Approximate Solution for the Electrohydrodynamic Flow in a Circular Cylindrical Conduit

This paper considers the nonlinear boundary value problem (BVP) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. The velocity field was solved using the new homotopy perturbation method (NHPM), considering the electrical field and strength of the nonlinearity. The approximate analytical procedure depends only on two components and polynomial initial condition. The analytical solution is obtained and the numerical results presented graphically. The effects of the Hartmann electric number Ha and the strength of nonlinearity α are discussed and presented graphically. We also compare this method with numerical solution (N.S) and show that the present approach is less computational and is applicable for solving nonlinear boundary value problem (BVP).


Introduction
The electrohydrodynamic flow of a fluid in an "ion drag" configuration in a circular cylindrical conduit is governed by a nonlinear second-order ordinary differential equation [1][2][3] d 2 w(r) dr 2 + 1 r dw(r) dr +H a 2 1 − w(r) 1 − αw(r) = 0, 0 < r < 1 (1) subject to the boundary conditions where w(r) is the fluid velocity, r is the radial distance from the center of the cylindrical conduit, H a is the Hartmann electric number, and the parameter α is a measure of the strength of the nonlinearity. In [1] McKee and his colleagues developed perturbation solutions in terms of the parameter α governing a nonlinear problem. McKee and his coworkers used a Gauss-Newton finite-difference solver combined with the continuation method and Runge-Kutta shooting method to provide numerical results for the fluid velocity over a large range of values of α. This was done for both large and small values of α. Paullet [2] proved the existence and uniqueness of a solution of BVP of electrohydrodynamic flow and in addition, discovered an error in the perturbative and numerical solutions given in [1] for large values of α. Very recently Mastroberardino [3] presented the approximate solution by homotopy analysis method (HAM) for the nonlinear BVP governed by electrohydrodynamic flow of a fluid in a circular cylindrical conduit.
In the present paper, we introduce a new computational method, namely, new homotopy perturbation method [4][5][6] for solving electrohydrodynamic flow of a fluid in a circular cylindrical conduit. It is interesting to note that the efficiency of the approach depends only on two components of the homotopy series. The method is an improvement of classical homotopy perturbation method [7][8][9][10][11][12]. In contrast to the HAM and HPM, in this method, it is not required to solve the functional equations in each iteration. Unlike the Adomian decomposition method (ADM) [13], the NHPM is free from the need to use Adomian polynomials.

Analysis of the Method
Let us consider the nonlinear differential equation where A is an operator, f is a known function, and u is a sought function. Assume that operator A can be written as where L is the linear operator and N is the nonlinear operator. Hence, (3) can be rewritten as follows: We define an operator H as where p ∈ [0, 1] is an embedding or homotopy parameter, v(z; p) : Ω × [0, 1] → , and u 0 is an initial approximation of solution of the problem in (3). Equation (6) can be written as We assume that the solution of equation H(v, p) can be written as a power series in embedding parameter p, as follows: Now, let us write (7) in the following form: By applying the inverse operator, L −1 to both sides of (9), we have Suppose that the initial approximation of (3) has the form where a n , n = 0, 1, 2, . . . are unknown coefficients and P n (z), n = 0, 1, 2, . . . are specific functions on the problem. By substituting (8) and (11) into (10), we get Equating the coefficients of like powers of p, we get following set of equations: Now, we solve these equations in such a way that v 1 (z) = 0. Therefore, the approximate solution may be obtained as

Analytical Solution
To obtain the solution of (1) by NHPM, we construct the following homotopy: Applying the inverse operator, L −1 (•) = r 0 ξ 0 (•)dηdξ to the both sides of (15), we obtain The solution of (16) to have the following form: Substituting (17) in (16) and equating the coefficients of like powers of p, we get following set of equations: Assuming w 0 (r) = 7 n=0 a n P n , P k = r k , a = W(0), solving the above equation for W 1 (r) leads to the result 4 ISRN Computational Mathematics  With vanishing W 1 (r), we have the following values for coefficients a i , i = 0, 1, . . . , 7 a 0 = −H a 2 (−1 + a + αa) 2(−1 + αa) , Therefore, we obtain the solutions of (1) as

Numerical Results and Concluding Remarks
In this paper we have studied electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit by using two-component homotopy perturbation method. Figures 1(a) and 1(b) and It has been noted that the nonlinearity confronted in this problem is in the form of a rational function and, thus, poses a significant challenge in regard to obtaining analytical solutions. Despite this fact, we have shown that the solutions obtained are convergent and that they compare extremely well with numerical solutions (N.S). It is interesting to note that NHPM yields convergent solutions for all of the cases considered. However, HPM yields divergent solutions for