# MHD Accelerated Flow of Maxwell Fluid in a Porous Medium and Rotating Frame

Authors: Faisal Salah; Zainal Abdul Aziz; Mahad Ayem; Dennis Ling Chuan Ching
Journal: ISRN Mathematical Physics
 (2013)

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## Abstract

The magnetohydrodynamic (MHD) and rotating flow of Maxwell fluid induced by an accelerated plate is investigated. The Maxwell fluid saturates the porous medium. Both constant and variable accelerated cases are considered. Exact solution in each case is derived by using Fourier sine transform. Many interesting available results in the relevant literature are obtained as the special cases of the present analysis. The graphical results are presented and discussed.


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## Body

## 1. Introduction

Several fluids including butter, cosmetics and toiletries, paints, lubricants, certain oils, blood, mud, jams, jellies, shampoo, soaps, soups, and marmalades have rheological characteristics and are referred to as the non-Newtonian fluids. The rheological properties of all these fluids cannot be explained by using a single constitutive relationship between stress and shear rate which is quite different than the viscous fluids [1, 2]. Such understanding of the non-Newtonian fluids forced researchers to propose more models of non-Newtonian fluids.In general, the classification of the non-Newtonian fluid models is given under three categories which are called the differential, the rate, and the integral types [3]. Out of these, the differential and rate types have been studied in more detail. In the present analysis we discuss the Maxwell fluid which is the subclass of rate-type fluids which take the relaxation phenomenon into consideration. It was employed to study various problems due to its relatively simple structure. Moreover, one can reasonably hope to obtain exact solutions from Maxwell fluid. This motivates us to choose the Maxwell model in this study. The exact solutions are important as these provide standard reference for checking the accuracy of many approximate solutions which can be numerical or empirical in nature. They can also be used as tests for verifying numerical schemes that are being developed for studying more complex flow problems [4–9].On the other hand, these equations in the non-Newtonian fluids offer exciting challenges to mathematical physicists for their exact solutions. The equations become more problematic, when a non-Newtonian fluid is discussed in the presence of MHD and porous medium. Despite this fact, various researchers are still making their interesting contributions in the field (e.g., see some recent studies [1–15]). Few investigations which provide the examination of non-Newtonian fluids in a rotating frame are also presented [1–19]. Such studies have special relevance in meteorology, geophysics, and astrophysics.To the best of our knowledge, no investigation has been reported so far which discusses the accelerated flows of non-Newtonian fluids in a rotating frame. This is the objective of the present study. Here, we examine the rotating and MHD flow induced by an accelerated plate. Two explicit examples of acceleration subject to a rigid plate are taken into account. Constitutive equations of a Maxwell fluid are used and modified Darcy’s law has been utilized. The exact solution to the resulting problem is developed by Fourier sine transform. With respect to physical applications, the graphs are plotted in order to illustrate the variations of embedded flow parameters. The mathematical results of many existing situations are shown as the special cases of the present study.


## 2. Formulation of the Problem

We choose a Cartesian coordinate system by considering an infinite plate atz=0. An incompressible fluid occupying the porous space is electrically conducting by exerting an applied magnetic field B° parallel to the z-axis. The electric field is not taken into consideration, the magnetic Reynolds number is small, and the induced magnetic field is not accounted for. The Lorentz force J×B° under these conditions is equal to -σB°2V. Here, J is the current density, V is the velocity field, and σ is electrical conductivity of fluid. Both plate and fluid possess solid body rotation with a uniform angular velocity Ω about the z-axis.The governing equations are(1)ρ[∂V∂t+(V·∇)V+2Ω×V+Ω×(Ω×r)]=-∇p+divS-σB°2V+R,(2)divV=0,
in which ρ is the fluid density, r is a radial vector with r2=x2+y2, p is the pressure, and R is Darcy’s resistance.The extra stress tensorS for a Maxwell fluid satisfies
(3)T=-pI+S,S+λ[dSdt-LS-SLT]=μA,
where T is the Cauchy stress tensor, I is the identity tensor, L is the velocity gradient, A=L+LT is the first Rivlin-Eriksen tensor, λ is the relaxation time, μ is dynamic viscosity of fluid and d/dt indicates the material derivative.According to Tan and Masuoka [4], Darcy’s resistance in an Oldroyd-B fluid satisfies the following expression:
(4)(1+λ∂∂t)R=-μϕk(1+λr∂∂t)V,
where λr is the retardation time, ϕ is the porosity, and k is the permeability of the porous medium. For Maxwell fluid λr=0, and hence,
(5)(1+λ∂∂t)R=-μϕkV.We seek a velocity field of the form:(6)V=(u(z,t),υ(z,t),w(z,t)),
which together with (2) yields w=0. Then, using (1) and (3) into (5) we arrive at
(7)ρ(∂u∂t-2Ωυ)=-∂p^∂x+∂Sxz∂z-σB°2u+Rx,ρ(∂υ∂t+2Ωu)=-∂p^∂y+∂Syz∂z-σB°2υ+Ry,
where
(8)(1+λ∂∂t)Sxz=μ∂u∂z,(1+λ∂∂t)Syz=μ∂υ∂z.Rx and Ry are x- and y-components of Darcy’s resistance R; and z-component of (1) indicates that p^≠p^(z), and modified pressure p^ is p^=p-(ρ/2)Ω2r2.Invoking (4) and (8) in (7) and then neglecting the pressure gradient, we arrive at
(9)ρ(1+λ∂∂t)(∂u∂t-2Ωυ)=μ∂2u∂z2-σB°2(1+λ∂∂t)u-μϕku,ρ(1+λ∂∂t)(∂υ∂t+2Ωu)=μ∂2υ∂z2-σB°2(1+λ∂∂t)υ-μϕkυ.
Mathematically when the pressure gradient is ignored, the equation of motion becomes simplified and manageably solvable, and physically the fluid is still in motion as required by the boundary conditions. We have T=-pI+S, and when the pressure gradient is negligible, the stress tensor T is equivalent to the extra stress tensor S, portraying the fluid motion.The initial and boundary conditions for a constant accelerated plate are(10)u=υ=0att=0,z>0,(11)u(0,t)=At,υ(0,t)=0fort>0,(12)u,∂u∂z,υ,∂υ∂z⟶0,asz⟶∞,t>0,
where A has a dimension of L/T2.


## 3. Solution for Constant Accelerated Flow

SettingF=u+iυ and introducing the following dimensionless quantities:
(13)ξ=Z(Aν2)1/3,τ=t(A2ν)1/3,G=F(νA)1/3,β=λ(A2ν)1/3,M=σB°2ρ(νA2)1/3,ω=(Ων1/3A2/3),1B=ϕk(ν2A)2/3,c=2iω+M.
The above problem statement reduces to
(14)β∂2G(ξ,τ)∂τ2+(1+βc)∂G(ξ,τ)∂τ+(c+1B)G(ξ,τ)=∂2G(ξ,τ)∂ξ2,ξ,τ>0,G(0,τ)=τ,τ>0,G(ξ,0)=∂G(ξ,τ)∂τ=0,ξ>0,G(ξ,τ),∂G(ξ,τ)∂ξ⟶0asξ⟶∞,τ>0.
After using Fourier sine transform, (14) becomes
(15)β∂2Gs(η,τ)∂τ2+(1+βc)∂Gs(η,τ)∂τ+(c+1B+η2)Gs(η,τ)=2πητ,η,τ>0,(16)Gs(η,0)=∂Gs(η,τ)∂τ=0,η>0,
in which Gs(η,τ) indicates the Fourier sine transform of G(ξ,τ).The solution of (15) satisfying condition (16) can be expressed as
(17)Gs(η,τ)=2πη(c+1/B+η2)Gs(η,τ)=×[(r3+c)r2er1τ-(r4+c)r1er2τ(c+1/B+η2)(r2-r1)β+τ-(1+βc)(c+1/B+η2)]
or
(18)Gs(η,τ)=2πη(c+1/B+η2)Gs(η,τ)=×[r12er2τ-r22er1τ(c+1/B+η2)(r2-r1)β+τ-(1+βc)(c+1/B+η2)]with(19)r1,r2=-(1+βc)±(1+βc)2-4β(c+(1/B)+η2)2β,r3,r4=(1-βc)±(1+βc)2-4β(c+(1/B)+η2)2β.
Inversion of Fourier sine transform in (17) gives
(20)G(ξ,τ)=τe-(1/B+c)ξG(ξ,τ)=-2π∫0∞[1+βc-r12er2τ-r22er1τ(r2-r1)β]G(ξ,τ)=×ηsin(ξη)(c+1/B+η2)2dη.
The above expression for hydrodynamic fluid M=0 in a nonporous space 1/B=0 is
(21)G(ξ,τ)=τe-(1+i)ωξG(ξ,τ)=-2π∫0∞[1+2iωβ-r52er6τ-r62er5τ(r6-r5)β]G(ξ,τ)×ηsin(ξη)(2iω+η2)2dη,(22)r5,r6=-(1+2iωβ)±(1+2iωβ)2-4β(2iω+η2)2β.
Letting ω=0 in (21) and (22), we arrive at
(23)G(ξ,τ)=τ-2π∫0∞[1-ra2erbτ-rb2eraτ(rb-ra)β]sin(ξη)η3dηra,rb=-1±1-4βη22β.
The velocity field G(ξ,τ) given by (21) is equivalent to that obtained by Fetecau et al. [3, Equation 25] and [5, Equation 31].Result (20) for a magnetohydrodynamic viscous fluid λ=0 in a porous space is
(24)G(ξ,τ)=τe-(1/B+c)ξG(ξ,τ)=-2π∫0∞[1-e(c+1/B+η2)τ]ηsin(ξη)(c+1/B+η2)2dη.
In the above expression, if we put c=0 and 1/B=0, we obtain the equivalent solution which was obtained by Fetecau et al. [5, Equation 22],
(25)G(ξ,τ)=τ-2π∫0∞[1-eη2τ]sin(ξη)η3dη.To see the variations of embedded flow parameters in the solution expressions, Figures1 to 5 have been displayed in order to illustrate such variations for the constant accelerated flow. Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.Velocity profiles for different values ofβ.
(a)
(b)


## 4. Solution for Variable Accelerated Flow

In this section, the problem statement consists of (9) to (10), (12) and
(26)u(0,t)=ℜt2,υ(0,t)=0fort>0,
where ℜ has a dimension of L/T3.The flow problem after defining in terms of the dimensionless quantities,(27)ζ=z(ℜν3)1/5,δ=t(ℜ2ν)1/5,S=F(ν2ℜ)1/5,P=λ(ℜ2ν)1/5,H=σB°2ρ(νℜ2)1/5,ψ=(Ων1/5ℜ2/5),1L=ϕk(ν3ℜ)2/5,D=2iψ+H,
becomes
(28)P∂2S(ζ,δ)∂δ2+(1+PD)∂S(ζ,δ)∂δ+[D+1L]S(ζ,δ)=∂2S(ζ,δ)∂ζ2;ζ,δ>0,S(0,δ)=δ2,δ>0,S(ζ,δ),∂S(ζ,δ)∂ζ⟶0asζ⟶∞;δ>0,S(ζ,0)=∂S(ζ,δ)∂δ=0,ζ>0.
Following similar methodology of solution as in the previous section, we obtain
(29)S(ζ,δ)=[ζ(1+PD)2(1+ζ(1/L)+D)4((1/L)+D)3δ2-ζ[δ(1+PD)+P]1/L+D+ζ(1+PD)2(1+ζ1/L+D)4(1/L+D)3]e-(1/L+D)ζ+4π∫0∞[((D+PD2-1L-η2)r7r10er8δ-r8r9er7δ+(r7er8δ-r8er7δ)(D+PD2-1L-η2))×(r8-r7)-1P((D+PD2-1L-η2)r7r10er8δ-r8r9er7δ]×ηsin(ζη)(D+1/L+η2)3dη+4Dπ∫0∞[(er8δ-er7δ)r8-r7P]ηsin(ζη)(D+1/L+η2)2dη,
where
(30)r7,r8=-(1+PD)±(1+PD)2-4P(D+1/L+η2)2P,r9,r10=(1-PD)±(1+PD)2-4P(D+1/L+η2)2P.The previous expression for hydrodynamic fluid in a nonrotatingD=0 and nonporous space 1/L=0 is
(31)S(ζ,δ)=δ2-4π(δ-P)+4π∫0∞ηsin(ζη)η3dη+4π∫0∞ηsin(ζη)η5dη+4π∫0∞[(r11er12δ-r12er11δ)r11r14er12δ-r12r13er11δ+(r11er12δ-r12er11δ)r12-r11P]×ηsin(ζη)η3dη,
where
(32)r11,r12=-1±1-4Pη22P,r13,r14=1±1-4Pη22P.For a magnetohydrodynamic viscous fluidλ=0→P=0 in a porous space, (31) takes the form
(33)S(ζ,δ)=δ2e-(1/L+D)ζS(ζ,δ)=+4π∫0∞[1-e(D+1/L+η2)δ]ηsin(ξη)(D+1/L+η2)3dηS(ζ,δ)=-4δπ∫0∞ηsin(ξη)(D+1/L+η2)2dη.In order to see the variations of embedded flow parameters in the solution expressions, Figures6, 7, 8, 9, and 10 have been displayed in order to illustrate such variations for the variable accelerated flow. Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.


## 5. Results and Discussion

This section concerns the variations of embedded flow parameters in the solution expressions. Hence, Figures1–10 have been displayed in order to illustrate such variations. We note that Figures 1–5 have been sketched for the constant accelerated flow, whereas Figures 6–10 are shown for the variable accelerated flow. Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.Velocity profiles for different values ofM.
(a)
(b)Velocity profiles for different values ofB.
(a)
(b)Velocity profiles for different values ofω.
(a)
(b)Velocity profiles for different values ofτ.
(a)
(b)Velocity profiles for different values ofP.
(a)
(b)Velocity profiles for different values ofH.
(a)
(b)Velocity profiles for different values ofψ.
(a)
(b)Velocity profiles for different values ofL.
(a)
(b)Velocity profiles for different values ofδ.
(a)
(b)The effect of dimensionless relaxation time parameterβ on the velocity profile is shown in Figure 1. It is noticed too that the magnitudes of an imaginary component of velocity are decreasing function of β, and thus r1, r2, r3, and r4 in G(ξ,τ). Figure 2 elucidates the behaviour of M on the velocity components. It is observed that the role of M on the magnitude of velocity component is qualitatively similar to that of β. This is in accordance with the fact that the Lorentz force acts as a resistance force to the flow under consideration. On the other hand, the boundary layer thickness decreases when both β and M increase. The effects of B and the velocity components are sketched in Figure 3. Clearly the magnitude of velocity decreases when B is increased; that is, when the medium becomes less porous (porosity ϕ decreases). Figure 4 plots the effects of the solid body rotation in terms of ω. Figure 4(a) indicates that the real part of velocity is reduced when ω increases. However, Figure 4(b) shows correspondingly a different behaviour, whereby the velocity first increases and then decreases with respect to ω. It is seen that the variation here is more oscillatory in character. The variation of τ on the magnitude of velocity components is sketched in Figure 5. It is found that the magnitude of velocity components increases with dimensionless time. Further, it is observed that the variations of parameters in constant accelerated flow (i.e., Figures 1–5) are qualitatively similar (i.e., in Figure 6, the magnitudes of an imaginary component of velocity are decreasing function of β) to the effects plotted for variable accelerated flow (i.e., Figures 6–10). However, the velocity profiles in constant accelerated flow and variable accelerated flow are not similar quantitatively. Comparison shows that the velocity profiles in variable accelerated flow are larger when compared to those of constant accelerated flow.


## 6. Conclusions

In this research, the constant accelerated flow and variable accelerated flow of non-Newtonian fluid in a rotating frame are examined. The exact solutions are first established via Fourier sine transform, and then the following observations have been noted through the graphs.(i)
The magnitude of velocity components in viscous fluid is greater when compared with Maxwell fluid in case of constant accelerated flow and variable accelerated flow.(ii)
The magnitude of velocity components for constant accelerated flow and variable accelerated flow and MHD fluid is less than that of hydrodynamic fluid.(iii)
The behaviours ofB on the magnitude of velocity components are quite opposite to those of M and β.(iv)
The salient features of the velocity profile with respect to variations of various embedded parameters in constant and variable accelerated cases are similar in a qualitative sense.



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