Analytic solutions of unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/sink

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Abstract

In this paper, we study the boundary layer flow and heat transfer on a permeable unsteady stretching sheet with non-uniform heat source/sink. The analytic solutions are obtained by using suitable similarity transformations and homotopy analysis method (HAM). Furthermore, the effects of unsteadiness parameter, Prandtl number and heat source/sink parameter on the dynamics are analyzed and discussed.

Introduction

Investigations of boundary layer flow and heat transfer are important due to its applications in industries and many manufacturing process. Crane [1] investigated the flow due to a stretching sheet with linear surface velocity and obtained the similarity solution to the problem. Vajravelu and Roper [2] studied the flow and heat transfer in a second grade fluid over a stretching sheet with viscous dissipation and internal heat generation or absorption. In Ref. [3], a second grade fluid has been analyzed for two heating process, namely PST-case and PHF-case. The series solutions were obtained through homotopy analysis method. Sarma [4] presented an analytical solution for heat transfer in a steady laminar flow of an incompressible viscoelastic fluid past a semi-infinite stretching sheet with power-law surface temperature, including the effects of viscous dissipation and internal heat generation or absorption. The flow of a power-law fluid with variable thermal conductivity and non-uniform heat source was studied and solved by the keller box method in Ref. [5].

These works were confined to the steady state conditions. Since the stretching sheet may varies with time, the aspect of unsteady stretching sheet becomes interesting in practical problems. Anderssona et al. [6] explored the heat transfer in a liquid film over an unsteady stretching sheet, and obtained the numerical solutions. Wang [7] gave the analytical solutions of this problem. Unsteady laminar boundary layer flow over a permeable stretching surface was analyzed in Ref. [8]. Mukhopadhyay [9], [10] extended it by assuming the viscosity and thermal diffusivity are linear functions of temperature and studied unsteady mix convection boundary layer flow of an incompressible viscous liquid through porous medium along a permeable surface, the thermal radiation effect on heat transfer was also considered. The effect of non-uniform heat source of laminar flow over an unsteady stretching sheet was explored in Ref. [11].

Since the study of heat source/sink effect on heat transfer is important in some cases, in the present paper we study the unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform source. The homotopy analysis method (HAM) [12], [13], [14] is employed to give analytic solutions. HAM was first proposed by Liao [12]. Many workers have used this method to solve various non-linear problem successfully, a list of the key references in the vast literature concerning this fields we refer to the recent papers [15], [16], [17], [18].

Section snippets

Boundary layer governing equations

Consider the unsteady, two dimensional incompressible viscous flow on a stretching permeable surface in a quiescent fluid, and the sheet is stretching with a velocity Uw=ax1-ct [8] in the positive x direction. Here a > 0, b > 0 and t<1c. The sheet surface temperature Tw=T+bx1-ct varies with the coordinate x and time t. The boundary layer governing equations are:ux+uy=0,ut+uux+vuy=ν2uy2,Tt+uTx+vTy=kρcp2Ty2-qρcp.The associated boundary layer conditions:u=Uw(x,t),v=vw=-νa1-ct1/2S0

Zero-order deformation equations

Solving Eqs. (8), (9), (10), (11) using HAM [3], [15], [16], [17], [18]. From the boundary conditions (10), (11), it is obvious to choose:f0(η)=S0+1-exp(-η),θ0(η)=exp(-η),as the initial approximations of f(η) and θ(η), respectively, and to choose:Lf[f(η;q)]=3Φ(η;q)η3+2Φ(η;q)η2,Lθ[ϕ(η;q)]=2Θ(η;q)η2+Θ(η;q)η,as the auxiliary linear operators, which have the following properties:Lf[c1+c2η+c3exp(-η)]=0,Lθ[c4+c5exp(-η)]=0,where ci (i = 1–5) are arbitrary constants.

Based on (8), (9), This paper

Convergence of the HAM solution

For an analytic solution obtained by the homotopy analysis method, its convergent depends on the auxiliary parameter h. If this parameter is properly chosen, the given solution is valid, as verified in previous works [3], [15], [16], [17], [18]. Since the interval for the admissible values of h corresponds to the line segments nearly parallel to the horizontal axis. Then we know that the admissible values for the parameter h is −1.4  h  −0.6 from Fig. 1, Fig. 2. In this paper we choose h = −1.2.

Results and discussion

Fig. 3, Fig. 4 show the effects of A and S0 on the dimensionless velocity f′(η). From Fig. 3, It can be seen that the increase of A decreases the velocity f′(η) when S0 = 1.5. Fig. 4 illustrates the effect of suction/injection number S0 on the velocity distribution. The increasing S0(>0) decreases the velocity, whereas in the case S0(<0) increases with increasing values of ∣S0∣.

The effects of A, A∗, B∗, S0 and Pr on the temperature profiles are shown in Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9. In

Conclusions

In this paper, we have investigated the unsteady boundary flow and heat transfer over a permeable surface with heat source/sink. Analytic solutions are obtained through HAM. It is found that the fluid velocity decreases as A and S0 increases. The thermal boundary layer thickness decreases with increasing A and S0, but increases with increasing A∗ and B∗. The values of −θ′(0) increase with the increase Pr, A and S0, however decrease with the increase of A∗ and B∗.

Acknowledgements

The work is supported by the National Natural Science Foundations of China (No. 50936003), The open Project of State Key Lab. for Adv. Metals and Materials (2009Z-02) and Research Foundation of Engineering Research Institute of USTB.

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