The radiative heat flux term is simplified by making use of the Roseland approximation as in eqn (1).

Where, α denote thermal diffusivity, k is thermal conductivity, σ denote Stefan–Boltzmann constant, K is absorption Coefficient, v is Kinematics viscosity, u,v denote velocity component, q_r is Radiative heat flux, β is Volumetric expansion coefficient(Temperature), G denote Gravitational acceleration, t is Time, U₀ denote wall velocity, T_w is Wall Temperature, T_∞ denote ambient temperature, c is suction parameter, R is radiation parameter, G_r denote local Grash of number, P_r is Prandtl number.

The appropriate boundary conditions are

We can define the similarity variables are as follows:

Here the length scale is defined as

Taking the Taylor series expansion of T⁴ and neglecting terms with higher powers, we have

Using the eqns. (7) and (8) into the eqns. (2) to (4) we obtained the non-linear ordinary differential equations are as follows:

The corresponding boundary conditions are as follows:

Here

The local wall shear stress (Skin friction) can be defined as

The Local surface heat flux

Solution of the Non-Linear Boundary problem using the Homotopy Analysis Method

This section deals with a basic strong analytic tool for non-linear problems, namely the Homotopy analysis method (HAM) which was generated by Liao [18], is employed to solve the nonlinear differential eqns. (9)–(11). The Homotopy analysis method is based on a basic concept in topology. Unlike perturbation techniques like [19], the Homotopy analysis method is independent of the small/large parameters. Unlike all other reported perturbation and non-perturbation techniques such as the artificial small parameter method [20], the δ-expansion method [21] and Adomianʼs decomposition method [22], the Homotopy analysis method provides us a simple way to adjust and control the convergence region and rate of approximation series. The Homotopy analysis method has been successfully applied to many nonlinear problems such as heat transfer [23], viscous flows [24], nonlinear oscillations [25], Thomas-Fermiʼs atom model [26], nonlinear water waves [27], etc. Such varied successful applications of the Homotopy analysis method confirm its validity for nonlinear problems in science and engineering. The Homotopy analysis method is a good technique when compared to other perturbation methods. The existence of the auxiliary parameter h in the Homotopy analysis method provides us with a simple way to adjust and control the convergence region of the solution series.

Basic concepts of the Homotopy analysis method [18-29]

Consider the following differential equation:

Where N is a nonlinear operator, t denotes an independent variable, u(t) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the conventional Homotopy method, Liao constructed the so-called zero-order deformation equation as:

Where p ∈ [0,1] is the embedding parameter, h≠0 is a nonzero auxiliary parameter, H(t) ≠ 0 is an auxiliary function, L an auxiliary linear operator, u₀(t) is an initial guess of u(t), φ (t; p) is an unknown function. It is important to note that one has great freedom to choose auxiliary unknowns in HAM. Obviously, when p=0 and p=1, it holds:

respectively. Thus, as P increases from 0 to 1, the solution φ (t; p) varies from the initial guess u₀(t) to the solution u (t).

Expanding φ (t; p) in Taylor series with respect to p, we have:

If the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the auxiliary function are so properly chosen, the series eqn.(18) converges at p=1 then we have:

Differentiating the eqn. (16) for m times with respect to the embedding parameter p, and then setting p=0 and finally dividing them by m!, we will have so-called m th order deformation equation as:

Where

and

Applying L^-1 on both side of eqn.(22), we get

In this way, it is easily to obtain u_m for m ≥ 1, at M^th order, we have

when M→ +∞, we get an accurate approximation of the original eqn.(16). For the convergence of the above method we refer the reader to Liao [19]. If an eqn. (16) admits unique solution, then this method will produce the unique solution.

Approximate analytical expressions of the non-linear differential eqns. (9) and (10) using Homotopy analysis method

In this section, we find the analytical expressions for the eqns. (9) and (10) with the help of the eqn.(10). We construct Homotopy for the eqns.(9) and (10) are as follows:

The approximate solution of the eqns.(27) and (28) are as follows:

The initial approximations are as follows:

Substituting the eqns. (29) and (30) into the eqns. (27) and (28) respectively we get

Comparing the coefficients of p⁰, p¹ in the eqns.(35) and (36) we get

Solving the eqns. (37)-(40) with the help of the eqns. (31)-(34) we get the following results:

Where

According to the HAM, we can conclude that

Substituting the eqns. (43) and (44) into an eqn. (46) and using the eqns. (41) and (42) into an eqn. (47) we get the following:

The analytical expression of the dimensionless skin friction using the eqn. (13) is given by Skin friction

The analytical expression of the Wall heat transfer rate using the eqn. (15) is given by

Results and Discussion

Figures 1 and 2 represents dimensionless temperature θ(y) versus dimensionless distance y. From figure 1, it is noted that the temperature increases when the radiation parameter R increases, and in some fixed values of the other dimensionless parameters. From figure 3, it is inferred that when the suction parameter c increases the corresponding temperature decreases in some fixed values of the other parameters.

Figures 3 and 4 represent dimensionless velocity f(y) versus dimensionless distance y. From figure 3, it is noted that the velocity increases when the radiation parameter R increases, and in some fixed values of the other dimensionless parameters. From figure 4, it is inferred that when the suction parameter c increases the corresponding velocity decreases in some fixed values of the other parameters.

Figure 5 represents wall heat transfer rate Nu versus radiation parameter R. From figure 5, it is observed that the wall heat transfer rate increases when the suction parameter increases, and in some fixed values of the other dimensionless parameters. Figure 6 represents wall shear stress versus radiation parameter R. From figure 6, it is inferred that the wall shear stress decreases when the suction parameter increases, and in some fixed values of the other dimensionless parameters.

Conclusion

In this paper the Homotopy analysis method is employed to mathematical study of a boundary layer flow with thermal radiation past a moving vertical porous plate. The approximate analytical expressions of the velocity and temperature profiles are derived mathematically and graphically using the Homotopy analysis method. The approximate analytical expressions of the wall shear stress and wall heat transfer rate are also derived using the analytical expressions for the velocity and temperature profiles. We also discussed the graphical representations of the wall shear stress and the wall heat transfer rate. The Homotopy analysis method can be easily extended to solve the other non-linear boundary value problems in physical and chemical sciences.

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