# MHD Flow of Micropolar Fluid over an Oscillating Vertical Plate Embedded in Porous Media with Constant Temperature and Concentration

Authors: Nadeem Ahmad Sheikh; Farhad Ali; Ilyas Khan; Muhammad Saqib; Arshad Khan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The present analysis represents the MHD flow of micropolar fluid past an oscillating infinite vertical plate embedded in porous media. At the plate, free convections are caused due to the differences in temperature and concentration. Therefore, the combined effect of radiative heat and mass transfer is taken into account. Partial differential equations are used in the mathematical formulation of a micropolar fluid. The system of dimensional governing equations is reduced to dimensionless form by means of dimensional analysis. The Laplace transform technique is applied to obtain the exact solutions for velocity, temperature, and concentration. In order to highlight the flow behavior, numerical computation and graphical illustration are carried out. Furthermore, the corresponding skin friction and wall couple stress are calculated.


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## Body

## 1. Introduction

The effect of MHD can be observed in several natural and artificial flows. MHD is the study of the interaction of conducting fluids with electromagnetic phenomena. Electrically conducting fluids in the presence of a magnetic field are of importance in many areas of technology and engineering such as MHD flow meters, MHD power generation, and MHD pumps. It is generally accepted that various astronomical bodies (e.g., Earth, the sun, Jupiter, magnetic stars, and pulsars) acquire fluid interiors and magnetic fields [1]. In the recent past, extensive consideration has been given to applications of MHD and heat transfer such as MHD generators, metallurgical processing, and geothermal energy extraction. The MHD flow with combined effects of heat and mass transfer has wide applications in science and technology, for example, quasi-solid bodies such as earth and in buoyancy induced flows in the atmosphere [2]. Prakash et al. [3] analyzed the effects of thermal diffusion and chemical reaction on MHD boundary layer flow of electrically conducting dusty fluid between two vertical heated plates. Shahid [4] obtained the exact solutions for MHD free convection flow of generalized viscous fluid over an oscillating plate. Raju and Sandeep [5] investigated the MHD flow of non-Newtonian fluid over a rotating cone with cross-diffusion. Therefore, the MHD flow with combined effect of heat and mass transfer has been a subject of concern of several researchers including [6–13].The flow of non-Newtonian fluids through porous medium has been of considerable interest in the last few decades. This interest is because several practical applications can be modeled as transport phenomena in a porous medium. These flows emerge at a large scale in a variety of industrial applications as well as in natural situations, that is, cooling of electronic components, casting and welding of manufacturing processes, food processing, packed-bed reactors, storage of nuclear waste material, oil recovery processes, thermal insulation engineering, pollutant dispersion in aquifers, ground water flows, geothermal extraction, fibrous insulation, industrial and agricultural water distribution, soil pollution, liquid metal flow through generic structures in alloy casting, and the dispersion of chemical contaminants in various processes in the chemical industry [14]. This area of research is of fundamental interest in view of the above applications, thereby generating the need for full understanding of transport processes through a porous medium. Abo-Eldahab and Ghonaim [15] obtained the numerical solutions by applying an efficient numerical technique based on the shooting method for convective flow of micropolar fluid in a porous medium. Kim [16] presented the analysis for the free convection with mass transfer flow of a micropolar fluid through a porous medium bounded by a semi-infinite vertical permeable plate in the presence of a transverse magnetic field. Modather et al. [17] analyzed the convective flow of a micropolar fluid over an infinite moving porous plate in a saturated porous medium in the presence of a transverse magnetic field and obtained the analytical solutions. Many researchers have considered the flow through a porous medium in their studies [18–25].Originally, the theory of a micropolar fluid was presented by Eringen in 1966 [26]. In the recent decades, the study of micropolar fluid is a popular field of research. In micropolar fluid, the randomly oriented particles are suspended in a viscous medium that can experience a rotation that can affect the hydrodynamics of the flow which makes it a distinctly non-Newtonian fluid. An excellent model for analyzing various complex fluids such as blood, polymeric fluids, and colloidal fluids was proposed in Eringen’s theory [26]. The study of free convection MHD flow of a micropolar fluid over an oscillating infinite vertical plate saturated porous medium has important applications in geophysics and engineering and technology. These applications included the boundary layer control in the field of aerodynamics, MHD generators, extraction of geothermal resources, petroleum resource extraction, and nuclear reactors [27]. Nazar et al. [28] used the Keller-Box method to obtain the numerical solutions for a steady flow of micropolar fluid over a stretching sheet. Nadeem et al. [29] considered the unsteady MHD flow of micropolar fluid near a forward stagnation point of two-dimensional plane surface. They solved the governing equations using homotopy analysis method. Many scholars studied micropolar fluids in various geometries applying different mathematical tools [30–36]. Recently, Khalid et al. [37] reported the exact solutions via Laplace transform technique for the analysis of combined effects of heat and mass transfer of micropolar fluid over an oscillating plate. Motivated by the above investigations, the present analysis is focused on the study of MHD flow of micropolar fluid over an oscillating vertical plate embedded in a porous medium in the presence of thermal radiation. Exact solutions are obtained using the Laplace transform technique.


## 2. Mathematical Formulation

The unsteady flow of an incompressible electrically conducting micropolar fluid in a porous medium is considered. Initially both the fluid and infinite vertical plate are in the state of rest with uniform temperatureT∞ and concentration C∞. At t=0+ the plate starts oscillations in its own plane with constant frequency ω and amplitude u0. The temperature and concentration of the plate are raised to Tw and Cw linearly with t. The fluid starts motion in x-direction along the plate and y-axis is taken normal to the plate. Magnetic field of uniform strength B0 is applied transverse to the plate and induced magnetic field is not taken into consideration due to enough small Reynold number. Physical geometry of the problem is presented in Figure 1.Figure 1
Physical sketch.By using the Boussinesq approximation, the governing equations of the problem under consideration are given as follows:(1)ρ∂u∂t=μ+α∂2u∂y2+α∂N∂y-σB02u-ϕ1μku+ρgβTT-T∞+ρgβCC-C∞,ρj∂N∂y=γ∂2N∂y2,ρcp∂T∂t=k1∂2T∂y2-∂qr∂y,∂C∂t=D∂2C∂y2,subject to the following initial and boundary conditions:(2)uy,0=0,Ny,0=0,Ty,0=T∞,Cy,0=C∞,u0,t=u0Htcos⁡ωt,N0,t=-n∂u0,t∂y,T0,t=Tw,C0,t=Cw,u∞,t=0,N∞,t=0,T∞,t=T∞,C∞,t=C∞.Here, (3)qr=-4σ13k2∂T4∂y,where σ1 is the Stefan-Boltzmann constant and k2 is the mean absorption coefficient. Considering small temperature difference between fluid temperature T and free stream temperature T∞, T4 is expanded in Taylor series about the free stream temperature T∞. Neglecting second- and higher-order terms in (T-T∞), we get(4)T4=4TT∞3-3T∞4.In the above equations, νis the kinematic viscosity, α is the vortex viscosity, ρ is the fluid density, μ is the dynamic viscosity, ϕ1(0<ϕ1<1) and k>0 are the porosity and permeability of the porous medium, respectively, σ is the electrical conductivity, g is the gravitational acceleration, βT is the coefficient of thermal expansion, and βC is the coefficient of concentration expansion. j is the microinertia per unit mass and γ is the spin gradient. T is the fluid temperature, cp is the specific heat capacity, k1 is the thermal conductivity, qr is radioactive flux, C is the species concentration, D is the mass diffusivity, ω is the frequency of the velocity of the wall, and H(t) is the unit step function. Three distinct values of 0≤n≤1, when n=0, which show N=0 at the wall represent concentrated particle flows in which the microelements near to the wall are unable to rotate; this situation is also called strong concentration of microelements. When n=1/2, it indicates the vanishing of antisymmetric part of the stress tensor and describes weak concentration of microelements. The case when n=1 is used for the modelling of turbulent boundary layer flows [37].We introduce the dimensionless variables,(5)v=uu0,ξ=u0νy,τ=u02νt,Ω=νNu02,θ=T-T∞Tw-T∞,Φ=C-C∞Cw-C∞,in (1)-(2), yielding(6)∂v∂τ=1+β∂2v∂ξ2+β∂Ω∂ξ-Keffv+Grθ+GmΦ,(7)∂Ω∂τ=1η∂2Ω∂ξ2,(8)∂θ∂τ=1Preff∂2θ∂ξ2,(9)Sc∂Φ∂τ=∂2Φ∂ξ2,(10)vξ,0=0,Ωξ,0=0,θξ,0=0,Φξ,0=0,v0,τ=Hτcos⁡ωτ,Ω0,τ=-n∂v0,τ∂ξ,θ0,τ=1,Φ0,τ=1,v∞,τ=0,Ω∞,τ=0,θ∞,τ=0,Φ∞,τ=0,where β=α/μ is the dimensionless micropolar fluid parameter, Gr=(νgβT/u03)Tw-T∞ is the thermal Grashof number, Gm=(νgβC/u03)Cw-C∞ is the mass Grashof number, 1/K=ν2ϕ1/ku02 is the permeability parameter of porous medium, M=νσB02/ρu02 is the magnetic parameter, η=μj/γ is the spin gradient, Pr=μcp/k1 is the Prandtl number, R=16σ1T∞3/3k1k2 is the radiation parameter, Sc=ν/D is the Schmidt number, and the other constants in calculi are Keff=M+1/K and Preff=Pr/(1+R).


## 3. Solution of the Governing Problem

Applying Laplace transforms to (6)–(9) and using corresponding initial conditions from (10), we get(11)d2v-dξ2-δq+Keffv-=Ω0βδ-δGrθ-q-δGmΦ-q,(12)d2θ-dξ2-Preffqθ-=0,(13)d2Φ-dξ2-ScqΦ-=0,(14)d2Ω-dξ2-ηqΩ-=0.The transformed boundary conditions are(15)v-0,q=qq2+ω2,θ-0,q=1q,Φ-0,q=1q,Ω-0,q=-n∂v-0,q∂ξ=Ω0,v-∞,q=0,θ-∞,q=0,Φ-∞,q=0,Ω-∞,q=0.With exact solutions of (12) and (13), taking corresponding boundary conditions from (15) into account, we arrive at(16)θ-ξ,q=1qe-ξPreff⁡q,Φ-ξ,q=1qe-ξSc⁡q.Inverting the Laplace transform of (16), we get(17)θξ,τ=erf⁡cξPreff2τ,Φξ,τ=erf⁡cξSc2τ.The solution of (11) and (14) with corresponding boundary conditions from (15) and in view of (16) is given by(18)v-ξ,q=v-1ξ,q+v-2ξ,q+v-3ξ,q-v-4ξ,q+v-5ξ,q,Ω-ξ,q=Ω-1ξ,q+Ω-2ξ,q,where(19)v-1ξ,q=A0f-ξδ,q,-iω,Keff+A1f-ξδ,q,iω,Keff-A2f-ξδ,q,0,Keff+A3f-ξδ,q,a11,Keff+A4f-ξδ,q,a12,Keff-A5f-ξδ,q,-δ1,Keff-A6f-ξδ,q,-δ2,Keff-A7f-ξδ,q,δ3,Keff,v-2ξ,q=A7f-ξη,q,δ3,0+A8f-ξη,q,a11,0+A9f-ξη,q,a12,0+A5f-ξη,q,-δ1,0+A6f-ξη,q,-δ2,0+A10f-ξη,q,iω,0+A11f-ξη,q,-iω,0,v3ξ,τ=λ0fξPreff,τ,0,0-λ0fξPreff,τ,a11,0+λ1fξSc,τ,0,0-λ1fξSc,τ,a12,0,v-4ξ,q=qA12g-ξδ,q,0,Keff+A13g-ξδ,q,δ3,Keff+A14g-ξδ,q,a11,Keff+A15g-ξδ,q,a12,Keff+A16g-ξδ,q,-δ1,Keff-A17g-ξδ,q,-δ2,Keff+A18g-ξδ,q,iω,Keff-A19g-ξδ,q,-iω,Keff,v-5ξ,q=q+KeffA12g-ξη,q,0,0+A13g-ξη,q,δ3,0+A14g-ξη,q,a11,0+A15g-ξη,q,a12,0+A16g-ξη,q,-δ1,0-A17g-ξη,q,-δ2,0+A18g-ξη,q,iω,0-A19g-ξη,q,-iω,0,Ω-1ξ,q=nδq+KeffL1f-ξη,q,0,0+L2f-ξη,q,a11,0+L3f-ξη,q,a12,0+L4f-ξη,q,-δ1,0+L5f-ξη,q,-δ2,0+L6f-ξη,q,iω,0+L7f-ξη,q,-iω,0,Ω-2ξ,q=nL8g-ξη,q,0,0+L9g-ξη,q,a11,0+L10g-ξη,q,a12,0+L11g-ξη,q,-δ1,0+L12g-ξη,q,-δ2,0+L13g-ξη,q,iω,0+L14g-ξη,q,δ3,0,f-ζ,q,λ,b=1q-λexp⁡-ζq+b,g-ζ,q,λ,b=q+bq-λexp⁡-ζq+b.For Ai,i=1,2,…,19, and Lj,j=1,2,…,14, see Appendix.The Laplace inverse transforms of (18) are given by(20)vξ,τ=v1ξ,τ+v2ξ,τ+v3ξ,τ-v4ξ,τ+v5ξ,τ,Ωξ,τ=Ω1ξ,τ+Ω2ξ,τ,where(21)v1ξ,τ=A0fξδ,τ,-iω,Keff+A1fξδ,τ,iω,Keff-A2fξδ,τ,0,Keff+A3fξδ,τ,a11,Keff+A4fξδ,τ,a12,Keff-A5fξδ,τ,-δ1,Keff-A6fξδ,τ,-δ2,Keff-A7fξδ,τ,δ3,Keff,v2ξ,τ=A7fξη,τ,δ3,0+A8fξη,τ,a11,0+A9fξη,τ,a12,0+A5fξη,τ,-δ1,0+A6fξη,τ,-δ2,0+A10fξη,τ,iω,0+A11fξη,τ,-iω,0,v3ξ,τ=λ0fξPreff,τ,0,0-λ0fξPreff,τ,a11,0+λ1fξSc,τ,0,0-λ1fξSc,τ,a12,0,v4ξ,τ=-12τπτ∗A12gξδ,τ,0,Keff+A13gξδ,τ,δ3,Keff+A14gξδ,τ,a11,Keff+A15gξδ,τ,a12,Keff+A16gξδ,τ,-δ1,Keff-A17gξδ,τ,-δ2,Keff+A18gξδ,τ,iω,Keff-A19gξδ,τ,-iω,Keff,v5ξ,τ=-12τπτe-Keffτ∗A12gξδ,τ,0,0+A13gξδ,τ,δ3,0+A14gξδ,τ,a11,0+A15gξδ,τ,a12,0+A16gξδ,τ,-δ1,0-A17gξδ,τ,-δ2,0+A18gξδ,τ,iω,0-A19gξδ,τ,-iω,0,Ω1ξ,τ=-nδ12τπτe-Keffτ∗L1fξη,τ,0,0+L2fξη,τ,a11,0+L3fξη,τ,a12,0+L4fξη,τ,-δ1,0+L5fξη,τ,-δ2,0+L6fξη,τ,iω,0+L7fξη,τ,-iω,0,Ω2ξ,τ=nL8gξη,τ,0,0+L9gξη,τ,a11,0+L10gξη,τ,a12,0+L11gξη,τ,-δ1,0+L12gξη,τ,-δ2,0+L13gξη,τ,iω,0+L14gξη,τ,δ3,0,fξ,τ,λ,b=exp⁡λτ2exp⁡-ξλ+berf⁡cξ2τ-λ+bτ+exp⁡ξλ+berf⁡cξ2τ+λ+bτ,gξ,τ,λ,b=λ+bexp⁡λτ2λ+bexp⁡-ξλ+berf⁡cξ2τ-λ+bτ-exp⁡ξλ+berf⁡cξ2τ+λ+bτ+1πτexp⁡-ξ24τ-bτ;here, f∗g shows the convolution product of f and g.The coefficient of skin frictionCf and coefficient of couple stress Cs are important physical quantities, which in dimensionless form are defined as follows:(22)Cf=21+1-nβ∂vξ,τ∂ξξ=0,Cs=∂Ωξ,τ∂ξξ=0.


## 4. Results and Discussion

Exact solutions are obtained and the results are plotted for various embedded parameters, namely, permeability parameterK, magnetic parameter M, radiation parameter R, Prandtl number Pr, thermal Grashof number Gr, mass Grashof number Gm, spin gradient parameter η, and micropolar fluid parameter β on velocity and microrotation distributions in order to highlight the flow behavior. Skin friction and couple stress coefficient are also calculated numerically and presented in tabular form.Figures2–15 are plotted to show the effects of different embedded parameters on velocity and microrotation profiles. From these figures, it is noticed that velocity, as well as the magnitude of microrotation, decreases with increasing magnetic parameter M. Physically, it is true because it gives rise to a resistive force called the Lorentz force. The velocity and microrotation profiles for different values of the permeability parameter K are also displayed. It is obvious from the figures that an increase in K decreases the drag force and causes the velocity and magnitude of microrotation to increase.Figure 2
Variation in velocity and microrotation for different value ofK and n.Figure 3
Variation in velocity and microrotation for different value ofM and n.Figure 4
Variation in velocity and microrotation for different value ofK and R.Figure 5
Variation in velocity and microrotation for different value ofM and R.Figure 6
Variation in velocity and microrotation for different value ofK and Pr.Figure 7
Variation in velocity and microrotation for different value ofM and Pr.Figure 8
Variation in velocity and microrotation for different value ofK and Gr.Figure 9
Variation in velocity and microrotation for different value ofM and Gr.Figure 10
Variation in velocity and microrotation for different value ofK and Gm.Figure 11
Variation in velocity and microrotation for different value ofM and Gr.Figure 12
Variation in velocity and microrotation for different value ofK and η.Figure 13
Variation in velocity and microrotation for different value ofM and η.Figure 14
Variation in velocity and microrotation for different value ofK and β.Figure 15
Variation in velocity and microrotation for different value ofM and β.Figures2 and 3 depicted the influence of n on the velocity and microrotation profiles, which relates to the shear stress and microgyration vector. The velocity and magnitude of microrotation show an increasing behavior for greater values of n. It is observed in Figures 4 and 5 that velocity and magnitude of microrotation decrease with increasing R; this can be explained by the fact that radiation diffuses energy and reduces temperature; consequently, the velocity and magnitude of microrotation decrease. Figures 6 and 7 depict the effect of Prandtl number Pr on the velocity and microrotation profiles. The values of Pr are chosen as 600 (light oil) and 1000 (glycerol). Physically greater values of Pr decrease thermal conductivity and enhance the viscous forces. Consequently, the fluid velocity and the magnitude of microrotation decrease. The influence of thermal Grashof number Gr and mass Grashof number Gm on velocity and microrotation is depicted from Figures 8–11. Increasing values of both these parameters are responsible for the rise in buoyancy forces and reducing viscous forces which result in an increase in fluid velocity and magnitude of microrotation. Figures 12 and 13 are sketched to show the effect of spin gradient viscosity parameter η on the velocity and microrotation profiles. In the case of the velocity profile, greater values of η decrease fluid velocity. However, microrotation increases with increasing η. The effect of β on the velocity and the microrotation profiles is shown in Figures 14 and 15. A decreasing behavior is observed for increasing values of β in both cases. In these figures, the comparison of micropolar fluid velocity with Newtonian fluid velocity is also sketched. The velocity of the micropolar fluid is smaller as compared to Newtonian fluid. It is true due to the fact that the boundary layer thickness of micropolar fluid is greater than the boundary layer thickness of the Newtonian fluid.In Table1, the effects of M, K, and R on the skin friction Cf and couple stress coefficient Cs are presented. It is observed that increasing values of K reduce Cf which as a result enhances the fluid velocity, while greater values of K give rise to Cs. The effects of different values of M on Cf are calculated. It is noted that Cf is the increasing function of M, which is quite opposite behavior of M to that of velocity as shown in Figures 2–15, whereas Cs decreases with the increase in M. It is clear from the table that, for increasing values of R, Cf increases. This behavior of R is quite opposite to that of velocity. The effect of R on Cs is noticed opposite to that of Cf.Table 1
Variations in coefficients of skin friction and couple stress for different values ofK, M, and R.
K M R C f C s 0.5 1 0.2 6.006 7.641 1 1 0.2 5.546 7.868 1.5 1 0.2 5.386 7.956 0.5 2 0.2 6.437 7.46 0.5 3 0.2 6.843 7.313 0.5 1 0.4 8.635 7.613 0.5 1 0.6 29.444 7.571


## 5. Conclusion

Unsteady MHD flow of micropolar fluid over an oscillating vertical plate embedded in a porous medium along with combined heat and mass transfer is studied. Exact solutions are obtained using the Laplace transform technique. The effects of different parameters on velocity and microrotation are discussed. The following main points are concluded from this work:(i)
An increase inM reduces the fluid velocity and magnitude of microrotation.(ii)
Velocity and magnitude of microrotation are directly related toK that causes fall in drag forces.(iii)
Velocity and magnitude of microrotation decrease asPr is increased.(iv)
For large values ofβ, the velocity decreases.(v)
The skin friction increases with increase inM.



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