(1-α0)τ)+α0(α0-1)απ∫0τ1sexp⁡(-ξ24α(τ-s)-α0s-(1-α0)(τ-s))×J1(2α0(α-1)s(τ-s))ds,(4.3)U(ξ,τ)=ξ2απ∫0τ(τ-s)ssexp⁡(-ξ24αs-(1-α0)s)ds-ξα0(α0-1)2απ∫0τ∫0w(τ-s)(s-w)wexp⁡(-ξ24α(s-w)-α0w-(1-α0)(s-w))×J1(2α0(α0-1)(s-w)w)dwds,(4.4)S(ξ,τ)=-απ∫0τ1sexp⁡(-ξ24αs-(1-α0)s)ds+α0(α0-1)απ×∫0τ∫0w1wexp⁡(-ξ24α(s-w)-α0w-(1-α0)(s-w))×J1(2α0(α0-1)w(s-w))dwds.Here we can see that (4.1) and (4.2) are identical to (3.20) and (3.19) from [14]. The criteria for getting (4.1) and (4.2) is different than one used in [14]. This is due to the fact that the inclusion of last term due to the porosity effect in the equation of motion makes it difficult to use the procedure followed in [14]. However, in the present situation we make it able to find the solutions (4.1) and (4.2) in a very easy and interesting way.(2) Making the limit asα→1(λr→λ) into (3.21), (3.22), and (3.25), we get the similar solutions for a Newtonian fluid performing the same motion. Thus the solutions (3.21) and (3.22) corresponding to the flow due to impulsive motion of the plate are given by
(4.5)U(ξ,τ)=ξK+12Kπ∫0τ1ssexp⁡(-(K+1)ξ24Ks-(M2+1K)s)ds,S(ξ,τ)=-K+1Kπτexp⁡(-(K+1)ξ24Kτ-(M2+1K)τ)-(KM2+1)(K+1)Kπ∫0τ1sexp⁡(-(K+1)ξ24Ks-(M2+1K)s)ds
and the solutions (3.25) due to uniform motion of the plate reduce to
(4.6)U(ξ,τ)=ξK+12πK∫0τ(τ-s)ssexp⁡(-ξ2(K+1)4Ks-(M2+1K)s)ds,S(ξ,τ)=-K+1Kπ∫0τ1sexp⁡(-(K+1)ξ24Ks-(M2+1K)s)ds-M2K+1(K+1)Kπ∫0τ(τ-s)sexp⁡(-(K+1)ξ24Ks-(M2+1K)s)ds.It is important to note that if we takeM=1/K=0 into (4.5) and make some suitable change of variables, we get the similar solutions as given in [14]([see (4.2) and (4.3)]).


## 5. Results and Discussion

This section includes the graphical illustrations of various results from the flow analyzed in this paper. The results have been interpreted for various values of permeability parameterKand MHD parameter M.Special attention has been focused on the permeability parameter K.It is found that the flow analysis strongly depends on these parameters. The graphical results are displayed for velocity and shear stress fields for two different flow cases, namely (i) flow due to impulsive motion of the plate and (ii) flow due to uniform motion of the plate. In all these figures the relaxation parameter λ, the retardation parameter λr and dimensionless time τare, respectively, chosen as 1.0, 2.0 and 0.4. Further, in these Figures 1–4 panels (a) and (b) are displayed for velocity and shear stress fields, respectively.Profiles of (a) velocity and (b) shear stress for different values ofK (impulsive motion of the plate).
(a)(b)Profiles of (a) velocity and (b) shear stress for different values ofM (impulsive motion of the plate).
(a)(b)Profiles of (a) velocity and (b) shear stress for different values ofK (uniform motion of the plate).
(a)(b)Profiles of (a) velocity and (b) shear stress for different values ofM (uniform motion of the plate).
(a)(b)Figures1 and 2 are prepared for velocity and shear stress fields for flow induced by the impulsive motion of the plate for various values of permeability parameter K magnetic parameter M. It is noted from Figure 1 that velocity and boundary layer thickness increases with increasing values of K. It may also be expected due to the fact that increasing values of K reduces the friction forces which assists the fluid considerably to move fast. Further as it was expected that the strongest shear stress occurs near the boundary and decreases rapidly with increasing distance from the plate. Figure 2 is sketched in order to explore the variations of magnetic parameter M. It is observed that the velocity and boundary layer thickness decreases upon increasing the values of M. It is because of the fact that the application of transverse magnetic field will result in a resistive type force (called Lorentz force) similar to drag force and upon increasing the values of M increases the drag force which leads to the deceleration of the flow. However, the magnitude of the shear stress increases with increasing values of M.The magnitude of the shear stress in the close regime of the boundary is larger as compared to the region away from the boundary.Figures3 and 4 have been sketched for the situation when the flow is driven by the uniform motion of the plate. It is observed that the variations of velocity and shear stress fields in these figures are qualitatively similar to the figures in case of impulsive motion of the plate. However, when analyzed carefully, it is found that these observations are not the same quantitatively.



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*Source: 408132-2011-06-13.xml*