Exponential Bernstein functions: an effective tool for the solution of heat transfer of a micropolar fluid through a porous medium with radiation

Abstract

In this study, an effective collocation method using new weighted orthogonal basis functions on the half-line, namely the exponential Bernstein functions, is proposed for simulating the solution of the heat transfer of a micropolar fluid through a porous medium with radiation. The governing equations and their associated boundary conditions can be written as a system of nonlinear ordinary differential equations. The presented approach does not require truncating or transforming the semi-infinite domain of the problem to a finite domain. In addition, this method reduces the solution of the problem to the solution of a system of algebraic equations. The effects of the coupling constant, radiation parameter and the permeability parameter on velocity and temperature profiles will be discussed in detail and shown graphically. A comparative study with the previous results of viscous fluid in the literature is made.

Appendix A: Basic idea of Liaos Homotopy analysis method (HAM) (Liao 2003)

Consider the following nonlinear differential equation in form of

$$\begin{aligned} N[u( t)]=0, \end{aligned}$$

(96)

where N is a nonlinear operator, t is an independent variable and $u\left( t\right)$ is the solution of equation. By means of generalizing the conventional Homotopy method, Liao constructed the so-called zero-order deformation equation as:

$$\begin{aligned} (1-q)L\left[ \Phi ( t;q) -u_{0}\left( t\right) \right] =qhH(t)N[\Phi ( t;q)], \end{aligned}$$

(97)

where $q\in \left[ 0,1\right]$ is an embedding parameter, $h\ne 0$ is a nonzero auxiliary parameter, $H\left( t\right) \ne 0$ is an auxiliary function, L is an auxiliary linear operator with the property

$$\begin{aligned} L\left[ f\right] =0 \quad \mathbf{when } f=0, \end{aligned}$$

(98)

$u_{0}(t)$ is an initial guess of u( t) and $\Phi \left( t;q\right)$ is an unknown function on the independent variables t and q. Obviously, when $q=0$ and $q=1$, it holds

$$\begin{aligned} \Phi ( t;0) =u_{0}\left( t\right) ,\quad \Phi ( t;1)=u\left( t\right) , \end{aligned}$$

(99)

respectively. Thus, as q increases from 0 to 1, the solution $\Phi ( t;q)$ varies from the initial guess $u_{0}(t)$ to the solution u(t) . Using the parameter q , we expand $\Phi \left( t;q\right)$ in Taylor series as follows:

$$\begin{aligned} \Phi ( t;q) =u_{0}(t)+\sum _{m=1}^{+\infty }u_{m}( t)q^{m}, \end{aligned}$$

(100)

where

$$\begin{aligned} u_{m}( t)=\frac{1}{m!}\frac{\partial ^{m}\Phi (t;q) }{\partial q^{m}}\Bigg |_{q=0}. \end{aligned}$$

(101)

Assume that auxiliary linear operator, the initial guess, the controlling convergence parameter h , and the auxiliary function H(t) are chosen such that the series (100) converges at $q=1$. So, from (99) and (100), we have

$$\begin{aligned} u(t) =u_{0}(t)+\sum _{m=1}^{+\infty }u_{m}( t). \end{aligned}$$

(102)

Define the vector

$$\begin{aligned} \mathbf {u}_{n}=\left\{ u_{0},u_{1},\ldots ,u_{n}\right\} . \end{aligned}$$

Differentiating (97) for m times with respect to the embedding parameter q and then setting $q=0$ and finally dividing them by m! , we have the so-called mth-order deformation equation

$$\begin{aligned} L\left[ u_{m}(t)-\chi _{m}u_{m-1}(t)\right] =hH(t)R_{m}\left( \mathbf {u}_{m-1}(t)\right) , \end{aligned}$$

(103)

where

$$\begin{aligned} R_{m}(\mathbf {u}_{m}(t)) =\frac{1}{(m-1)!}\frac{\partial ^{m-1}\left\{ N \Phi ( t;q)\right\} }{\partial q^{m-1}}\Bigr |_{q=0}, \end{aligned}$$

(104)

and

$$\begin{aligned} \chi _{m}=\Big \{ \begin{array}{ll} 0, &{}\quad m\le 1, \\ 1, &{}\quad m>1. \end{array} \end{aligned}$$

Applying $L^{-1}$ on both sides of (103), we get

$$\begin{aligned} u_m(t)=\chi _{m}u_{m-1}(t)+hL^{-1}[H(t)R_{m}(\mathbf {u}_{m-1}(t))]. \end{aligned}$$

(105)

In this way, it is easy to obtain $u_m$ for $m\ge 1$, at Mth order; we have

$$\begin{aligned} u(t) =\sum _{m=0}^{M}u_{m}( t). \end{aligned}$$

(106)

When $M\rightarrow +\infty$, we get an accurate approximation of the original Eq. (96). For the convergence of the previous method, we refer the reader to Liao (2003). Rashidi and Mohimanian Pour (2011) implemented the above method for solving heat transfer problem of a micropolar fluid through a porous medium with radiation.