Abstract
A class of nonlinear convection-diffusion equation is studied in this paper. The partial differential equation is converted into nonlinear ordinary differential equation by introducing a similarity transformation. The asymptotic analytical solutions are obtained by using double-parameter transformation perturbation expansion method (DPTPEM). The influences of convection functional coefficient and power law index on the heat transport characteristics are discussed and shown graphically. The comparison with the numerical results is presented and it is found to be in excellent agreement. The method and technique used in this paper have the significance in studying other engineering problems.
1. Introduction
Convection-diffusion equations arose from various fields of applied sciences such as heat transfer problems in a draining film [1] or a nanofluid filled enclosure [2], radial transport in a porous medium [3], and water transport in soils [4] and have received extensive attentions during the past several decades. For convection-diffusion equations, only few cases with special initial or boundary value conditions have analytical solutions. Therefore, most of the main concerns were the study of the qualitative properties of the solutions [5–13] and numerical study [14–23] for convection-diffusion equations. However, the very important approximate solution of convection-diffusion equation has not been well solved. In this paper we present similarity solutions for a class of convection-diffusion equation, which are then solved by DPTPEM. The method was first proposed by Yan Zhang et al. [24, 25]. In this paper, we can find that the DPTPEM can be successfully used to solve the studied problem, and the approximate solutions obtained by DPTPEM agree very well the numerical solutions obtained by bvp4c with Matlab [26].
2. Mathematical Formulation
In this paper, we consider a class of nonlinear convection-diffusion equation as follows [20]:subject to the following initial value conditions:where , is positive power law exponent. Convection functional coefficient is assumed to be a real-valued continuous differential function defined on , , for . is the heat density per unit area.
3. Converting into Nonlinear Ordinary Differential Equation Boundary Value Problems
Introduce the following similarity variables [20]:The transformed convection-diffusion equation together with the initial value conditions given by (1)-(2) can be written as
Conversely, if is a solution of (4)-(5), then must be a similarity solution to (1)-(2).
Let be a solution to (4)-(5). If is strictly increasing in , then the function inverse to exists. Here holds on . Then (4) can be changed into the following nonlinear ordinary differential equation:and integration from to 1 yieldsSetCombine (7) and (8) to getThe corresponding boundary conditions arewhere is heat diffusion flux. The derivation process indicates that only the positive solutions of (9)-(10) are physically significant.
4. A Brief Introduction to DPTPEM
The method was first proposed by Yan Zhang and Liancun Zheng [24, 25].
We consider a class of ordinary differential equation initial or boundary value problem without small parameter as follows:
where is order differential operator without small parameter , is a constant, and is the region containing the origin.
Introduce an embedded parameter transformation as follows:where is an artificial small parameter, is independent variable parameter, is dependent variable parameter, and and are undetermined constants. Substituting this transformation (12) into (11), we can obtain the following nonlinear initial value problem:
where is a function of , is the region containing the origin. The solution of is expressed in form of power series by applying perturbation expansion method
Combine (12) and (14) to get the solution of (11) in terms of power series. The undetermined parameters are determined according to the boundary conditions of the original problem.
The basic idea of the DPTPEM consists of three steps. Firstly, introducing an artificial small parameter , the independent variable and the unknown function are transformed simultaneously, and the problem is transformed into a new one related to small parameter . Secondly, the transformed new differential equation is expanded in the form of power series of and decomposed into the sum of several solution components. Then, by using known initial or boundary value conditions, we try to find out the solution components of each order separately and combine them to get the solution of the new problem. Thirdly, by substituting the solution of the new equation into the transformation equation (12), the small parameter is eliminated and the solution of the original problem is obtained.
5. Approximate Analytical Results of (9)-(10) and Discussion
The initial condition is assumed as follows:In order to solve (9), we transform the dependent variable and independent variable as follows according to formula (12):where is an artificial small parameter.
Let and substituting this transformation (16) into (9)-(10), we can obtain the following nonlinear initial value problems:Expanding in a power series development, we can getThe solution of (19) can be obtained by expanding in a power series development near as follows:Substituting (20) into (19) and equating the coefficients of , we can get the following expressions:
Then, we can obtain in the following.
Combine (16), (20), and the above ,… to get solution of (9)-(10).
Similarly, let ; we obtain approximate analytical solution of (9)-(10) as follows:
Let ; approximate analytical solution of (9)-(10) is presented as follows:
It is obvious that we can promptly obtain the value of by applying in (23)-(25) for each fixed . Based on (23)-(25) and the corresponding we can easily obtain the graph of the heat diffusion flux distribution for different and . The results are presented in Tables 1–5 and Figures 1–5.
It can be seen from Figures 1–3 that the heat diffusion flux decreases with increase of for specific , the physical meaning is that heat diffusion flux is a decreasing function of , which means that the profiles exhibited by a big power law index possess a smaller diffusion. Figures 4 and 5 indicate the heat diffusion flux decrease with the decrease of .
In order to verify the efficiency and reliability of approximate analytical solution obtained by using DPTPEM, a comparison of approximate analytical solution and numerical one obtained by bvp4c with Matlab is presented in Tables 6 and 7 and Figures 6 and 7. It is obvious that excellent agreement exists for approximate analytical solution and numerical one.
6. Conclusions
In the paper, a class of nonlinear convection-diffusion was studied. The partial differential equation and corresponding initial value conditions were transformed into a class of singular nonlinear boundary value problems of ordinary differential equation when similarity variables were introduced. An efficient approximate analytical method named DPTPEM was applied to solve these nonlinear problems. The reliability and effectiveness of the DPTPEM were verified by comparing approximate results with the numerical solutions. The effects of convection functional coefficient and power law index on transfer behavior were presented. The results show that heat diffusion flux is an increasing function of convection functional coefficient and a decreasing function of power law index .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The research was supported by a grant from the National Natural Science Foundation of China (no. 11501496) and the Natural Science Basic Research Plan in Shaanxi Province of China (no. 2014JQ2-1003).