# Modeling and Analysis of MHD Oscillatory Flows of Generalized Burgers’ Fluid in a Porous Medium Using Fourier Transform

Authors: Mazhar Hussain; Mubashir Qayyum; Sidra Afzal
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, exact solutions of unsteady oscillatory generalized Burgers’ fluid are proposed for three different cases using Fourier transform approach. The fluid is electrically conducting under the influence of uniform transverse magnetic field and passing through the porous medium. MHD flows are induced by imposed periodic pressure gradients with smaller oscillations. Closed form solutions are obtained using Fourier sine transform, and several existing results are recovered as limiting cases. Furthermore, effects of different fluid parameters on the velocity profile are studied graphically. Analysis reveals that magnetic field and porosity parameter increases the velocity profile in case of Oldroyd-B and generalized Burgers’ fluid. It is also observed that magnetic field has more prominent effect on Burgers’ fluid as compared to Oldroyd-B fluid, while porosity parameter showed noticeable effect on Oldroyd-B fluid as compared to Burgers’ fluid.


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## Body

## 1. Introduction

The mechanics of non-Newtonian fluids is much complicated and more nonlinear in comparison with that of the Newtonian fluids. Many problems dealing with the flow of non-Newtonian fluids have been studied by engineers and mathematicians in various geometrical configurations. Examples of these fluids are paste, gel, shampoo, soap, bead dough etc. The analysis of such flows finds important applications in engineering practice, particularly in chemical industries. A huge variety of consumer goods are now made from injection-molded plastics which often contain high concentrations of glass or carbon fibers. Many modern paints and lubricants contain polymer additives which are added to enhance their flow properties or the quality of the finished products. Also, many food stuffs (e.g. tomato sauce) and biological fluids (e.g. blood) are non-Newtonian.Due to variety of fluids, there are several constitutive equations of non-Newtonian fluids in the literature. Among these the viscoelastic fluids have acquired the special status. In the literature, numerous works have been done to investigate the behavior of viscoelastic fluids. Some interesting flows in this direction are discussed by Rajagopal [1], Shamsuddin et al. [2, 3], Anwar et al. [4], Tan and Masuoka [5], Venkatesan and Ganesan [6], Fetecau and Vien [7], Fetecau and Fetecau [8], Lee [9], Fetecau and Agop [10], Ali et al. [11], Hayat et al. [12–14], and Nanganthran et al. [15].All the above investigations have dealt with second grade and Oldroyd-B fluid models. Very little work have been reported to flow of a Burgers’ fluid. In light of the above literature review, it is noted that generalized Burgers’ fluid model with oscillatory motion in porous medium has not been solved using Fourier transform method. Novelty of this study is further elaborated through Table1. Moreover, recent studies on Burgers’ fluid flow are also being considered in literature. Gangadhar et al. [16] have investigated Burgers’ fluid with convective heating. Generalized Burgers’ nanofluid with Cattaneo–Christov Relations is studied by Shahzad et al. [17] using Galerkin finite element method. Hayat et al. [18, 19] reported some analytical results for flows of a Burgers’ fluid under varying conditions. Khan et al. [20] investigated Burgers’ fluid under magnetization with stagnation point flow. The Burgers’ fluid is a viscoelastic fluid which has been used to characterize cheese, soil, asphalt etc [21–25]. This model has also been used in calculating the transient creep properties of the earth’s mantle and in the modeling of high temperature viscoelasticity of fine-grained poly-crystalline olivine [26, 27]. The Burgers’ fluid model has also been generalized and Waqas et al. [28] discussed some heat flux models to flow in this direction.Table 1
Present study in comparison with existing work in literature.
GeneralizedMHDPorousOscillatoryFourierBurgers’ fluidmediumflowtransformFetecau and Vieru [32]NoYesYesNoNoAkram et al. [33]NoNoNoYesNoAli et al. [34]NoYesYesYesNoHayat et al. [35]YesNoNoNoYesPresentYesYesYesYesYesIn order to solve fluid models various methodologies are used in literature. Fourier transform method is employed in this study to obtain exact solutions of the modeled problems. The Fourier transform can express an arbitrary a-periodic function as an infinite integral over a continuous range of frequencies. Firstly it was used in the treatment of single pulse phenomena by electrical engineers. The Fourier transform and the related operations of convolution and correlation has applications in optics, acoustics, scattering and diffraction of x-rays, neutrons and electrons, and a-periodic effects in electrical circuits. In various recent studies, Fourier transform method is used to solve mathematical problems effectively. Abro et al. [29] studied thermal effects on micro polar fluid with MHD and porosity by utilizing Fourier sine transform scheme. Numerical solution of Burgers’ equation was presented by Egidi et al. [30] by using Fourier transforms. Liu et al. [31] employed different transforms to study MHD flow and heat transfer of generalized Burgers’ fluid.The main objective of this paper is to analyze the unsteady flows of generalized Burgers’ fluid induced by the cosine and sine oscillations of an infinite plate in a porous medium. The flow caused by an oscillating pressure gradient is also considered. The fluid is electrically conducting and occupies the porous structure. Graphical analysis for fluid velocity is given against varying fluid parameters. Generalization of fluid model into different fluids such as Newtonian, second grade, Maxwell, Oldroyd-B, and generalized fluid with varying material constants is depicted in tabular form. Such flows in porous media are important in enhanced oil recovery, paper and textile coating, and composite manufacturing processes. The whole analysis is given in the presence of a constant magnetic field which is very important. Finally, the results are discussed with the help of several graphs.


## 2. Governing Equations

The Cauchy stressT in a generalized Burgers’ fluid is [28].
(1)T=−pI+S,(2)1+λ1δδt+λ2δ2δt2S=μ1+λ3δδt+λ4δ2δt2A,where −pI indicates the indeterminate spherical stress, A=L+LT is first Rivlin-Ericksen tensor, μ is the dynamic viscosity, L is the velocity gradient, λ1 and λ3<λ1 are the relaxation and retardation times, S is the extra stress tensor, λ2,λ4 are material constants and δ/δt is the upper convected time derivative defined by
(3)δSδt=dSdt+LS−SLT,where d/dt is the material derivative and
(4)δ2Sδt2=δδtδSδt.The flows under consideration have the following propertiesUnsteady velocity iny-direction and fluid is at rest at t=0.
(i)
Uniform magnetic field is assumed along with Darcian Porous mediumThe fluid velocity vector in this case is(5)V=uy,ti,where i and u are the unit vector and velocity parallel to the x-axis, respectively. The velocity field (5) automatically satisfies the incompressibility condition. Since u is a function of y and t, the stress field will also depend upon y and t. Now, Eq.(2) together with the initial condition (the fluid being at rest up to the moment t=0)
(6)Sy,0=0yields Sxz=Syz=Syy=Szz=0 and
(7)Sxx+λ1∂Sxx∂t−2Sxy∂u∂y+λ2∂2Sxx∂t2−4∂Sxy∂t∂u∂y−2Sxy∂2u∂t∂y=−2μλ3∂u∂y2−6μλ4∂u∂y∂2u∂t∂y,1+λ1∂∂t+λ2∂2∂t2Sxy=μ1+λ3∂∂t+λ4∂2∂t2∂u∂y.The balance of linear momentum for MHD fluid in a porous medium is(8)ρdVdt=−∇p+divS+J×B+r,where ρ is the fluid density, J is the current density, B=B0+b is the total magnetic field, B0 is applied magnetic field, b is the induced magnetic fields and r is the Darcy resistance for a generalized Burgers’ fluid in the porous medium. Neglecting the displacement current, the Maxwell equations and Ohms’ law are
(9)divB=0,curlB=μmJ,curlE=−∂B∂t,J=σE+V×B,where E is the electric field, μ is the magnetic permeability and σ is the electrical conductivity. For small magnetic Reynolds number, the induced magnetic field is neglected. It is also assumed that E=0.The Darcy’s law holds for flows of viscous fluid with low speed, in an unbounded porous medium. This law gives relation among velocity and pressure drop induced by frictional drag while ignoring boundary effects on the flow. A direct proportionality between induced pressure and Darcian velocity is observed by this law. Brinkman proposed an equation describing the locally averaged flow for the porous medium with boundaries. In literature, there are various modified Darcy’s law applied for viscous flows. Very little attention has been given to macroscopic models for viscoelastic flows in a porous medium. The following law for both relaxation and retardation phenomenon in an unbounded porous medium holds [5]:
(10)1+λ1∂∂t∇p=−μk1+λ3∂∂tVD,in which k is permeability, VD=ϕV is the Darcian velocity and ϕ is the porosity of porous medium. Note that for λ1=λ3=0, Eq. (10) reduces to well known Darcy’s law of viscous fluid.By the analogy with constitutive Eq (2), the following law for unidirectional flow of a generalized Burgers’ fluid has been suggested:
(11)1+λ1∂∂t+λ2∂2∂t2∇p=−μϕk1+λ3∂∂t+λ4∂2∂t2V.The pressure gradient in above equation can also be interpreted as a measure of resistance to flow in the bulk of porous medium andr is a measure of flow resistance offered by the solid matrix. Thus r can be inferred from Eq.(11) to satisfy the following equation:
(12)1+λ1∂∂t+λ2∂2∂t2r=−μϕk1+λ3∂∂t+λ4∂2∂t2V.Upon use of the stated assumptions, Eq.(8) yields
(13)1+λ1∂∂t+λ2∂2∂t2∂u∂t=1+λ3∂∂t+λ4∂2∂t2∂2u∂y2−σB02ρ1+λ1∂∂t+λ2∂2∂t2u−νϕk1+λ3∂∂t+λ4∂2∂t2u,where the pressure gradient in x direction has been ignored and ν is the kinematic viscosity.


## 3. Stokes’ Second Problem

This section deals with the MHD flow of a generalized Burgers’ fluid in a porous spacey>0.The fluid is bounded by a rigid boundary at y=0. Initially, both fluid and boundary are at rest. For y=0, the boundary starts to oscillate in its own plane. In absence of pressure gradient, the equation which governs the flow is (13). The appropriate boundary and initial conditions are
(14)u0,t=U0cosωtoru0,t=U0sinωt,t>0,(15)uy,y⟶0,∂u∂y⟶0asy⟶∞,t≥0(16)uy,0=∂uy,0∂t=∂2uy,0∂t2=0,y≥0in which ω is the imposed frequency.In order to find the solution we define the Fourier sine transform pair as(17)u~ξ,t=2π∫0∞uy,tsinξydy,uy,t=2π∫0∞u¯ξ,tsinξydξ.


### 3.1. Whenu0,t=U0cosωt and λ2≠0

Taking Fourier sine transform of Eqs. (13)–(15) and then solving them in the ξ-plane, we get the expression for the starting solution as follows
(18)u¯ξ,t=u¯tξ,t+u¯sξ,t,where u¯tξ,t and u¯sξ,t indicate the transient and steady state solutions, respectively, and are given by
(19)u¯tξ,t=−2/πU0ξνm2m3−ω2F0−ω2F1−ω2m2+m3F2em1tm1−m2m1−m3F3+2/πU0ξνω2−m1m3F0−ω2F1+ω2m1+m3F2em2tm2−m1m2−m3F3−2/πU0ξνm1m2−ω2F0−ω2F1−ω2m1+m2F2em3tm3−m1m3−m2F3,(20)u¯sξ,t=2πU0νωsinωtξF2F3+2πU0νcosωtξF0−ω2F1F3,where
(21)F0=νξ2E1+σB02ρ+νϕk,F1=−E2+σB02ρE3+νϕkE4,F2=E5−σB02ρE2,F3=νξ2+σB02ρ+νϕk−ω2E62+ω2E72,E1=1−λ4ω22+λ3ω2,E2=λ3−λ1−λ2λ3ω2+λ1λ4ω2,E3=λ2+λ4−λ1λ3ω2−λ2λ4ω2,E4=2λ4−λ42ω2−λ32,E5=1−ω2λ2+λ4+ω2λ1λ3+λ2λ4ω2,E6=λ1+βξ2+σB02ρλ2+ϕkβ,E7=1+αξ2+σB02ρλ1+ϕkα−λ2ω2,m1=−d3λ2−21/3−d2+3eλ23λ2−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3+−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3321/3λ2,m2=−d3λ2−1+i3−d2+3eλ2322/3λ2−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3−1−i3−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3621/3λ2,m3=−d3λ2+1−i3−d2+3eλ2322/3λ2−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3−1+i3−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3621/3λ2,d=λ1+βξ2+σB02ρλ2+ϕkβ,e=1+αξ2+σB02ρλ1+ϕkα,f=νξ2+σB02ρ+ϕkβ,α=νλ3,β=νλ4.Fourier inversion of Eqs. (19) and (20) yields
(22)uty,t=−2πU0ν∫0∞ξm2m3−ω2F0−ω2F1−ω2m2+m3F2em1tsinξydξm1−m2m1−m3F3+2πU0ν∫0∞ξω2−m1m3F0−ω2F1+ω2m1+m3F2em2tsinξydξm2−m1m2−m3F3−2πU0ν∫0∞ξm1m2−ω2F0−ω2F1−ω2m1+m2F2em3tsinξydξm3−m1m3−m2F3,(23)usy,t=2πU0νωsinωt∫0∞ξF2sinξydξF3+2πU0νcosωt∫0∞ξF0−ω2F1sinξydξF3.Note that for large timesuty,t⟶0 and usy,t becomes [see appendix]
(24)usy,t=U0exp−Aycosωt−By,(25)2A2=ω2E2+ϕ/kνE1+σB02/ρE5νE12+ν2ω2E5−σB02/ρE2ν2E12+ω2E2+ϕ/kνE1+σB02/ρE5νE1,(26)2B2=ω2E2+ϕ/kνE1+σB02/ρE5νE12+ν2ω2E5−σB02/ρE2ν2E12−ω2E2+ϕ/kνE1+σB02/ρE5νE1.Introducing the following dimensionless quantities(27)y~=ω2νy,u~=uU0,t~=ωt,λ~1=λ1ω,λ~2=λ2ωλ~3=λ3ω,λ~4=λ4ω,M2=σB02ρω,1K=νϕkωEq.(24) takes the following form
(28)u~sy,t=exp−A~y~cost~−B~y~.where
(29)A~=E~2+1/KE~1+M2E~5E~12+E~5+M2E~2E~12+E~2+1/KE~1+M2E~5E~11/2,B=E~2+1/KE~1+M2E~5E~12+E~5+M2E~2E~12−E~2+1/KE~1+M2E~5E~11/2.E~1=1−λ~42+λ~32,E2=λ~3−λ~1+λ~1λ~4−λ~2λ~3,E~5=1−λ~2+λ~4+λ~1λ~3+λ~2λ~4


### 3.2. Foru0,t=U0cosωt and λ2=0

For this case we have the following expression(30)uty,t=2πU0ν∫0∞ξr2F0−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er1tsinξydξr1−r2F8−ω2F92+ω2F102+2πU0ν∫0∞ξr1F0−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er2tsinξydξr2−r1F8−ω2F92+ω2F102,(31)usy,t=2πU0νωsinωt∫0∞ξF7−σB02/ρF4sinξydξF8−ω2F92+ω2F102+2πU0νcosωt∫0∞ξF0−ω2−F4+σB02/ρF5+νϕ/kF6F8−ω2F92+ω2F102sinξydξ,where
(32)r1=−F10+F102−4F8F92F9,(33)r2=−F10−F102−4F8F92F9,(34)F4=λ3−λ1+λ1λ4ω2,F5=λ4−λ1λ3,F6=2λ4−λ42ω2−λ32,(35)F7=1−ω2λ4+ω2λ1λ3,F8=νξ2+σB02ρ+νϕk,F9=λ1+βξ2+ϕkβ,(36)F10=1+αξ2+σB02ρλ1+ϕkα.Adopting the same methodology of solution as for Eqs. (23) and (31), gives
(37)u~s1y,t=exp−A~1y~cost~−B~1y~,where
(38)A~1=F~4+1/KE~1+M2F~7E~12+F~7−M2F~4E~12+F~4+1/KE~1+M2F~7E~11/2,B~1=F~4+1/KE~1+M2F~7E~12+F~7−M2F~4E~12−F~4+1/KE~1+M2F~7E~11/2,F~4=λ~3−λ~1+λ~1λ~4,F~7=1−λ~4+λ~1λ~3.It is worth mentioning to note that forλ4=M=ϕ=0 the Eq. (37) reduces to the solutions for an Oldroyd-B fluid. Moreover, Eq. (37) recovers the results of second grade fluid [8] when λ1=λ4=M=ϕ=0 and νλ3=α1/ρ (α1 is the material parameter of second grade fluid).


### 3.3. Foru0,t=U0sinωt and λ2≠0

Employing the similar procedure as for the case ofU0cosωt, the transient and steady state solutions are
(39)uty,t=−2πU0νω∫0∞ξω2−m2m3F2−m2+m3F0−ω2F1em1tsinξydξm1−m2m1−m3F3+2πU0νω∫0∞ξm1m3−ω2F2+m1+m3F0−ω2F1em2tsinξydξm2−m1m2−m3F3−2πU0νω∫0∞ξω2−m1m2F2−m1+m2F0−ω2F1em3tsinξydξm3−m1m3−m2F3,(40)usy,t=2πU0νωcosωt∫0∞ξF2sinξydξF3+2πU0νsinωt∫0∞ξF0−ω2F1sinξydξF3,The Eq. (40) in dimensionless variables now gives
(41)u~sy,t=exp−A~y~sint~−B~y~.


### 3.4. Foru0,t=U0sinωt and λ2=0

Here we have(42)uty,t=−2πU0νω∫0∞ξr2F7−σB02/ρF4+F0−ω2−F4+σB02/ρF5+νϕ/kF6er1tsinξydξr1−r2F8−ω2F92+ω2F102+2πU0νω∫0∞ξr1F7−σB02/ρF4+F0−ω2−F4+σB02/ρF5+νϕ/kF6er2tsinξydξr1−r2F8−ω2F92+ω2F102,usy,t=−2πU0νωcosωt∫0∞ξF7−σB02/ρF4sinξydξF8−ω2F92+ω2F102+2πU0νsinωt∫0∞ξF0−ω2−F4+σB02/ρF5+νϕ/kF6sinξydξF8−ω2F92+ω2F102,After finding the above integral, the solution in dimensionless variables is obtained as follows(43)u~s1y,t=exp−A~1y~sint~−B~1y~The above equation also reduces to the result of second grade fluid [8] for λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 3.1. Whenu0,t=U0cosωt and λ2≠0

Taking Fourier sine transform of Eqs. (13)–(15) and then solving them in the ξ-plane, we get the expression for the starting solution as follows
(18)u¯ξ,t=u¯tξ,t+u¯sξ,t,where u¯tξ,t and u¯sξ,t indicate the transient and steady state solutions, respectively, and are given by
(19)u¯tξ,t=−2/πU0ξνm2m3−ω2F0−ω2F1−ω2m2+m3F2em1tm1−m2m1−m3F3+2/πU0ξνω2−m1m3F0−ω2F1+ω2m1+m3F2em2tm2−m1m2−m3F3−2/πU0ξνm1m2−ω2F0−ω2F1−ω2m1+m2F2em3tm3−m1m3−m2F3,(20)u¯sξ,t=2πU0νωsinωtξF2F3+2πU0νcosωtξF0−ω2F1F3,where
(21)F0=νξ2E1+σB02ρ+νϕk,F1=−E2+σB02ρE3+νϕkE4,F2=E5−σB02ρE2,F3=νξ2+σB02ρ+νϕk−ω2E62+ω2E72,E1=1−λ4ω22+λ3ω2,E2=λ3−λ1−λ2λ3ω2+λ1λ4ω2,E3=λ2+λ4−λ1λ3ω2−λ2λ4ω2,E4=2λ4−λ42ω2−λ32,E5=1−ω2λ2+λ4+ω2λ1λ3+λ2λ4ω2,E6=λ1+βξ2+σB02ρλ2+ϕkβ,E7=1+αξ2+σB02ρλ1+ϕkα−λ2ω2,m1=−d3λ2−21/3−d2+3eλ23λ2−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3+−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3321/3λ2,m2=−d3λ2−1+i3−d2+3eλ2322/3λ2−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3−1−i3−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3621/3λ2,m3=−d3λ2+1−i3−d2+3eλ2322/3λ2−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3−1+i3−2d3+9deλ2−27fλ22+4−d2+3eλ23+−2d3+9deλ2−27fλ221/3621/3λ2,d=λ1+βξ2+σB02ρλ2+ϕkβ,e=1+αξ2+σB02ρλ1+ϕkα,f=νξ2+σB02ρ+ϕkβ,α=νλ3,β=νλ4.Fourier inversion of Eqs. (19) and (20) yields
(22)uty,t=−2πU0ν∫0∞ξm2m3−ω2F0−ω2F1−ω2m2+m3F2em1tsinξydξm1−m2m1−m3F3+2πU0ν∫0∞ξω2−m1m3F0−ω2F1+ω2m1+m3F2em2tsinξydξm2−m1m2−m3F3−2πU0ν∫0∞ξm1m2−ω2F0−ω2F1−ω2m1+m2F2em3tsinξydξm3−m1m3−m2F3,(23)usy,t=2πU0νωsinωt∫0∞ξF2sinξydξF3+2πU0νcosωt∫0∞ξF0−ω2F1sinξydξF3.Note that for large timesuty,t⟶0 and usy,t becomes [see appendix]
(24)usy,t=U0exp−Aycosωt−By,(25)2A2=ω2E2+ϕ/kνE1+σB02/ρE5νE12+ν2ω2E5−σB02/ρE2ν2E12+ω2E2+ϕ/kνE1+σB02/ρE5νE1,(26)2B2=ω2E2+ϕ/kνE1+σB02/ρE5νE12+ν2ω2E5−σB02/ρE2ν2E12−ω2E2+ϕ/kνE1+σB02/ρE5νE1.Introducing the following dimensionless quantities(27)y~=ω2νy,u~=uU0,t~=ωt,λ~1=λ1ω,λ~2=λ2ωλ~3=λ3ω,λ~4=λ4ω,M2=σB02ρω,1K=νϕkωEq.(24) takes the following form
(28)u~sy,t=exp−A~y~cost~−B~y~.where
(29)A~=E~2+1/KE~1+M2E~5E~12+E~5+M2E~2E~12+E~2+1/KE~1+M2E~5E~11/2,B=E~2+1/KE~1+M2E~5E~12+E~5+M2E~2E~12−E~2+1/KE~1+M2E~5E~11/2.E~1=1−λ~42+λ~32,E2=λ~3−λ~1+λ~1λ~4−λ~2λ~3,E~5=1−λ~2+λ~4+λ~1λ~3+λ~2λ~4


## 3.2. Foru0,t=U0cosωt and λ2=0

For this case we have the following expression(30)uty,t=2πU0ν∫0∞ξr2F0−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er1tsinξydξr1−r2F8−ω2F92+ω2F102+2πU0ν∫0∞ξr1F0−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er2tsinξydξr2−r1F8−ω2F92+ω2F102,(31)usy,t=2πU0νωsinωt∫0∞ξF7−σB02/ρF4sinξydξF8−ω2F92+ω2F102+2πU0νcosωt∫0∞ξF0−ω2−F4+σB02/ρF5+νϕ/kF6F8−ω2F92+ω2F102sinξydξ,where
(32)r1=−F10+F102−4F8F92F9,(33)r2=−F10−F102−4F8F92F9,(34)F4=λ3−λ1+λ1λ4ω2,F5=λ4−λ1λ3,F6=2λ4−λ42ω2−λ32,(35)F7=1−ω2λ4+ω2λ1λ3,F8=νξ2+σB02ρ+νϕk,F9=λ1+βξ2+ϕkβ,(36)F10=1+αξ2+σB02ρλ1+ϕkα.Adopting the same methodology of solution as for Eqs. (23) and (31), gives
(37)u~s1y,t=exp−A~1y~cost~−B~1y~,where
(38)A~1=F~4+1/KE~1+M2F~7E~12+F~7−M2F~4E~12+F~4+1/KE~1+M2F~7E~11/2,B~1=F~4+1/KE~1+M2F~7E~12+F~7−M2F~4E~12−F~4+1/KE~1+M2F~7E~11/2,F~4=λ~3−λ~1+λ~1λ~4,F~7=1−λ~4+λ~1λ~3.It is worth mentioning to note that forλ4=M=ϕ=0 the Eq. (37) reduces to the solutions for an Oldroyd-B fluid. Moreover, Eq. (37) recovers the results of second grade fluid [8] when λ1=λ4=M=ϕ=0 and νλ3=α1/ρ (α1 is the material parameter of second grade fluid).


## 3.3. Foru0,t=U0sinωt and λ2≠0

Employing the similar procedure as for the case ofU0cosωt, the transient and steady state solutions are
(39)uty,t=−2πU0νω∫0∞ξω2−m2m3F2−m2+m3F0−ω2F1em1tsinξydξm1−m2m1−m3F3+2πU0νω∫0∞ξm1m3−ω2F2+m1+m3F0−ω2F1em2tsinξydξm2−m1m2−m3F3−2πU0νω∫0∞ξω2−m1m2F2−m1+m2F0−ω2F1em3tsinξydξm3−m1m3−m2F3,(40)usy,t=2πU0νωcosωt∫0∞ξF2sinξydξF3+2πU0νsinωt∫0∞ξF0−ω2F1sinξydξF3,The Eq. (40) in dimensionless variables now gives
(41)u~sy,t=exp−A~y~sint~−B~y~.


## 3.4. Foru0,t=U0sinωt and λ2=0

Here we have(42)uty,t=−2πU0νω∫0∞ξr2F7−σB02/ρF4+F0−ω2−F4+σB02/ρF5+νϕ/kF6er1tsinξydξr1−r2F8−ω2F92+ω2F102+2πU0νω∫0∞ξr1F7−σB02/ρF4+F0−ω2−F4+σB02/ρF5+νϕ/kF6er2tsinξydξr1−r2F8−ω2F92+ω2F102,usy,t=−2πU0νωcosωt∫0∞ξF7−σB02/ρF4sinξydξF8−ω2F92+ω2F102+2πU0νsinωt∫0∞ξF0−ω2−F4+σB02/ρF5+νϕ/kF6sinξydξF8−ω2F92+ω2F102,After finding the above integral, the solution in dimensionless variables is obtained as follows(43)u~s1y,t=exp−A~1y~sint~−B~1y~The above equation also reduces to the result of second grade fluid [8] for λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 4. Modified Stokes’ Second Problem

Here, we consider the MHD flow between two infinite plates distanced apart. The lower plate at y=0 oscillates in its own plane for t>0 while the upper plate at y=d is stationary. The problem which governs the flow consists of Eqs. (13) and (14)
(44)ud,t=0;t∈R,(45)uy,0=∂uy,0∂t=∂2uy,0∂t2=0,0<y<dFollowing the same method of solution as in the previous section we have.


### 4.1. Foru0,t=U0cosωt and λ2≠0

(46)uty,t=−2dU0ν∑n=1∞λnn2n3−ω2F11−ω2F1−ω2n2+n3F2en1tsinλnyn1−n2n1−n3F12+2dU0ν∑n=1∞λnω2−n1n3F11−ω2F1+ω2n1+n3F2en2tsinλnyn2−n1n2−n3F12−2πU0ν∑n=1∞λnn1n2−ω2F11−ω2F1−ω2n1+n2F2en3tsinλnyn3−n1n3−n2F12,usy,t=2dU0νωsinωt∑n=1∞λnF2sinλnyF12+2dU0νcosωt∑n=1∞λnF11−ω2F1sinλnyF12where
(47)F11=νλn2E1+σB02ρ+νϕk,F12=νλn2+σB02ρ+νϕk−ω2E92+ω2E102,E9=λ1+βλn2+σB02ρλ2+ϕkβ,E10=1+αλn2+σB02ρλ1+ϕkα−λ2ω2n1=−d3λ2−21/3−d12+3e1λ23λ2−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3+−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3321/3λ2,n2=−d13λ2+1+i3−d12+3e1λ2322/3λ2−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3−1−i3−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3621/3λ2,n3=−d13λ2+1−i3−d12+3e1λ2322/3λ2−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3−1+i3−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1eλ2−27f1λ221/3621/3λ2,d1=λ1+βλn2+σB02ρλ2+ϕkβ,e1=1+αλn2+σB02ρλ1+ϕkα,f1=νλn2+σB02ρ+ϕkν.E9=λ1+βλn2+σB02ρλ2+ϕkβ,E10=1+αλn2+σB02ρλ1+ϕkα−λ2ω2.


### 4.2. Foru0,t=U0cosωt and λ2=0

we have(48)uty,t=2dU0ν∑n=1∞λnr4F11−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er3tsinλnyr3−r4F13−ω2F142+ω2F152+2dU0ν∑n=1∞λnr3F11−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er4tsinλnyr4−r3F13−ω2F142+ω2F152,(49)usy,t=2dU0νωsinωt∑n=1∞λnF7−σB02/ρF4sinλnyF13−ω2F142+ω2F152+2dU0νcosωt∑n=1∞λnF11−ω2−F4+σB02/ρF5+νϕ/kF6sinλnyF13−ω2F142+ω2F152,where
(50)r3=−F15+F152−4F13F142F14,r4=−F15−F152−4F13F142F14,F13=νλn2+σB02ρ+νϕk,F14=λ1+βλn2+ϕkβ,F15=1+αλn2+σB02ρλ1+ϕkα.Note that the results of second grade fluid can be obtained by choosingλ1=λ4=M=ϕ=0 and νλ3=α1/ρ in Eq. (49).


### 4.3. Foru0,t=U0sinωt and λ2≠0

we obtain(51)uty,t=−2dU0νω∑n=1∞λnω2−n2n3F2−n2+n3F11−ω2F1en1tsinλnyn1−n2n1−n3F12+2dU0νω∑n=1∞λnn1n3−ω2F2+n1+n3F11−ω2F1en2tsinλnyn2−n1n2−n3F12−2dU0νω∑n=1∞λnω2−n1n2F2−n1+n2F11−ω2F1en3tsinλnyn3−n1n3−n2F12,usy,t=−2dU0νωcosωt∑n=1∞λnF2sinλnyF12+2dU0νsinωt∑n=1∞λnF11−ω2F1sinλnyF12.


### 4.4. Foru0,t=U0sinωt and λ2=0

we get(52)uty,t=−2dU0νω∑n=1∞λnr4F7−σB02/ρF4+F11−ω2−F4+σB02/ρF5+νϕ/kF6×er3tsinλnyr3−r4F13−ω2F142+ω2F152+2dU0νω∑n=1∞λnr3F7−σB02/ρF4+F11−ω2−F4+σB02/ρF5+νϕ/kF6×er4tsinξyr3−r4F13−ω2F142+ω2F152,usy,t=−2dU0νωcosωt∑n=1∞λnF7−σB02/ρF4sinλnyF13−ω2F142+ω2F152+2dU0νsinωt∑n=1∞λnF11−ω2−F4+σB02/ρF5+νϕ/kF6F13−ω2F142+ω2F152sinλny.The above equation gives the solution of second grade fluid [8] for λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 4.1. Foru0,t=U0cosωt and λ2≠0

(46)uty,t=−2dU0ν∑n=1∞λnn2n3−ω2F11−ω2F1−ω2n2+n3F2en1tsinλnyn1−n2n1−n3F12+2dU0ν∑n=1∞λnω2−n1n3F11−ω2F1+ω2n1+n3F2en2tsinλnyn2−n1n2−n3F12−2πU0ν∑n=1∞λnn1n2−ω2F11−ω2F1−ω2n1+n2F2en3tsinλnyn3−n1n3−n2F12,usy,t=2dU0νωsinωt∑n=1∞λnF2sinλnyF12+2dU0νcosωt∑n=1∞λnF11−ω2F1sinλnyF12where
(47)F11=νλn2E1+σB02ρ+νϕk,F12=νλn2+σB02ρ+νϕk−ω2E92+ω2E102,E9=λ1+βλn2+σB02ρλ2+ϕkβ,E10=1+αλn2+σB02ρλ1+ϕkα−λ2ω2n1=−d3λ2−21/3−d12+3e1λ23λ2−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3+−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3321/3λ2,n2=−d13λ2+1+i3−d12+3e1λ2322/3λ2−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3−1−i3−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3621/3λ2,n3=−d13λ2+1−i3−d12+3e1λ2322/3λ2−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1e1λ2−27f1λ221/3−1+i3−2d13+9d1e1λ2−27f1λ22+4−d12+3e1λ23+−2d13+9d1eλ2−27f1λ221/3621/3λ2,d1=λ1+βλn2+σB02ρλ2+ϕkβ,e1=1+αλn2+σB02ρλ1+ϕkα,f1=νλn2+σB02ρ+ϕkν.E9=λ1+βλn2+σB02ρλ2+ϕkβ,E10=1+αλn2+σB02ρλ1+ϕkα−λ2ω2.


## 4.2. Foru0,t=U0cosωt and λ2=0

we have(48)uty,t=2dU0ν∑n=1∞λnr4F11−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er3tsinλnyr3−r4F13−ω2F142+ω2F152+2dU0ν∑n=1∞λnr3F11−ω2−F4+σB02/ρF5+νϕ/kF6−ω2F7−σB02/ρF4er4tsinλnyr4−r3F13−ω2F142+ω2F152,(49)usy,t=2dU0νωsinωt∑n=1∞λnF7−σB02/ρF4sinλnyF13−ω2F142+ω2F152+2dU0νcosωt∑n=1∞λnF11−ω2−F4+σB02/ρF5+νϕ/kF6sinλnyF13−ω2F142+ω2F152,where
(50)r3=−F15+F152−4F13F142F14,r4=−F15−F152−4F13F142F14,F13=νλn2+σB02ρ+νϕk,F14=λ1+βλn2+ϕkβ,F15=1+αλn2+σB02ρλ1+ϕkα.Note that the results of second grade fluid can be obtained by choosingλ1=λ4=M=ϕ=0 and νλ3=α1/ρ in Eq. (49).


## 4.3. Foru0,t=U0sinωt and λ2≠0

we obtain(51)uty,t=−2dU0νω∑n=1∞λnω2−n2n3F2−n2+n3F11−ω2F1en1tsinλnyn1−n2n1−n3F12+2dU0νω∑n=1∞λnn1n3−ω2F2+n1+n3F11−ω2F1en2tsinλnyn2−n1n2−n3F12−2dU0νω∑n=1∞λnω2−n1n2F2−n1+n2F11−ω2F1en3tsinλnyn3−n1n3−n2F12,usy,t=−2dU0νωcosωt∑n=1∞λnF2sinλnyF12+2dU0νsinωt∑n=1∞λnF11−ω2F1sinλnyF12.


## 4.4. Foru0,t=U0sinωt and λ2=0

we get(52)uty,t=−2dU0νω∑n=1∞λnr4F7−σB02/ρF4+F11−ω2−F4+σB02/ρF5+νϕ/kF6×er3tsinλnyr3−r4F13−ω2F142+ω2F152+2dU0νω∑n=1∞λnr3F7−σB02/ρF4+F11−ω2−F4+σB02/ρF5+νϕ/kF6×er4tsinξyr3−r4F13−ω2F142+ω2F152,usy,t=−2dU0νωcosωt∑n=1∞λnF7−σB02/ρF4sinλnyF13−ω2F142+ω2F152+2dU0νsinωt∑n=1∞λnF11−ω2−F4+σB02/ρF5+νϕ/kF6F13−ω2F142+ω2F152sinλny.The above equation gives the solution of second grade fluid [8] for λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 5. Time-Periodic Plane Poiseuille Flow

In this section the flow between the two stationary plates is induced by an oscillating pressure gradient in thex-direction. Initially the fluid and plates are at rest. The pressure gradient is of the following form
(53)∂p∂x=−ρQcosωtor∂p∂x=−ρQsinωt.The flow is governed by Eq. (45) and
(54)1+λ1∂∂t+λ2∂2∂t2∂u∂t+1+λ1∂∂t+λ2∂2∂t2∂p∂x=ν1+λ3∂∂t+λ3∂2∂t2∂2u∂y2−σB02ρ1+λ1∂∂t+λ2∂2∂t2u−νϕk1+λ3∂∂t+λ3∂2∂t2u,u0,t=ud,t=0;t∈R.The solutions here are given by.


### 5.1. When∂p/∂x=−ρQcosωt and λ2≠0

then(55)uty,t=−4dQ∑n=1∞q2q3−ω2F16−ω2q2+q3F17eq1tsinλ2n−1yq1−q2q1−q3F18λ2n−1+4dQ∑n=1∞ω2−q1q3F16+ω2q1+q3F17eq2tsinλ2n−1yq2−q1q2−q3F18λ2n−1−4dQ∑n=1∞q1q2−ω2F16−ω2q1+q2F17eq3tsinλ2n−1yq3−q1q3−q2F18λ2n−1,usy,t=4Qdcosωt∑n=1∞F16sinλ2n−1yλ2n−1F18+4ωQdsinωt∑n=1∞F17sinλ2n−1yλ2n−1F18where
(56)q1=−d23λ2−21/3−d22+3e2λ23λ2−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3+−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3321/3λ2,q2=−d23λ2−1+i3−d22+3e2λ2322/3λ2−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3−1−i3−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3621/3λ2,q3=−d23λ2+1−i3−d22+3e2λ2322/3λ2−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e1λ2−27f2λ221/3−1+i3−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3621/3λ2,F16=νλ2n−12E5+σB02ρE11+νϕE5k,F17=νλ2n−12E2+E11+νϕkE2,F18=νλ2n−12+σB02ρ+νϕk−ω2E122+ω2E132E13=1−ω2λ22−λ12ω2,E12=λ1+βλ2n−12+σB02ρλ2+ϕkβ,E13=1+αλ2n−12+σB02ρλ1+ϕkα−λ2ω2d2=λ1+βλ2n−12+σB02ρλ2+ϕkβ,e2=1+αλ2n−12+σB02ρλ1+ϕkα,f2=νλ2n−12+σB02ρ+ϕkν.


### 5.2. When∂p/∂x=−ρQcosωt and λ2=0

then(57)uty,t=4dQ∑n=1∞r6νλ2n−12F7+σB02/ρF19+νϕ/kF7−ω2νλ2n−12F4+F19+νϕ/kF4er5tsinλ2n−1yr5−r6F20−ω2F212+ω2F222λ2n−1+4dQ∑n=1∞r5νλ2n−12F7+σB02/ρF19+νϕ/kF7−ω2νλ2n−12F4+F19+νϕ/kF4er6tsinλ2n−1yr6−r5F20−ω2F212+ω2F222λ2n−1,(58)usy,t=4Qdcosωt∑n=1∞νλ2n−12F7+σB02/ρF19+νϕ/kF7λ2n−1F20−ω2F212+ω2F222sinλ2n−1y+4wQdsinωt∑n=1∞νλ2n−12F4+F19+νϕ/kF4sinλ2n−1yλ2n−1F20−ω2F212+ω2F222,where
(59)r5=−F22+F222−4F21F202F21,r4=−F22−F222−4F21F202F21F19=1+λ12ω2,F20=νλ2n−12+σB02ρ+νϕk,F21=λ1+βλ2n−12+ϕkβ,F22=1+αλ2n−12+σB02ρλ1+ϕkα.The solution of second grade fluid [8] can be deduced from Eq. (58) by taking λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


### 5.3. For∂p/∂x=−ρQsinωt and λ2≠0

we have(60)uty,t=−4dωQ∑n=1∞ω2−q2q3F17−q2+q3F16eq1tsinλ2n−1yq1−q2q1−q3F18λ2n−1+4dωQ∑n=1∞q2q3−ω2F17−q2+q3F16eq2tsinλ2n−1yq2−q1q2−q3F18λ2n−1−4dωQ∑n=1∞ω2−q1q2F17−q1+q2F16eq1tsinλ2n−1yq3−q1q3−q1F18λ2n−1,usy,t=4Qdsinωt∑n=1∞F16sinλ2n−1yλ2n−1F18+4ωQdcosωt∑n=1∞F17sinλ2n−1yλ2n−1F18.


### 5.4. For∂p/∂x=−ρQsinωt and λ2=0

we get(61)uty,t=4dQ∑n=1∞r6νλ2n−12F4+σB02/ρF19+νϕ/kF4+νλ2n−12F7+F19+νϕ/kF7er5tsinλ2n−1yr5−r6F20−ω2F212+ω2F222λ2n−1+4dQ∑n=1∞r5νλ2n−12F4+σB02/ρF19+νϕ/kF4+νλ2n−12F7+F19+νϕ/kF7er6tsinλ2n−1yr5−r6F20−ω2F212+ω2F222λ2n−1,usy,t=4Qdsinωt∑n=1∞νλ2n−12F7+σB02/ρF19+νϕ/kF7λ2n−1F20−ω2F212+ω2F222sinλ2n−1y−4wQdcosωt∑n=1∞νλ2n−12F4+F19+νϕ/kF4sinλ2n−1yλ2n−1F20−ω2F212+ω2F222.The above solution yields the result of second grade fluid [8] when λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 5.1. When∂p/∂x=−ρQcosωt and λ2≠0

then(55)uty,t=−4dQ∑n=1∞q2q3−ω2F16−ω2q2+q3F17eq1tsinλ2n−1yq1−q2q1−q3F18λ2n−1+4dQ∑n=1∞ω2−q1q3F16+ω2q1+q3F17eq2tsinλ2n−1yq2−q1q2−q3F18λ2n−1−4dQ∑n=1∞q1q2−ω2F16−ω2q1+q2F17eq3tsinλ2n−1yq3−q1q3−q2F18λ2n−1,usy,t=4Qdcosωt∑n=1∞F16sinλ2n−1yλ2n−1F18+4ωQdsinωt∑n=1∞F17sinλ2n−1yλ2n−1F18where
(56)q1=−d23λ2−21/3−d22+3e2λ23λ2−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3+−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3321/3λ2,q2=−d23λ2−1+i3−d22+3e2λ2322/3λ2−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3−1−i3−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3621/3λ2,q3=−d23λ2+1−i3−d22+3e2λ2322/3λ2−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e1λ2−27f2λ221/3−1+i3−2d23+9d2e2λ2−27f2λ22+4−d22+3e2λ23+−2d23+9d2e2λ2−27f2λ221/3621/3λ2,F16=νλ2n−12E5+σB02ρE11+νϕE5k,F17=νλ2n−12E2+E11+νϕkE2,F18=νλ2n−12+σB02ρ+νϕk−ω2E122+ω2E132E13=1−ω2λ22−λ12ω2,E12=λ1+βλ2n−12+σB02ρλ2+ϕkβ,E13=1+αλ2n−12+σB02ρλ1+ϕkα−λ2ω2d2=λ1+βλ2n−12+σB02ρλ2+ϕkβ,e2=1+αλ2n−12+σB02ρλ1+ϕkα,f2=νλ2n−12+σB02ρ+ϕkν.


## 5.2. When∂p/∂x=−ρQcosωt and λ2=0

then(57)uty,t=4dQ∑n=1∞r6νλ2n−12F7+σB02/ρF19+νϕ/kF7−ω2νλ2n−12F4+F19+νϕ/kF4er5tsinλ2n−1yr5−r6F20−ω2F212+ω2F222λ2n−1+4dQ∑n=1∞r5νλ2n−12F7+σB02/ρF19+νϕ/kF7−ω2νλ2n−12F4+F19+νϕ/kF4er6tsinλ2n−1yr6−r5F20−ω2F212+ω2F222λ2n−1,(58)usy,t=4Qdcosωt∑n=1∞νλ2n−12F7+σB02/ρF19+νϕ/kF7λ2n−1F20−ω2F212+ω2F222sinλ2n−1y+4wQdsinωt∑n=1∞νλ2n−12F4+F19+νϕ/kF4sinλ2n−1yλ2n−1F20−ω2F212+ω2F222,where
(59)r5=−F22+F222−4F21F202F21,r4=−F22−F222−4F21F202F21F19=1+λ12ω2,F20=νλ2n−12+σB02ρ+νϕk,F21=λ1+βλ2n−12+ϕkβ,F22=1+αλ2n−12+σB02ρλ1+ϕkα.The solution of second grade fluid [8] can be deduced from Eq. (58) by taking λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 5.3. For∂p/∂x=−ρQsinωt and λ2≠0

we have(60)uty,t=−4dωQ∑n=1∞ω2−q2q3F17−q2+q3F16eq1tsinλ2n−1yq1−q2q1−q3F18λ2n−1+4dωQ∑n=1∞q2q3−ω2F17−q2+q3F16eq2tsinλ2n−1yq2−q1q2−q3F18λ2n−1−4dωQ∑n=1∞ω2−q1q2F17−q1+q2F16eq1tsinλ2n−1yq3−q1q3−q1F18λ2n−1,usy,t=4Qdsinωt∑n=1∞F16sinλ2n−1yλ2n−1F18+4ωQdcosωt∑n=1∞F17sinλ2n−1yλ2n−1F18.


## 5.4. For∂p/∂x=−ρQsinωt and λ2=0

we get(61)uty,t=4dQ∑n=1∞r6νλ2n−12F4+σB02/ρF19+νϕ/kF4+νλ2n−12F7+F19+νϕ/kF7er5tsinλ2n−1yr5−r6F20−ω2F212+ω2F222λ2n−1+4dQ∑n=1∞r5νλ2n−12F4+σB02/ρF19+νϕ/kF4+νλ2n−12F7+F19+νϕ/kF7er6tsinλ2n−1yr5−r6F20−ω2F212+ω2F222λ2n−1,usy,t=4Qdsinωt∑n=1∞νλ2n−12F7+σB02/ρF19+νϕ/kF7λ2n−1F20−ω2F212+ω2F222sinλ2n−1y−4wQdcosωt∑n=1∞νλ2n−12F4+F19+νϕ/kF4sinλ2n−1yλ2n−1F20−ω2F212+ω2F222.The above solution yields the result of second grade fluid [8] when λ1=λ4=M=ϕ=0 and νλ3=α1/ρ.


## 6. Results and Discussion

In this section we discuss the graphical results of velocity profiles due to the oscillations of the plate att>0.


### 6.1. Analysis of Results

The difference between velocity profiles of Oldroyd-B fluid and generalized Burgers’ fluid is shown for different values ofM and K. Figure 1 is constructed to describe the effects of M on the velocity profiles in two fluid cases. It is evident from this figure that the fluid velocity decreases in Oldroyd-B fluid and generalized Burgers’ fluids by increasing M. Figure 2 shows the influence of permeability of porous medium K on the velocity profile in the presence of magnetic field parameter M. By increasing K the velocity profiles for both fluid increase. Figure 3 is displayed for the variation of K in the absence of M. It can be seen from this figure that the velocity also increases for both cases of Oldroyd-B and generalized Burgers’ fluid. The comparison of steady state solution us in various fluid models is shown in Table 2.Figure 1
Profile of normalized steady state velocityuy for various values of M.Figure 2
Profile of normalized steady state velocityuy for various values of K.Figure 3
Profile of normalized steady state velocityuy for various values of K at M=0.Table 2
Comparison of velocity in different fluids whent=1.5 and y=0.5.
Types of fluidMaterial constantsusforM=1,K=0.4usforM=0,K⟶∞usforM=0,K=0.4usforM=1,K⟶∞Newtonianλi=0,i=1,2,3,40.1313570.327710.1587580.227262Second gradeλ1=λ2=λ4=0,λ3=1.50.1561160.5423390.1844180.726136Maxwellλ2=λ3=λ4=0,λ1=10.1958150.4886690.1987660.322845Oldroyd-Bλ2=λ4=0,λ1=1,λ3=1.50.3332880.3381830.2096390.444708G. Burgers’λ1=1,λ2=2,λ3=1.5,λ4=30.1365380.3013060.1868460.2033335


### 6.2. Discussion of Results

It is observed from Figure1 that influence of magnetic field is more prominent on the velocity profiles in Oldroyd -B fluid when compared with a generalized Burgers’ fluid. The effects of K on the velocity in presence of magnetic field M, in Figure 2 are more significant in Oldroyd-B fluid case as compared to generalized Burgers’ fluid. Similarly, when K is varied in absence of magnetic field M in Figure 3, Oldroyd-B fluid shows more notable behavior in comparison with generalized Burgers’ fluid. Hence it is concluded that Oldroyd-B fluid shows significant increase in steady state velocity when porosity is increased in presence of magnetic field and similar behavior is observed when magnetic field is neglected.In Table2 it is clearly seen that the value of steady state solution us in Oldroyd-B fluid is greatest and smallest in viscous fluid when M≠0 and K≠0. Moreover, the value of us in Maxwell fluid is greater than that of second grade and generalized Burgers’ fluids. However, us is maximum for hydrodynamic second grade fluid and minimum in hydrodynamic generalized Burgers’ fluid when permeability of the porous medium is much larger. In this case, us for Maxwell fluid is large when compared with Newtonian and Oldroyd-B fluid. For M=0 and K≠0 the behavior of us is similar to the case of M≠0 and K≠0. However, it is found that us for M=0, K≠0 is large for all fluids except Oldroyd-B fluid when compared with M≠0, K≠0. For M=1 and K⟶∞ the behavior of us is similar to that of M=1 and K⟶∞. As a result, it can be concluded that steady state velocity gives highest value for Oldroyd B fluid in all cases when compared with Newtonian or Maxwell fluid, whereas generalized Burgers’ fluid shows minimum value of us.


## 6.1. Analysis of Results

The difference between velocity profiles of Oldroyd-B fluid and generalized Burgers’ fluid is shown for different values ofM and K. Figure 1 is constructed to describe the effects of M on the velocity profiles in two fluid cases. It is evident from this figure that the fluid velocity decreases in Oldroyd-B fluid and generalized Burgers’ fluids by increasing M. Figure 2 shows the influence of permeability of porous medium K on the velocity profile in the presence of magnetic field parameter M. By increasing K the velocity profiles for both fluid increase. Figure 3 is displayed for the variation of K in the absence of M. It can be seen from this figure that the velocity also increases for both cases of Oldroyd-B and generalized Burgers’ fluid. The comparison of steady state solution us in various fluid models is shown in Table 2.Figure 1
Profile of normalized steady state velocityuy for various values of M.Figure 2
Profile of normalized steady state velocityuy for various values of K.Figure 3
Profile of normalized steady state velocityuy for various values of K at M=0.Table 2
Comparison of velocity in different fluids whent=1.5 and y=0.5.
Types of fluidMaterial constantsusforM=1,K=0.4usforM=0,K⟶∞usforM=0,K=0.4usforM=1,K⟶∞Newtonianλi=0,i=1,2,3,40.1313570.327710.1587580.227262Second gradeλ1=λ2=λ4=0,λ3=1.50.1561160.5423390.1844180.726136Maxwellλ2=λ3=λ4=0,λ1=10.1958150.4886690.1987660.322845Oldroyd-Bλ2=λ4=0,λ1=1,λ3=1.50.3332880.3381830.2096390.444708G. Burgers’λ1=1,λ2=2,λ3=1.5,λ4=30.1365380.3013060.1868460.2033335


## 6.2. Discussion of Results

It is observed from Figure1 that influence of magnetic field is more prominent on the velocity profiles in Oldroyd -B fluid when compared with a generalized Burgers’ fluid. The effects of K on the velocity in presence of magnetic field M, in Figure 2 are more significant in Oldroyd-B fluid case as compared to generalized Burgers’ fluid. Similarly, when K is varied in absence of magnetic field M in Figure 3, Oldroyd-B fluid shows more notable behavior in comparison with generalized Burgers’ fluid. Hence it is concluded that Oldroyd-B fluid shows significant increase in steady state velocity when porosity is increased in presence of magnetic field and similar behavior is observed when magnetic field is neglected.In Table2 it is clearly seen that the value of steady state solution us in Oldroyd-B fluid is greatest and smallest in viscous fluid when M≠0 and K≠0. Moreover, the value of us in Maxwell fluid is greater than that of second grade and generalized Burgers’ fluids. However, us is maximum for hydrodynamic second grade fluid and minimum in hydrodynamic generalized Burgers’ fluid when permeability of the porous medium is much larger. In this case, us for Maxwell fluid is large when compared with Newtonian and Oldroyd-B fluid. For M=0 and K≠0 the behavior of us is similar to the case of M≠0 and K≠0. However, it is found that us for M=0, K≠0 is large for all fluids except Oldroyd-B fluid when compared with M≠0, K≠0. For M=1 and K⟶∞ the behavior of us is similar to that of M=1 and K⟶∞. As a result, it can be concluded that steady state velocity gives highest value for Oldroyd B fluid in all cases when compared with Newtonian or Maxwell fluid, whereas generalized Burgers’ fluid shows minimum value of us.


## 7. Concluding Remarks

In current study MHD flow of generalized Burgers’ fluid is mathematically modeled in a porous medium. The exact solutions for three flow problems are developed. The results for various fluids in a porous space can be obtained as a special case of the present analysis by choosing appropriate values of the involved parameters. The comparison of the steady state velocity has been shown for five different fluids. Key conclusions of this study are:(i)
In case of Oldroyd-B fluid decrease inus is more prominent when magnetic field is increased(ii)
Steady state velocity increases more prominently with increasingK incase of Oldroyd-B fluid in presence of magnetic field when compared with generalized Burgers’ fluidOldroyd-B fluid shows higher steady state velocity as compared to Newtonian or Maxwell fluid for all values ofM and K.Generalized Burgers’ fluid shows lower steady state velocity as compared to Newtonian or Maxwell fluid for all values ofM and K.The existing results of second grade fluid [8] can be deduced by selecting λ1=λ2=λ4=M=ϕ=0 and νλ3=α1/ρ. Moreover, this work can be extended in future by considering different flow geometries and boundary conditions for various non-Newtonian fluids.



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