Solution of the Boundary Layer Equation of the Power-Law Pseudoplastic Fluid Using Differential Transform Method

The boundary layer equation of the pseudoplastic fluid over a flat plate is considered. This equation is a boundary value problem (BVP) with the high nonlinearity and a boundary condition at infinity. To solve such problems, powerful numerical techniques are usually used. Here, through using differential transform method (DTM), the BVP is replaced by two initial value problems (IVP) and then multi-step differential transform method (MDTM) is applied to solve them. The differential equation and its boundary conditions are transformed to a set of algebraic equations, and the Taylor series of solution is calculated in every sub domain. In this solution, there is no need for restrictive assumptions or linearization. Finally, DTM results are compared with the numerical solution of the problem, and a good accuracy of the proposed method is observed.


Introduction
Boundary layer equation of the Newtonian fluid over a flat plate is one of the classic issues of mechanical engineering. Because of high level of application of non-Newtonian fluids in industry, the boundary layer equation of the non-Newtonian fluids is being noted by engineers. On the other hand, nonlinearity of these equations attracted the interest of mathematicians to evaluate the power and accuracy of the approximate and numerical methods. One of the most significant non Newtonian models is the power law one. In this model, shear thinning or shear thickening property of the fluid is considered and the shear stress has a nonlinear relation with shear rate, while shear stress has a linear relation with shear rate for the Newtonian fluid.
As we pointed out above, boundary layer problems are among the nonlinear ones and most of them do not have an exact analytical solution. So, numerical and approximate methods are used by researchers to solve such equations. The most known numerical method used to solve boundary layer problems is the shooting one. Based on this method, boundary value problem is transformed to an initial value problem with unknown initial values. After that, the problem is replaced with a system of first-order ordinary differential equations and usually is solved through Runge-Kutta method. Shooting method is appropriate to solve many boundary value problems, but this one is not useful for solving some BVPs because of the instability of the solution.
Approximate techniques like decomposition method (DM), homotopy analysis method (HAM), homotopy perturbation method (HPM), and variational iteration method (VIM) are good substitutes for numerical methods. During the recent years, boundary layer problems have been solved using some of these methods, such as HAM [1][2][3][4][5][6][7][8][9], HPM [10][11][12][13][14], VIM [15][16][17], and DM [18][19][20][21]. In most of the researches, some modifications were introduced to overcome the nonlinearity and the boundary condition at infinity. Differential transform method (DTM) is also one of the other approximate methods to solve differential equations. Here, DTM and multi-step DTM are used to solve the boundary layer equation of the pseudoplastic fluid. This method was introduced by Zhou [22] to solve initial value problems in analysis of the electrical circuits. After that, DTM is applied to differential algebraic equation [23,24], partial differential equation [25][26][27][28][29][30], integral equation [31][32][33], ordinary differential equation [34][35][36][37][38], and fractional differential equation [39][40][41][42]. The method is an iterative technique to find the Taylor series solution of the problem. There is no need for the high calculation cost to determine the coefficients of Taylor series, which is the reverse of the standard Taylor series method. This paper is organized as follows. In Section 2, DTM and multi-step DTM are introduced. In Section 3, the boundary layer equation of the pseudoplastic fluid is extracted from continuity and momentum equations as presented in [43]. In Section 4, the problem is solved and the results are illustrated as some figures and tables.

Differential Transform Method
The differential transform is defined as follows: where ( ) is an arbitrary function and ( ) is the transformed function. The inverse transformation is as follows: Substituting (1) into (2), we have The function ( ) is usually considered as a series with limited terms and (2) can be rewritten as where represents the number of Taylor series' components. Usually, through elevating this value, we can increase the accuracy of the solution.
Although the DTM series solution is a good approximate of the exact solution, but this series is diverged for greater areas. There are two ways to overcome this problem. One is the using pade approximate technique. Usually, Pade approximate gives us more exact information about the behavior of the solution (see [44]). Another one is to use multi-step DTM. Based on this one, solution domain is divided to some sub domains.
To solve a differential equation in the domain [0, ] using multi-step DTM, this domain is divided to sections. We suppose the sub domains are equal and length of each sub domain is = / . So, there is a separate function for every sub domain as follows: where = ( − 1) . Multi-step DTM for every sub domain is defined as The inverse multi-step DTM is Some of the properties of DTM and multi-step DTM are shown in Table 1. These properties are extracted from (1) and (6).

Mathematical Formulation
Two-dimensional boundary layer equations for an incompressible fluid are where is the fluid density, is the shear stress, and and are the velocities in direction of and , respectively. For the non-Newtonian power-law fluid, the shear stress is calculated through the following relation: where and are the flow consistency and power-law index, respectively.
Mathematical Problems in Engineering 3 Here, the following dimensionless parameters are used: where is the plate length, ∞ is the far field velocity, and Re is generalized Reynolds number.

Solution of the Problem
In this section, we try to solve (17) using DTM and multistep DTM. The solution consists of two stages; first through mathematical relations and applying DTM, the value of (0) is found. After that, boundary layer equation of the pseudoplastic fluid is solved as an initial value problem (IVP) using multi-step DTM. (Obtaining (0)). The BVP (17) can be transformed to an initial value problem with the replacement of the unknown initial condition (0) = instead of the boundary condition at infinity (∞) = 1:

Applying DTM
where By applying the DTM on (18) at = 0, the recursive relation is derived: Regarding (19), Taking differential transform on (21), we have From (19), the value of (0) can be obtained: Combining (20), (22), and (23), the coefficients of the Taylor series solution of (18) can be calculated: In the same manner, the coefficients ( ) are Substituting (24) and (26) in (2), the ( ) and ( ) are obtained. These functions have a relation as follows: Regarding (28) and the boundary condition at infinity in (17), the value of can be obtained:
To solve boundary layer problems, the domain [0, ∞) is replaced by [0, ∞ ). But ∞ should be great enough that the solution is not to be dependent on. The solution domain should be divided to equal parts ( = ∞ / ). So, we have where Applying multi-step DTM on (30) and (31), we have So And for ≥ 1 ( + 1) The initial conditions of the problem are considered for first sub domain ( = 1). Regarding (6), (25), and (31), initial conditions transformed as follows: The boundary conditions of each sub domain are the continuity of the ( ), ( ), and ( ). These boundary conditions can be obtained from (7): And we have for The value of the ( ∞ ) can be calculated by differentiating from (7): The unknown parameter ( ) is calculated from (29). Now, (18) is solved with a similar process like (25) using multi-step DTM. The only difference is that the condition (0) = is replaced by the condition (0) = 1.
In Figure 1 ( ) is illustrated for different values of and compared with the numerical solution and results of Lemieux et al. [43]. The numerical solution of the problem is done by the MATLAB software. Figures 2 and 3 show the variation of ( ) and ( ), respectively. The most important step of this scheme is to choose the appreciate finite value of ∞ . Thus to estimate this value, we start with an initial guess for ∞  Mathematical Problems in Engineering Table 3: The solution of ( ) for = 0.7 when ∞ = 10 and = 10. and solve the problem. The solution process is repeated with another larger value of ∞ until the values of (0) and (0) differ regarding the desired accuracy.
In Table 2 the values of the (0) = are compared with the numerical solution and results of Lemieux et al. [43]. The approximate solution of the problem is presented in Table 3 for a special case ( = 0.7).

Conclusion
In this paper, the boundary layer equation of the pseudoplastic fluid over a flat plate was extracted from the boundary layer theory as presented in [43]. This equation is a boundary value problem with high nonlinearity and boundary condition at infinity. Using DTM, the BVP transformed to a pair of initial value problems and multi-step DTM is used to solve them.
Using the method, the differential equation and boundary conditions are transformed into a recurrence set of equations. Finally, the coefficients of power series are obtained based on the solution of this set of equations. The results validated with the numerical solution, and a good accuracy is observed. The proposed method overcame nonlinearity without using linearization or restrictive assumptions.