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

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

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𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑈𝑑𝑈𝜕𝑑𝑥+𝜈²𝑣𝜕𝑦²−𝜎𝐵²𝑢𝜌(𝑢−𝑈), (2.1) 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 are𝑢=𝑈_𝑤,𝑣=𝑉_𝑤(𝑥),at𝑦=0,𝑢⟶𝑈,as𝑦⟶∞, (2.2) where𝑈(𝑥)=𝑎𝑥^𝑚,𝐵(𝑥)=𝐵₀𝑥^(𝑚−1)/2, (2.3) where 𝑉_𝑤 is the porous velocity and 𝑈_𝑤 is the surface velocity.

Denoting by Ψ the stream function and defining𝜂=𝑚+12𝑈𝜈𝑥𝑦,𝜓=𝑚+12√𝜈𝑥𝑈𝑓𝜂,𝑢=𝑈𝑓^′𝜂,𝑣=−𝑚+12𝜈𝑈𝑥𝑓𝜂+𝑚−1𝑚+1𝜂𝑓^′𝜂,𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓,𝜕𝑥 (2.4) the continuity equations are automatically satisfied and the other equations yield𝑑³𝑓𝑑𝜂³𝑑+𝑓²𝑓𝑑𝜂²+𝛽1−𝑑𝑓𝑑𝜂²−𝑀²𝑑𝑓𝑓(0)𝑑𝜂−1=0,=𝛾,𝑓^′(0)𝑓=𝜆,_′(∞)=1. (2.5) Here primes denote the differentiation with respect to 𝜂 and 𝛽,𝑀,Pr, and 𝛽 are the dimensionless numbers. These are defined as𝑀²=2𝜎𝐵²₀𝜌𝑎(1+𝑚),𝛽=2𝑚𝑈𝑚+1,𝜆=^𝑤𝑈,𝛾=−𝑉_𝑤(𝑚+1)𝑣𝑈2𝑥^−1/2. (2.6)

3. Homotopy Analysis Solutions

The velocity can be expressed as the set of base function𝜂^𝑘exp−𝑛𝜂,𝑘,𝑛𝜂≥0 (3.1) by selecting the initial guess𝑓₀𝜂=𝛾+𝜂−1−𝜆𝛿1−𝛿exp−𝜂 (3.2) and the auxiliary linear operator ℒ_𝑓ℒ_𝑓𝑓=𝑑³𝑓𝑑𝜂³𝑑+𝛿²𝑓𝑑𝜂², (3.3) withℒ_𝑓𝐶₁+𝐶₂exp2𝜂+𝐶₃exp−𝜂=0, (3.4) in which 𝐶_𝑖,(𝑖=1−3) are the arbitrary constants. If 𝑝∈[0,1] is the embedding parameter and ℏ_𝑓, is the nonzero auxiliary parameter, then the zeroth-order deformation problem is given as follows: 1−𝑝ℒ_𝑓𝑓𝜂;𝑝−𝑓₀𝜂=𝑝ℏ_𝑓𝑁_𝑓𝑓,𝑓𝑓𝜂;𝑝0;𝑝=𝛾,^′0;𝑝=𝜆,𝑓^′∞;𝑝=1, (3.5) in which the nonlinear operator 𝑁_𝑓 is of the following form:𝑁_𝑓𝑓=𝜕𝜂;𝑝³𝑓𝜂;𝑝𝜕𝜂³+𝑓𝜕𝜂;𝑝²𝑓𝜂;𝑝𝜕𝜂²𝜕𝑓+𝛽1−𝜂;𝑝𝜕𝜂²−𝑀²𝜕𝑓𝜂;𝑝.𝜕𝜂−1 (3.6) For 𝑝=0 and 𝑝=1, the aforementioned zeroth-order deformation equation has the solutions:𝑓𝜂;0=𝑓₀𝜂,𝑓𝜂;1=𝑓𝜂. (3.7) when 𝑝 increases from 0 to 1, 𝑓(𝜂;𝑝) vary from 𝑓₀(𝜂) to the exact solutions 𝑓(𝜂). In view of Taylor’s theorem and (3.7), we can write𝑓𝜂;𝑝=𝑓₀𝜂+^∞_𝑚=1𝑓_𝑚𝜂𝑝^𝑚, (3.8) where𝑓_𝑚𝜂=1𝜕𝑚!^𝑚𝑓𝜂;𝑝𝜕𝑝^𝑚_𝑝=0. (3.9) The auxiliary parameter is so properly chosen that the series (3.8) converge at 𝑝=1. Hence𝑓𝜂=𝑓₀𝜂+^∞_𝑚=1𝑓_𝑚𝜂. (3.10) The 𝑚th-order deformation problem isℒ_𝑓𝑓_𝑚𝜂−𝜒_𝑚𝑓_𝑚−1𝜂=ℏ_𝑓ℛ^𝑓_𝑚𝜂,ℛ^𝑓_𝑚𝑓=𝜕𝜂;𝑝³𝑓_𝑚−1𝜂;𝑝𝜕𝜂³+𝑓𝜕𝜂;𝑝²𝑓_𝑚−1𝜂;𝑝𝜕𝜂²𝜕𝑓+𝛽1−^𝑚−1𝜂,𝑝𝜕𝜂²−𝑀²𝜕𝑓_𝑚−1𝜂,𝑝,𝑓𝜕𝜂−1_𝑚(0)=𝑓^′_𝑚(0)=𝑓^′_𝑚(∞)𝜒=0,_𝑚=0,𝑚≤1,1,𝑚>1. (3.11)

4. Analysis and Discussion

Here convergence is checked at 20th order of approximations. Figures 1 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.

Table 2 

[10,10] homotopy-Pade approximation at 𝑚=−3/5,𝜆=0,𝛾=0.

Figure 1 

The ℎ curve of 𝑓′′(0) at 20th order of approximation for 𝑀=5 and 𝛿=6.

Figure 2 

The ℎ 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 parameter 𝛾=1,0,−1 for 𝜆=2.

Figure 4 

Velocity profiles for suction/injection parameter 𝛾=2,0,−2 for 𝜆=2.

Figure 5 

Skin friction coefficient for different values of stretching parameter 𝛽=4,3,2,1,0.

Figure 6 

Shear stress 𝑓′′(0) versus the stretching parameter 𝛽 for various values of the wall suction/injection parameter 𝛾=1,0,−1.

Figure 7 

Shear stress 𝑓′′(0) versus the stretching parameter 𝛽 for various values 𝜆=1,0.5,0and −0.5.

Acknowledgments

The authors are thankful to Professor Tasawar Hayat for useful discussion of the paper. M. Hussain acknowledges the financial support provided by Higher Education Commission (HEC) of Pakistan.