DIFFERENTIAL TRANSFORMATION METHOD FOR ANALYSIS OF NONLINEAR FLOW AND MASS TRANSFER THROUGH A CHANNEL FILLED WITH A POROUS MEDIUM

In this paper, mass transfer and chemical reaction effects on laminar viscous flow through a porous channel with moving or stationary walls are studied. The governing partial differential equations of the physical problem are transformed into a set of coupled nonlinear ordinary differential equations using similarity transformation. The coupled nonlinear ordinary differential equations are solved using differential transform method (DTM). The results obtained through the approximate analytical method are compared with the results of numerical method and high accuracy of the present approximate analytical solution is observed. The valuable achievement of the present study is imbedding a precise and efficient analytical method for the flow of viscous fluid in a porous channel with a chemical reaction. Also, the effects of some pertinent parameters such as Reynolds number, Darcy number, Schmidt number and suction/injection parameter on velocity components, heat transfer, concentration, and Sherwood distribution are presented in this work.


INTRODUCTION
Most of scientific problems in fluid mechanics and dynamics are innately nonlinear. All of these problems are displayed by partial or ordinary differential equations. The problem of flow and mass transfer in a porous media with a chemical reaction is an example of system of coupled nonlinear differential equations which can be solved by analytical methods such as DTM.
The flow in the porous media with a chemical reaction frequently occurs in many physical problems and engineering applications such as filtration processes, combustion systems, geothermal energy extraction systems, oil and gas production and chemical engineering [1][2][3][4]. In review of the importance of this problem, the flow characteristics have been investigated by numerous authors. Beg and Makinde [5] analyzed the laminar flow of an upper-convected Maxwell (UCM) viscoelastic fluid with species diffusion in a Darcian high-permeability porous channel using 6th order Rung-Kutta method. Hayat and Abbas [6] studied the flow of UCM fluid in a porous channel with chemical reaction in which the effects of Deborah number, Reynolds number, Schmidt number and chemical reaction parameter on the velocity and concentration distribution are examined. Rundora and Makinde [7] investigated the thermal effects of suction or injection on an unsteady reactive variable viscosity third grade fluid in a porous media subject to convective boundary condition using finite difference method. The obtained results reveal that the velocity and temperature profile are both decreased by increasing injection/suction Reynolds number. Chinyoka and Makinde [8] presented the transient solution of the flow of a reactive variable viscosity fluid in a circular pipe with a porous wall using semi-implicit finite difference method. Srinivas and Muthuraj [9] studied the influence of MHD mixed convection flow through a vertical asymmetric channel with chemical reaction. The momentum, energy and concentration equations of the problem have been linearized using long-wavelength approximation. In addition, other works [10-15] also investigated the flow and heat transfer in the porous medium for different conditions. The effects of internal heat generation and chemical reaction on free convective of polar fluid in a porous medium is studied by Patil and Kulkarni [16]. The results show that the presence of chemical reaction and internal heat source affect the flow field considerably. Reddy et al. [17] solved the heat and mass transfer equations of asymmetric laminar flow in a porous channel with expanding or contracting walls having different permeability in the presence of chemical reaction. Matin and Pop [18] investigated the fully developed flow of nanofluid through a porous channel with the constant heat flux wall in the presence of the chemical reaction. A Brinkman model and the clear fluid compatible model are applied to derive the governing equations. The heat and mass transfer flow of chemically reactive dusty viscoelastic fluid in a porous channel with convective boundary condition was investigated by Sivaraj and Kumar [19]. They deduced that the velocity and temperature profiles decrease by increasing the radiation parameter and the velocity and concentration distribution also decrease by increasing chemical reaction parameter. Srinivas et al. [20] examined the effects of mass transfer and chemical reaction on laminar flow in a porous channel subject to different boundary condition at the walls. Mahdy [21] studied coupled heat and mass transfer problem on double-diffusive convection from a vertical truncated in porous medium in the presence of chemical reaction with variable viscosity and also the effect of heat generation or absorption on the problem was examined.
Most of the physical problems are nonlinear and do not have an exact analytical solution. So, numerical and approximate methods are used by researchers to solve such equations. The numerical methods are often costly and time consuming to get a complete form of results, because it gives the solution at the discrete points. Furthermore, in the numerical solution the stability and convergence should be considered to avoid divergence or inappropriate results. Approximate techniques like decomposition method (DM), Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM) are good substitutes for numerical methods. During the recent years, some of the nonlinear engineering problems have been solved using some of these methods, such as HAM [22][23][24][25], HPM [26][27][28], VIM [29][30][31] and DM [32][33][34]. In most of the researches, some modifications introduced to overcome the nonlinearity.
Differential transform method (DTM) is also one of the other approximate methods to solve differential equations. This method was introduced by Zhou [35] to solve initial value problems in analysis of the electrical circuits. After that, DTM is applied on differential algebraic equations [36,37], partial differential equations [38][39][40][41][42][43], integral equations [44][45][46], ordinary differential equations [47][48][49][50] and fractional differential equations [51][52][53][54]. The method is an iterative technique to find the Taylor series solution of the problem. In this method, there is no need to the high calculation cost to determine the coefficients of Taylor series. The main objective of the present work is to analytically study the effects of heat transfer and chemical reaction on viscous laminar flow in a porous channel using differential transformation method. The results obtained in this research in comparison with numerical solution show that the method is efficient and accurate.

PROBLEM DESCRIPTION
Consider steady, laminar, two-dimensional flow of a viscous incompressible fluid through a porous medium in the presence of chemical reaction. The flow regime studying is shown in Fig. 1 including a porous channel with different boundary conditions in a coordinate system. The axis is taken along the direction of the flow and the y is normal to it. By choosing polar coordinates, the governing equations are: Where u and v are the velocity components of x and y direction,  is the density,  is the fluid viscosity,  is the porosity of the porous medium and K is the permeability of the porous media. The related boundary conditions for the flow are: , Introducing the following transformations to facilitate the solution: 2 , , , , Re.
Re.   [55], the momentum equation is reduced to: Re.
Eliminating the pressure from the Eq. (10) gives: And then, the boundary condition for the nonlinear ODE can be expressed as follows: Also, the governing mass-spices equation for this problem is expressed as follows: where D is the species diffusivity, C is the concentration field and 1 k is the reaction rate constant of a homogeneous first-order chemical reaction. The appropriate boundary conditions of the problem are: Here the Schmidt number ( Sc ), the chemical reaction parameter ( g ) and 1 K are denoted by:

THE BASIC PRINCIPLE OF DIFFERENTIAL TRANSFORMATION METHOD
The differential transform is defined as follows: is an arbitrary function, and X(k) is the transformed function. The inverse transformation is as follows The function x(t) is usually considered as a series with limited terms and Eq. (1), can be rewritten as: Where, q represents the number of Taylor series' components. Usually, through elevating this value, we can increase the accuracy of the solution. Some of the properties of DTM shown in Table 1. These properties are extracted from Eqs. 0 and (1).

APPLICATION OF DTM TO THE FLOW PROBLEM
In this section, we try to solve the Eqs.0 and 0 using DTM. The solution consists of two stages, first through mathematical relations and applying DTM properties, the Taylor series of solution is found. After that, the boundary conditions applied on solution to obtain the unknown parameters.       In order to validate the present solution of the problem and find the accuracy, we will compare our solution and numerical results. Numerical solution of the problem is done with Maple package. The available methods in this software are a combination of the base scheme (midpoint or trapezoid), and a method enhancement scheme (deferred corrections or Richardson extrapolation). The numerical solution method is capable of handling both linear and nonlinear BVPs with fixed, periodic, and even nonlinear boundary conditions. A good agreement between the results of present technique and numerical solution is observed in Table 2, which confirms the validity of the proposed method. As it can be seen, error of the method is about in order of 1e-4 to 1e-3.

RESULTS AND DISCUSSION
Laminar flow and mass transfer of viscous fluid through a porous channel with moving or stationary wall in the presence of the chemical reaction is considered. The solution is obtained for velocity and mass concentration distributions versus governing parameter such as Reynolds number, Darcy number and suction/injection parameter.
The results for this simulation were obtained for the case where   Schmidt number is defined as the ratio of momentum diffusivity (viscosity) and mass diffusivity and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. It is seen that by decreasing the mass diffusivity, the concentration rate decreased due to the higher mass transfer rate.

CONCLUSION
In this paper an analytical approach called Differential transformation method (DTM) has been applied to solve the problem of laminar flow and mass transfer of viscous fluid in a porous channel in the presence of chemical reaction with the moving or stationary walls. As a main outcome from the present study, the results are in excellent agreement with numerical ones. This method is accurate, efficient and powerful technique for solving the coupled nonlinear differential problems. Also, the effects of different parameters on velocity profile, concentration distribution and Sherwood number distribution have been presented. According to the results, increasing the Reynolds number causes to increase the fluid concentration. It is observed for the case of by decreasing the mass diffusivity (increasing Schmidt number), the fluid concentration is decreased due to higher mass transfer rate.