MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2019/3467276 3467276 Research Article Similarity Solution and Heat Transfer Characteristics for a Class of Nonlinear Convection-Diffusion Equation with Initial Value Conditions http://orcid.org/0000-0002-6997-2540 Xu Yunbin 1 Vespri Vincenzo School of Mathematics and Statistics Yulin University Shaanxi Yulin 719000 China yulinu.edu.cn 2019 2352019 2019 25 02 2019 07 05 2019 16 05 2019 2352019 2019 Copyright © 2019 Yunbin Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 k(z) and power law index n 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.

National Natural Science Foundation of China 11501496 Natural Science Basic Research Plan in Shaanxi Province of China 2014JQ2-1003
1. Introduction

Convection-diffusion equations arose from various fields of applied sciences such as heat transfer problems in a draining film  or a nanofluid filled enclosure , radial transport in a porous medium , and water transport in soils  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  and numerical study  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 .

2. Mathematical Formulation

In this paper, we consider a class of nonlinear convection-diffusion equation as follows :(1)ut=xunuxn-1ux+t-n/n+1kuux,x,tG,subject to the following initial value conditions:(2)u0,t=0,t>0;ux,0=1,x>0.where G={(x,t);x>0,t>0}, n is positive power law exponent. Convection functional coefficient k(z) is assumed to be a real-valued continuous differential function defined on [0,1], k(0)=0, k(z)>0 for z>0. D(u)=-unuxn-1ux is the heat density per unit area.

3. Converting into Nonlinear Ordinary Differential Equation Boundary Value Problems

Introduce the following similarity variables :(3)ux,t=uη,η=xt-1/n+1.The transformed convection-diffusion equation together with the initial value conditions given by (1)-(2) can be written as(4)uηnuηn-1uη+ηn+1+kuηuη=0(5)u0=0,u+=1.

Conversely, if u(η) is a solution of (4)-(5), then u(x,t) must be a similarity solution to (1)-(2).

Let u=u(η) be a solution to (4)-(5). If u(η) is strictly increasing in [0,+), then the function η=η(u) inverse to u=u(η) exists. Here η(u)=(u(η))-1 holds on (0,1). Then (4) can be changed into the following nonlinear ordinary differential equation:(6)unηu-n=-1n+1ηu+ku,and integration from u to 1 yields(7)ηu=uu11n+1ηs+ksds-1/n.Set(8)hz=z11n+1ηs+ksds.Combine (7) and (8) to get(9)hz=-1n+1zh-1/nz-kz.The corresponding boundary conditions are(10)h0=0,h1=0.where h(z) 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 ε as follows:(11)P:Lux=fx,x=x1,x2,,xmΩ,ujx=aj,j=0,1,,n-1,xΩΩ.

where L is n order differential operator without small parameter ε, aj is a constant, and Ω is the region containing the origin.

Introduce an embedded parameter transformation as follows:(12)ux=εagξ+k=0n-1uk0k!xk,x=εbξ,where ε is an artificial small parameter, ξ is independent variable parameter, g is dependent variable parameter, and a and b are undetermined constants. Substituting this transformation (12) into (11), we can obtain the following nonlinear initial value problem:(13)P0:Lεgε,ξ=hξ,ξ=ξ1,ξ2,,ξmΩ1,gj0=0,j=0,1,,n-1.

where h(ξ) is a function of ξ, Ω1 is the region containing the origin. The solution of P0 is expressed in form of power series by applying perturbation expansion method(14)gξ,ε=k=0εkgkξk.

Combine (12) and (14) to get the solution u 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 ε, the independent variable x and the unknown function u(x) are transformed simultaneously, and the problem is transformed into a new one related to small parameter ε. Secondly, the transformed new differential equation is expanded in the form of power series of ε 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 g(ξ) of the new problem. Thirdly, by substituting the solution of the new equation into the transformation equation (12), the small parameter ε is eliminated and the solution of the original problem is obtained.

5. Approximate Analytical Results of (<xref ref-type="disp-formula" rid="EEq9">9</xref>)-(<xref ref-type="disp-formula" rid="EEq10">10</xref>) and Discussion

The initial condition is assumed as follows:(15)h0=α.In order to solve (9), we transform the dependent variable and independent variable as follows according to formula (12):(16)hz=ε2gξ+α,z=εξ,where ε is an artificial small parameter.

Let k(z)=z and substituting this transformation (16) into (9)-(10), we can obtain the following nonlinear initial value problems:(17)gξ+1n+1εξε2gξ+α-1/n+1=0,(18)g0=0,g0=0.Expanding [ε2g(ξ)+α]-1/n in a power series development, we can get(19)gξ+1n+1εξ1α1/n-1nα1/n+1ε2gξ+n+12n2α1/n+2ε4g2ξ-n+12n+16n3α1/n+3ε6g3ξ+n+12n+13n+124n4α1/n+4ε8g4ξ++1=0The solution of (19) can be obtained by expanding g(ξ) in a power series development near ε=0 as follows:(20)gξ=g0ξ+εg1ξ+ε2g2ξ+ε3g3ξ+ε4g4ξ+ε5g5ξ+ε6g6ξ+ε7g7ξ+ε8g8ξ+ε9g9ξ+ε10g10ξ+Substituting (20) into (19) and equating the coefficients of εi, we can get the following expressions:(21)ε0:g0ξ+1=0,ε1:g1ξ+ξn+1α1/n=0,ε2:g2ξ=0,ε3:g3ξ-ξg0ξnn+1α1/n+1=0,ε4:g4ξ-ξg1ξnn+1α1/n+1=0,ε5:g5ξ-ξg2ξnn+1α1/n+1+ξg02ξ2n2α1/n+2=0,ε6:g6ξ-ξg3ξnn+1α1/n+1+ξg0ξg1ξn2α1/n+2=0,ε7:g7ξ-ξg4ξnn+1α1/n+1+ξ2g0ξg2ξ+g12ξ2n2α1/n+2-2n+1ξg03ξ6n3α1/n+3=0,ε8:g8ξ-ξg5ξnn+1α1/n+1+ξg0ξg3ξ+g1ξg2ξn2α1/n+2-2n+1ξg02ξg1ξ2n3α1/n+3=0,ε9:g9ξ-ξg6ξnn+1α1/n+1+ξ2g0ξg4ξ+2g1ξg3ξ+g22ξ2n2α1/n+2-2n+1ξg02ξg2ξ+g0ξg12ξ2n3α1/n+3+2n+13n+1ξg04ξ24n4α1/n+4=0,ε10:g10ξ-ξg7ξnn+1α1/n+1+ξg0ξg5ξ+g1ξg4ξ+g2ξg3ξn2α1/n+2-2n+1ξ3g02ξg3ξ+6g0ξg1ξg2ξ+g13ξ6n3α1/n+3+2n+13n+1ξg03ξg1ξ6n4α1/n+4=0,.

Then, we can obtain gi(ξ)(i=0,1,2,) in the following.(22)g0ξ=-12ξ2,g1ξ=-16n+1α1/nξ3,g2ξ=0,g3ξ=-140nn+1α1/n+1ξ5,g4ξ=-1180nn+12α2/n+1ξ6,g5ξ=-1336n2α1/n+2ξ7,g6ξ=-10n+136720n2n+12α2/n+2ξ8,g7ξ=-5n+725920n2n+13α3/n+2+2n+13456n3α1/n+3ξ9,g8ξ=-70n+61151200n+1n3α2/n+3ξ10,g9ξ=-280n2+590n+3192217600n3n+13α3/n+3+6n2+5n+142240n4α1/n+4ξ11,g10ξ=-15120n2+18636n+607895800320n+1n4α2/n+4+1200n2+2670n+1530102643200n3n+14α4/n+3ξ12,.

Combine (16), (20), and the above gi(ξ)(i=0,1,2,…) to get solution h(z) of (9)-(10).(23)h=α-12z2-16n+1α1/nz3-140nn+1α1/n+1z5-1180nn+12α2/n+1z6-1336n2α1/n+2z7-10n+136720n2n+12α2/n+2z8-5n+725920n2n+13α3/n+2+2n+13456n3α1/n+3z9-70n+61151200n+1n3α2/n+3z10-280n2+590n+3192217600n3n+13α3/n+3+6n2+5n+142240n4α1/n+4z11-15120n2+18636n+607895800320n+1n4α2/n+4+1200n2+2670n+1530102643200n3n+14α4/n+3z12+

Similarly, let k(z)=z2; we obtain approximate analytical solution of (9)-(10) as follows:(24)h=α-13+16n+1α1/nz3-190nn+1α1/n+1+1180nn+12α2/n+1z6-11296n2α1/n+2+5n+66480n2n+12α2/n+2+5n+725920n2n+13α3/n+2z9-2n+121384n3α1/n+3+120n+89855360n+1n3α2/n+3+60n2+119n+60855360n3n+13α3/n+3+40n2+89n+513421440n3n+14α4/n+3z12+

Let k(z)=z3; approximate analytical solution of (9)-(10) is presented as follows:(25)h=α-16n+1α1/nz3-14z4-1180nn+12α2/n+1z6-1168nn+1α1/n+1z7-5n+725920n2n+13α3/n+2z9-7n+815120n2n+12α2/n+2z10-13520n2α1/n+2z11-40n2+89n+513421440n3n+14α4/n+3z12+

It is obvious that we can promptly obtain the value of α by applying h(1)=0 in (23)-(25) for each fixed n. Based on (23)-(25) and the corresponding α we can easily obtain the graph of the heat diffusion flux distribution for different n and k(z). The results are presented in Tables 15 and Figures 15.

Values of α for n=0.2, 0.6, 1.0, 2.5, 5.0, 10.0, and k(z)=z.

n k ( z ) α
0.2 z 1.013
0.6 z 0.7815
1.0 z 0.6792
2.5 z 0.5701
5.0 z 0.5342
10.0 z 0.5169

Values of α for n=0.2,0.6,1.0,2.5,5.0,10.0, and k(z)=z2.

n k ( z ) α
0.2 z 2 0.9122
0.6 z 2 0.6532
1.0 z 2 0.5352
2.5 z 2 0.4090
5.0 z 2 0.3691
10.0 z 2 0.3506

Values of α for n=0.2,0.6,1.0,2.5,5.0,10.0, and k(z)=z3.

n k ( z ) α
0.2 z 3 0.8643
0.6 z 3 0.5927
1.0 z 3 0.4670
2.5 z 3 0.3300
5.0 z 3 0.2870
10.0 z 3 0.2676

Values of α for n=2.0 and k(z)=z,z2,z3.

n k ( z ) α
2.0 z 0.5885
2.0 z 2 0.4300
2.0 z 3 0.3529

Values of α for n=8.0 and k(z)=z,z2,z3.

n k ( z ) α
8.0 z 0.5212
8.0 z 2 0.3551
8.0 z 3 0.2723

Flux distribution for n=0.2,0.6,1.0,2.55.0,10.0,k(z)=z.

Flux distribution for n=0.2,0.6,1.0,2.55.0,10.0,k(z)=z2.

Flux distribution for n=0.2,0.6,1.0,2.55.0,10.0,k(z)=z3.

Flux distribution for n=2.0,k(z)=z,z2,z3.

Flux distribution for n=8.0,k(z)=z,z2,z3.

It can be seen from Figures 13 that the heat diffusion flux h(z) decreases with increase of n for specific k(z), the physical meaning is that heat diffusion flux h(z) is a decreasing function of n, which means that the profiles exhibited by a big power law index n possess a smaller diffusion. Figures 4 and 5 indicate the heat diffusion flux h(z) decrease with the decrease of k(z).

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.

Comparison of h(z) for n=10 and k(z)=z.

z D P T P E M b v p 4 c
0.0 0.5169 0.5169
0.1 0.5119 0.5119
0.2 0.4968 0.4968
0.3 0.4715 0.4715
0.4 0.4359 0.4359
0.5 0.3899 0.3899
0.6 0.3334 0.3334
0.7 0.2663 0.2663
0.8 0.1884 0.1884
0.9 0.0997 0.0997
1.0 0 0

Comparison of h(z) for n=2 and k(z)=z3.

z D P T P E M b v p 4 c
0.0 0.3529 0.3551
0.1 0.3528 0.3550
0.2 0.3518 0.3540
0.3 0.3483 0.3506
0.4 0.3405 0.3427
0.5 0.3255 0.3278
0.6 0.3000 0.3023
0.7 0.2601 0.2624
0.8 0.2008 0.2031
0.9 0.1164 0.1186
1.0 0 0

The comparison of the results of approximate analytical solution and numerical solution for n=10.0,k(z)=z.

The comparison of the results of approximate analytical solution and numerical solution for n=2,k(z)=z3.

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 k(z) and power law index n on transfer behavior were presented. The results show that heat diffusion flux h(z) is an increasing function of convection functional coefficient k(z) and a decreasing function of power law index n.

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).

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