# Analytic Solution for MHD Falkner-Skan Flow over a Porous Surface

Authors: Fatheah A. Hendi; Majid Hussain
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper discusses the MHD Falkner-Skan flow over a porous surface.  The solution to nonlinear  problem is  first constructed and analyzed  for the emerging parameters


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## Body

## 1. Introduction

Motivated by significant applications in packed bed reactor, geothermal system, extractions of crude oil, water or nuclear pollution, and so forth, the wedge flow over shaped bodies has attracted the attention of various researchers as the early formulation given by Falkner and Skan [1]. Later, Asaithambi [2] analyzed the Falkner-Skan equation by using finite difference scheme. Magnetohydrodynamics effects on the Falkner Skan wedge flow are studied by Abbasbandy and Hayat [3, 4]. They used Hankel-Pade and homotopy analysis methods for the derivation of the solutions. Rajagopal et al. [5] discussed the Falkner-Skan flow of a non-Newtonian fluid. Massoudi and Ramezan [6] extended the idea of Rajagopal et al. [5] for suction and blowing cases. Forced convection boundary layer flow over a wedge with uniform suction or injection is analyzed by Yih [7]. Kuo [8] discussed the heat transfer for the Falkner-Skan wedge flow by using differential transformation method. Numerical treatment for the Falkner-Skan wedge flow of a power law fluid saturating the porous space has been discussed by Kim [9]. More recently, Hayat et al. [10] extended the idea of Kim [9] to study the mixed convection flow. In this paper, Falkner-Skan flow over a porous surface is considered. Case of uniform suction/blowing is taken into account. Stream function formulation and suitable transformations reduce the arising problem to ordinary differential equation which has been solved by a homotopy analysis method HAM [11–23]. Finally the solutions are sketched and analyzed.


## 2. Problem Statement

We consider the two-dimensional Falkner-Skan flow in an incompressible magnetohydrodynamic (MHD) viscous fluid. Under the usual boundary layer approximations, the equations which govern the Falkner-Skan flow are(2.1)∂u∂x+∂v∂y=0,u∂u∂x+v∂u∂y=UdUdx+ν∂2v∂y2-σB2uρ(u-U),
where u and v are the velocity components in the x and y directions, respectively, ρ is the fluid density, ν is the kinematic viscosity of the fluid, σ is the electrical conductivity, U is the characteristic velocity, and a uniform magnetic field B is applied. The induced magnetic field is neglected in view of small magnetic Reynolds number. The boundary conditions are(2.2)u=Uw,v=Vw,aty=0,u⟶U(x),asy⟶∞,
where(2.3)U(x)=axm,B(x)=B0x(m-1)/2,
where Vw is the porous velocity and Uw is the surface velocity.Denoting byΨ the stream function and defining(2.4)η=m+12Uνxy,ψ=m+12νxUf(η),u=Uf′(η),v=-m+12νUx[f(η)+m-1m+1ηf′(η)],u=∂ψ∂y,v=-∂ψ∂x,
the continuity equations are automatically satisfied and the other equations yield(2.5)d3fdη3+fd2fdη2+β[1-(dfdη)2]-M2(dfdη-1)=0,f(0)=γ,f′(0)=λ,f′(∞)=1.
Here primes denote the differentiation with respect to η and β,M,Pr, and β are the dimensionless numbers. These are defined as(2.6)M2=2σB02ρa(1+m),β=2mm+1,λ=UwU,γ=-Vw((m+1)vU2x)-1/2.


## 3. Homotopy Analysis Solutions

The velocity can be expressed as the set of base function(3.1){ηkexp⁡(-nη),k,nη≥0}
by selecting the initial guess(3.2)f0(η)=γ+η-1-λδ(1-δexp⁡(-η))
and the auxiliary linear operator ℒf(3.3)Lf(f)=d3fdη3+δd2fdη2,
with(3.4)Lf[C1+C2exp⁡(2η)+C3exp⁡(-η)]=0,
in which Ci,(i=1-3) are the arbitrary constants. If p∈[0,1] is the embedding parameter and ℏf, is the nonzero auxiliary parameter, then the zeroth-order deformation problem is given as follows:
(3.5)(1-p)Lf[f̂(η;p)-f0(η)]=pℏfNf[f̂(η;p)],f̂(0;p)=γ,f̂′(0;p)=λ,f̂′(∞;p)=1,
in which the nonlinear operator Nf is of the following form:(3.6)Nf[f̂(η;p)]=∂3f̂(η;p)∂η3+f̂(η;p)∂2f̂(η;p)∂η2+β[1-(∂f̂(η;p)∂η)2]-M2(∂f̂(η;p)∂η-1).
For p=0 and p=1, the aforementioned zeroth-order deformation equation has the solutions:(3.7)f̂(η;0)=f0(η),f̂(η;1)=f(η).
when p increases from 0 to 1, f̂(η;p) vary from f0(η) to the exact solutions f(η). In view of Taylor’s theorem and (3.7), we can write(3.8)f̂(η;p)=f0(η)+∑m=1∞fm(η)pm,
where(3.9)fm(η)=1m!∂mf̂(η;p)∂pm|p=0.
The auxiliary parameter is so properly chosen that the series (3.8) converge at p=1. Hence(3.10)f(η)=f0(η)+∑m=1∞fm(η).
The mth-order deformation problem is(3.11)Lf[fm(η)-χmfm-1(η)]=ℏfRmf(η),Rmf[f̂(η;p)]=∂3f̂m-1(η;p)∂η3+f̂(η;p)∂2f̂m-1(η;p)∂η2+β[1-(∂f̂m-1(η,p)∂η)2]-M2(∂f̂m-1(η,p)∂η-1),fm(0)=fm′(0)=fm′(∞)=0,χm={0,m≤1,1,m>1.


## 4. Analysis and Discussion

Here convergence is checked at 20th order of approximations. Figures1 and 2 show the ℏ curves at various values of the emerging parameters. The range of ℏ curves is -0.8≤ℏ≤-0.4 for Figure 1 and -0.8≤ℏ≤-0.4 for Figure 2, respectively. It is obvious from these figures that auxiliary parameter h is necessary for the convergence purposes. Tables 1 and 2 are made to give the comparison with numerical results at [10,10] order of approximation for various values of magnetic parameter M and Falkner-Skan flow parameter m. The results are similar to those of Abbasbandy and Hayat [4]. Figure 3 is plotted for various values of suction/injection parameter γ. For positive values of γ, the velocity decreases whereas for negative values the velocity increases. In Figure 4 with the fixed value of slip condition λ=2, the peak value of velocity is high when compared with Figure 3. The skin friction versus γ is sketched in Figure 5 for various values of β with slip velocity equal to zero. It is obvious that with an increase in β the skin friction increases. Figure 6 is prepared for different values of γ. It is shown that f′′(0) decreases for injection and increases for suction. The skin friction versus stretching parameter β is displayed in Figure 7. It is noticed that for negative values of slip velocity, the skin friction increases.Table 1
[10,10] homotopy-Pade approximation at m=2,λ=0,γ=0.
MPresent resultsNumerical results by Abbasbandy and Hayat [4]11.7194651.7194654022.4394982.4394983355.1909595.190959451010.09677510.096775455050.01944050.01944071100100.009721100.00972170Table 2
[10,10] homotopy-Pade approximation at m=-3/5,λ=0,γ=0.
MPresent resultsNumerical results by Abbasbandy and Hayat [4]32.2733882.2733883643.4881483.4881485754.6007544.60075494109.8064649.806464201514.87167414.871674842019.90393719.90393701Figure 1
Theh curve of f′′(0) at 20th order of approximation for M=5 and δ=6.Figure 2
Theh curve of f′′(0) at 20th order of approximation for m=-3/5,M=3 and δ=3.Figure 3
Velocity profiles for suction/injection parameterγ=1,0,-1 for λ=2.Figure 4
Velocity profiles for suction/injection parameterγ=2,0,-2 for λ=2.Figure 5
Skin friction coefficient for different values of stretching parameterβ=4,3,2,1,0.Figure 6
Shear stressf′′(0) versus the stretching parameter β for various values of the wall suction/injection parameter γ=1,0,-1.Figure 7
Shear stressf′′(0) versus the stretching parameter β for various values λ=1,0.5,0and -0.5.



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*Source: 123185-2012-03-29.xml*