# Magnetohydrodynamics and Radiation Effects on Unsteady Convection Flow of Micropolar Fluid Past a Vertical Porous Plate with Variable Wall Heat Flux

Authors: M. Gnaneswara Reddy
Journal: ISRN Thermodynamics
 (2012)

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## Abstract

An analysis is presented for the problem of the unsteady two-dimensional laminar flow of a viscous incompressible micropolar fluid past a vertical porous plate in the presence of a transverse magnetic field and thermal radiation with variable heat flux. The free stream velocity follows an exponentially increasing or decreasing small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which it absorbs the micropolar fluid with a suction velocity varying with time. The Rosseland approximation is used to describe radiative heat transfer in the limit of optically thick fluids. The effects of flow parameters and thermophysical properties on the flow temperature fields across the boundary layer are investigated. The method of solution can be applied for small perturbation approximation. Numerical results of velocity profiles of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient and the couple stress coefficient at the wall are prepared with various values of the fluid properties.


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## Body

## 1. Introduction

The classical Navier-Stokes theory does not describe adequately the flow properties
of polymeric fluids, colloidal suspensions, and fluids containing certain additives.
Eringen [1] proposed the theory of micropolar fluids
which shows microrotation effects as well as microinertia. The theory of
thermomicropolar fluids was developed by Eringen [2]
by extending his theory of micropolar fluid. Physically, they represent fluids
consisting of randomly oriented particles suspended in a viscous medium. The flow
characteristics of the boundary layer of micropolar fluid over a semi-infinite plate
in different situations have been studied by many authors in (Ahmadi [3], Peddieson and McNitt [4], Takhar and Soundagekhar [5], and
Gorla [6]). The study of the flow and heat transfer
for a micropolar fluid past porous plate embedded in a porous medium has attracted
the interest of many investigators in view of its applications in many engineering
problems such as oil exploration, chemical catalytic reactors, thermal insulation,
and geothermal energy extractions (Raptis [7]).
Sharma and Gupta [8] considered thermal convection in
micropolar fluids in porous medium. Kim [9] presented
an analysis of an unsteady convection flow of a micropolar fluid past a vertical
porous plate embedded in a porous medium.The study of magnetohydrodynamics (MHD) plays an important role in agriculture,
engineering, and petroleum industries. The problem of free convection under the
influence of a magnetic field has attracted the interest of many researchers in view
of its applications in geophysics and astrophysics. In addition, this type of flow
has attracted the interest of many investigators in view of its applications in many
engineering problems such as MHD generators, plasma studies, nuclear reactors, and
geothermal energy extractions. Kim [10] studied
unsteady MHD convection flow of polar fluids past a semi-infinite vertical-moving
porous plate in a porous medium. Soundalgekar [11]
obtained approximate solutions for the two-dimensional flow of an incompressible,
viscous fluid past an infinite porous vertical plate with constant suction velocity
normal to the plate, the difference between the temperature of the plate and the
free stream is moderately large causing the free convection currents. Raptis [12] studied mathematically the case of unsteady
two-dimensional natural convective heat transfer of an incompressible, electrically
conducting viscous fluid via a highly porous medium bound by an infinite vertical
porous plate. El-Hakiem et al. [13] studied the
effect of viscous and Joule heating on MHD-free convection flow with variable plate
temperature in a micropolar fluid in the presence of uniform transverse magnetic
field using the Kellerbox implicit scheme. Rahman and Sattar [14] analyzed magnetohydrodynamic convective flow of a micropolar fluid
past a continuously moving vertical porous plate in the presence of heat
generation/absorption.In the context of space technology and in the processes involving high temperatures,
the effects of radiation are of vital importance. Recent developments in hypersonic
flights, missile reentry, rocket combustion chambers, power plants for
interplanetary flight, and gas-cooled nuclear reactors, have focused attention on
thermal radiation as a mode of energy transfer and emphasized the need for improved
understanding of radiative transfer in these processes. Raptis [15] studied the flow of a micropolar fluid past a
continuous moving plate by the presence of radiation. Abo-Eldahab and Ghonaim [16] analyzed the radiation effects on heat transfer of
a micropolar fluid through a porous medium. Kim and Fedorov [17] considered the case of mixed convection flow of a micropolar fluid
past a semi-infinite moving vertical porous plate with varying suction velocity
normal to the plate in the presence of radiation.The purpose of this work is to study radiation effects on unsteady
Magnetohydrodynamics (MHD) mixed convection flow of micropolar fluid with variable
heat flux has received a little attention. We also consider the free stream to
consist of a mean velocity over which is superimposed an exponentially varying with
time. The effects of various governing parameters on the velocity, microrotation and
temperature, skin-friction coefficient, and the wall couple stress coefficient are
shown in figures and table and discussed in detail.


## 2. Mathematical Analysis

An unsteady two-dimensional mixed convection flow of a laminar incompressible
micropolar fluid past a vertical porous plate in the presence of a transverse
magnetic field and thermal radiation with variable heat flux is considered. Thex*-axis is taken along the vertical porous plate in an upward direction and y*-axis is taken normal to the plate. The acceleration of gravity g is in a direction opposite to x*-coordinate. The transversely applied magnetic field and magnetic
Reynolds number are very small and hence the induced magnetic field is negligible.
Viscous and Darcy resistance terms are taken into account as the constant
permeability porous medium. It is also assumed here that the size of holes in the
porous plate is significantly larger than a characteristic microscopic length scale
of the micropolar fluid to simplify formulation of the boundary conditions. Due to
the assumption of infinite plate, the flow variables are functions of y* and the time t* only. It is also assumed that the free stream to consist of a mean
velocity and temperature over which are superimposed an exponentially varying with
time. Under the usual Boussenesq’s approximation, the equation of continuity,
linear momentum, microrotation, and energy can be written ascontinuity equation:(1)∂v*∂y*=0,linear momentum equation:(2)∂u*∂t*+v*∂u*∂y*=-1ρ∂p*∂x*+(ν+νr)∂2u*∂y*2+gβf(T-T∞)-νK*u*-σB02ρu*+2νr∂ω*∂y*,angular momentum equation:(3)ρj*(∂ω*∂t*+v*∂ω*∂y*)=γ∂2ω*∂y*2,energy equation:(4)∂T∂t*+v*∂T∂y*=α(∂2T∂y*2-1k∂qr∂y*). The third term on the right-hand side of the momentum equation
(2) denotes thermal buoyancy effect.By using the Rosseland approximation, Brewster [18],
the radiative heat flux in the y* direction is given by (5)qr=-4σs∂T43ke∂y*, where σs is the Stefan-Boltzmann constant and ke is the mean absorption coefficient, respectively.It should be noted that by Rosseland approximation, we limit our analysis to
optically thick fluids. If the temperature differences within the flow are
sufficiently small, then (4) can be
linearized by expanding T4 into the Taylor series about T∞ and neglecting higher terms to take the form: (6)T4≅4T∞3T-3T∞4. It is assumed that the porous plate moves with constant velocity in
the longitudinal direction, and the free steam velocity follows an exponentially
increasing or decreasing small perturbation law. It is also assumed that the plate
temperature and suction velocity vary exponentially with time.Under these assumptions, the appropriate boundary conditions for the velocity,
microrotation, temperature, and concentration fields are(7)u*=0,ω*=-12∂u*∂y*,∂T∂y*=-qwk(1+εBen*t*)aty*=0,u*⟶U∞*=U0(1+εen*t*),ω*⟶0,(8)T⟶T∞asy*⟶∞, where B is a real positive constant and ε is a small parameter such that εB≤1, T∞ is the free stream dimensional temperature, n* is constant, ε is small less than unity and U0 is a scale of free steam velocity. The boundary condition for
microrotation variable ω* describes its relationship with the surface stress.It is clear from (1) that the suction
velocity normal to the plate is either a constant or a function of time. Hence it is
assumed in the form: (9)v*=-V0(1+εAen*t*), where A is a real constant, ε and εA are small less than unity, and V0 is the scale of suction velocity which is a nonzero positive constant.
The negative sign indicates that the suction is towards the plate. Outside the
boundary layer, (2) gives (10)-1ρdp*dx*=dU∞*dt*+νK*U∞*+σB02ρU∞* In order to write the governing equations and the boundary
conditions dimensionless form, the following nondimensional quantities are
introduced (11)u=u*U0,v=v*V0,y=V0y*ν,U∞=U∞*U0,ω=νU0V0ω*,t=t*V02ν,θ=(T-T∞)kV0(Tw-T∞)qw,j=V02ν2j*,n=n*V02ν,Gr=νβfg(Tw-T∞)U0V02,(12)Pr=νρCpk=να=μCpk,R=kke4σsT∞3. Furthermore, the spin-gradient viscosity γ, which gives some relationship between the coefficients of viscosity and
microinertia, is defined as (13)γ=(μ+Λ2)j*=μj*(1+12β),β=Λμ. In view of (5), (6) and (9)–(13), the
governing equations (2)–(4) reduce to the following nondimensional
form: (14)∂u∂t-(1+εAent)∂u∂y=dU∞dt+(1+β)∂2u∂y2+Grθ+N(U∞-u)+2β∂ω∂y,∂ω∂t-(1+εAent)∂ω∂y=1η∂2ω∂y2,∂θ∂t-(1+εAent)∂θ∂y=1Γ∂2θ∂y2, where η=μj*/γ=2/(2+β),N=(M+(1/K)),andΓ=(1-(4/(3R+4)))Pr.The boundary conditions (7) are then given
by the following dimensionless equations: (15)u=0,∂θ∂y=-(1+εBent),ω=-12∂u∂yaty=0,u⟶U∞,θ⟶0,ω⟶0asy⟶∞.


## 3. Solution of the Problem

In order to reduce the above system of partial differential equations to a system of
ordinary differential equations in dimensionless form, we perform an asymptotic
analysis by representing the linear velocity, microrotation, and temperature in the
neighborhood of the porous plate as(16)u=u0(y)+εentu1(y)+O(ε2)+⋯,ω=ω0(y)+εentω1(y)+O(ε2)+⋯,θ=θ0(y)+εentθ1(y)+O(ε2)+⋯. Substituting (16) into
(14), and equating the harmonic and
nonharmonic terms, and neglecting the higher-order terms of O(ε2), we obtain the following pairs of equations for (u0,ω0,θ0) and (u1,ω1,θ1): (17)(1+β)u0′′+u0′-Nu0=-N-Grθ0-2βω0′,(1+β)u1′′+u1′-(N+n)u1=-(N+n)-Au0′-Grθ1-2βω1′,ω0′′+ηω0′=0,ω1′′+ηω1′-nηω1=-Aηω0′,θ0′′+Γθ0′=0,θ1′′+Γθ1′-nΓθ1=-AΓθ0′. In the above equations, prime indicates differentiation with respect
to y only.The corresponding boundary conditions can be written as(18)u0=0,u1=0,ω0=-12u0′,ω1=-12u1′,θ0′=-1,θ1′=-Baty=0,u0=1,u1=1,ω0⟶0,ω1⟶0,θ0⟶0,θ1⟶0,asy⟶∞. Solving (17) under the
boundary conditions (18), we get (19)u0(y)=1+a1e-h2y+a2e-Γy+a3e-ηy,u1(y)=1+b1e-h1y+b2e-h2y+b3e-h3y+b4e-h4y+b5e-Γy+b6e-ηy,ω0(y)=c1e-ηy,ω1(y)=c2e-h1y-Aηnc1e-ηy,θ0(y)=1Γe-Γy,θ1(y)=1h4(B+AΓn)e-h4y-Ane-Γy, where the exponential indices and coefficients are given in the
appendix.By virtue of (16), we obtain the velocity,
microrotation, and temperature profiles as follows: (20)u(y,t)=1+a1e-h2y+a2e-Γy+a3e-ηy+εeδt(1+b1e-h1y+b2e-h2y+b3e-h3y+b4e-h4y+b5e-Γy+b6e-ηy),ω(y,t)=c1e-ηy+εeδt(c2e-h1y-Aηnc1e-ηy),θ(y,t)=1Γe-Γy+εeδt(1h4(B+AΓn)e-h4y-Ane-Γy). Knowing the velocity field in the boundary layer, we can calculate
the skin-friction coefficient Cf at the porous plate, which in the nondimensional form is given by (21)τw*=μ∂u*∂y*|y*=0,Cf=τw*ρU0V0=∂u∂y|y=0=-(a1h1+a2Γ+a3η)-εent(b1h1+b2h2+b3h3+b4h4+b5Γ+b6η). Knowing the microrotation in the boundary layer, we can calculate
the couple stress coefficient Cm at the porous plate, which in the nondimensional form is given by (22)Mw=γ∂w*∂y*|y*=0,Cm=MwμjU0=∂w∂y|y=0=1+εent((B+AΓn)-AΓn).


## 4. Results and Discussion

The formulation of the problem that accounts for the effects of MHD and radiation on
flow of an incompressible, micropolar fluid along a vertical porous plate has been
carried out in the preceding sections. This enables us to carry out the numerical
computations for the velocity, microrotation, and temperature for various values of
the flow and material parameters. In the present study the following default
parameter values are adopted for computations:β=0.2, t=1.0, n=0.1, ε=0.001, A=0.5, Gr=2.0, M=2.0, K=1.0, Pr=0.71, R=2.0, and B=0.1. All graphs therefore correspond to these values unless specifically
indicated on the appropriate graph. Also, the boundary condition for y→∞ is replaced by where ymax is a sufficiently large value of y where the velocity profile u approaches the relevant free stream velocity. We choose ymax=8 and a step size Δy=0.001.The effect of viscosity ratioβ on the translational velocity and microrotation profiles across the
boundary layer are presented in Figure 1. It is
noteworthy that the velocity distribution is greater for a Newtonian fluid (β=0) with given parameters, as compared with micropolar fluids until its
peak value reaches. The translational velocity shows a decelerating nature near the
porous plate as β-parameter increases, and then decays to the relevant free stream
velocity. In addition, the magnitude of microrotation at the wall is decreased as β-parameter increases. However, the distributions of microrotation across
the boundary layer do not show consistent variations with increment of β-parameter.Velocity and microrotation profiles for various values ofβ.
(a)
(b)The translational velocity and the microrotation profiles against spanwise coordinatey for different values of Grashof number Gr are described in Figure 2. The thermal
Grashof number signifies the relative effect of the thermal buoyancy force to the
viscous hydrodynamic force. It is observed that an increase in Gr leads to a rise in the values of velocity, but decreases due to
microrotation. Here the positive values of Gr corresponds to a cooling of the surface by natural convection. In
addition, the curves show that the peak value of velocity increases rapidly near the
wall of the porous plate as Gr increases, and then decays to the free stream velocity.Velocity and microrotation profiles for various values ofGr.
(a)
(b)For different values of the magnetic field parameterM, the translational velocity and microrotation profiles are plotted in
Figure 3. It is obvious that the effect of
increasing values of M-parameter results in a decreasing velocity distribution across the
boundary layer. The results also show that the magnitude of microrotation on the
porous plate is decreased as M-parameter increases.Velocity and microrotation profiles for various values ofM.
(a)
(b)For various values of the permeability parameterK, the profiles of the translational velocity and the microrotation across
the boundary layer are shown in Figure 4. Clearly as K increases, although the corresponding profiles of the microrotation is
difficult to show clearly due to very little variation, the translational velocity
and the microrotation profiles tend to increase.Velocity and microrotation profiles for various values ofK.
(a)
(b)Figure5 shows the translational velocity and the
temperature profiles across the boundary layer for different values of Prandtl
number Pr. The Prandtl number defines the ratio of momentum diffusivity to thermal
diffusivity. The results show the effect of increasing values Pr on decreasing the translational velocity at the wall and then approach
to the free stream boundary layer conditions. Typical variations of the temperature
profiles along the spanwise coordinate y are shown Figure 5(b) for different values
of Prandtl number Pr. The results show that an increase of Prandtl number results in
decreasing the thermal boundary layer thickness and more uniform temperature
distribution across the boundary layer. The reason is that smaller values of Pr are
equivalent to increasing the thermal conductivities, and therefore heat is able to
diffuse away from the heated surface more rapidly than for higher values of Pr. Hence, the boundary layer is thicker and rate of heat transfer is
reduced, as gradients have been reduced.Velocity and temperature profiles for various values ofPr.
(a)
(b)For various values of the radiation parameterR, the velocity and temperature profiles are plotted in Figure 6. It is obvious that an increase in the radiation
parameter R results in decreasing velocity and temperature within the boundary
layer, as well as a decreased thickness of the velocity and temperature boundary
layers. This is because the large R-values correspond to an increased dominance of conduction over radiation
thereby decreasing buoyancy force (thus, vertical velocity) and thickness of the
thermal and momentum boundary layers.Velocity and temperature profiles for various values ofR.
(a)
(b)Numerical values for functions proportional to shear stress and wall couple stress
are given in Table1. It is noted that both the
skin-friction coefficient and the wall couple stress coefficient decreases as Pr increases. From Table 1, it is noticed
that as Gr increases, the skin friction and wall couple stress coefficient
increases. It is obvious that an increase in the magnetic field parameter or the
radiation parameter reduces the skin friction and wall couple stress
coefficient.Table 1
Values ofCf and Cm for various values of Gr,M, Pr, and R with β=0.2, t = 1.0, n = 0.1, ε=0.001, A=0.5, K=1.0, and B=0.1.
Gr
M
Pr
R
C
f
C
m
2.0
2.0
0.71
2.0
6.9567
3.3146
4.0
2.0
0.71
2.0
10.5384
5.0936
2.0
3.0
0.71
2.0
6.7482
3.1875
2.0
2.0
2.0
2.0
4.5211
2.1690
2.0
2.0
0.71
5.0
6.1079
2.9172


## 5. Conclusions

An analysis is presented for the problem of unsteady two-dimensional laminar flow of
a viscous incompressible micropolar fluid past a vertical porous plate in the
presence of a transverse magnetic field and thermal radiation with variable heat
flux. Numerical results are presented to illustrate the details of the MHD
convective radiative flow and heat transfer characteristics and their dependence on
the fluid properties and flow conditions. We may conclude that the translational
velocity across the boundary layer and the magnitude of microrotation at the wall
are decreased with increasing values ofM and K. Also, the translational velocity across the boundary layer and
temperature at the wall are decreased with increasing values of M, Pr, and R, while they show opposite trends with increasing values of Gr.



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*Source: 146263-2012-07-05.xml*