Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 232698, 9 pages

doi:10.1155/2012/232698

Research Article

Using Lie Symmetry Analysis to Solve a Problem That Models Mass Transfer from a Horizontal Flat Plate

W. Sinkala and M. Chaisi

Department of Applied Mathematics, Walter Sisulu University, Private Bag X1, Mthatha 5117, South Africa

Received 16 July 2012; Revised 5 November 2012; Accepted 30 November 2012

Academic Editor: Hung Nguyen-Xuan

Copyright © 2012 W. Sinkala and M. Chaisi. 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.

Abstract.

We use Lie symmetry analysis to solve a boundary value problem that arises in chemical engineering, namely, mass transfer during the contact of a solid slab with an overhead flowing fluid. This problem was earlier tackled using Adomian decomposition method (Fatoorehchi and Abolghasemi 2011), leading to the Adomian series form of solution. It turns out that the application of Lie group analysis yields an elegant form of the solution. After introducing the governing mathematical model and some preliminaries of Lie symmetry analysis, we compute the Lie point symmetries admitted by the governing equation and use these to construct the desired solution as an invariant solution.

1. Introduction

A simplified model for mass transfer phenomenon from a horizontal flat plate fixed along a laminar fluid flow is considered [1]. Imagine a solid slab placed horizontally on the -axis as shown in Figure 1. The free stream velocity is denoted by (m/s), is the initial concentration of molecules of species (mol/m³), is the concentration of molecules of species at the plate interface (mol/m³), is the momentum (or hydrodynamic) boundary layer thickness (m), and is the concentration boundary layer thickness (m). Molecules of species from the solid slab diffuse along the -axis only to be swept downstream by the fluid flow in the hydrodynamic boundary layer. Subject to a number of assumptions (given in [1]), mass balance for species over an infinitesimal element based on the Cartesian coordinates leads to the PDE

for the concentration of molecules of species , with the associated boundary conditions

where , and and are constants. Precisely

where is the diffusion coefficient (m²/s) and is the kinematic viscosity (m²/s). The rest of the paper is organised as follows. In Section 2 we present preliminaries of Lie symmetry analysis. Section 3 is the thrust of the paper. In this section we determine Lie point symmetries admitted by the governing PDE and subsequently the basis generator of the one-dimensional Lie symmetry algebra admitted by the boundary value problem (BVP). We then construct the solution to the BVP as an invariant solution. We give concluding remarks in Section 4.

Figure 1: Molecules of species from the solid slab diffuse along the -axis only to be swept downstream by fluid flow in the hydrodynamic boundary layer.

2. Lie Symmetry Analysis of Differential Equations

There are many good introductions to Lie symmetry analysis of differential equations [2–6]. We will however indulge in a little introduction of the subject for completeness and to whet the appetite of the readers who could be new to the subject. Consider a scalar PDE of order ,

with independent variables, , and one dependent variable , where represents all the th order partial derivatives of with respect to . An element of the set represented by is denoted by , for . We want to define invariance of (2.1) under a one-parameter Lie group of transformations in the parameter

, acting on the -space. The group (2.2) has as its infinitesimal generator

where

and the th extension given by

where and for . The th extension of the group (2.2) is generated by the following (th extended) infinitesimal generator:

with the explicit formulas for the extended infinitesimals given recursively by

for with , where is the total derivative operator defined by

with summation over a repeated index.

We say that (2.1) admits (or is invariant under) the Lie group of point transformations (2.2) if (2.1) has the same form in the new variables and , that is,

When this is the case we sometimes (loosely) simply say that (the infinitesimal generator defined in (2.3)) is a symmetry of (2.1). Invariance of a differential equation under a given Lie group of transformations is neatly characterised by the infinitesimal criterion. Equation (2.1) is invariant under the Lie group of transformations (2.2) if and only if

The infinitesimal criterion (2.10) provides a key to the explicit determination of symmetry groups admitted by differential equations. Using a straightforward algorithm based on (2.10) one obtains infinitesimals of the Lie group of point transformations that leaves a given differential equation invariant. The tedious algebraic calculations involved in this process are today done easily, often automatically, thanks to powerful Computer Algebra Systems (CAS) like Maple and Mathematica, Maxima and Reduce, and the many specific packages for performing symmetry analysis of differential equations [7–13].

An admissible Lie group characterises symmetry properties of a differential equation and is used for, among other things, complete integration (in the case of ODEs) or construction of special exact solutions of the differential equation (see, e.g., [14–17]). For boundary (initial) value problems (BIVPs) there is a principle, the invariance principle, which states that if a BIVP is invariant under a given group, then one should seek the solution to the problem among the functions invariant under the admitted group. To state this principle concretely consider a BIVP

where is a given manifold. Suppose that (2.11) admits one-parameter symmetries

We say that (2.11)-(2.12) is invariant under a symmetry

for some constants , provided that

(i)the manifold is invariant under ;(ii)the boundary (initial) condition is invariant under restricted to .

We now state the invariance principle [15, 18]: if the BIVP (2.11)-(2.12) admits one-parameter symmetries , one should seek the solution of the problem among the functions invariant under . Successful application of the invariance principle to solve BIVPs has been reported in a number of papers (see, e.g., [17, 19–21]).

3. Solution of the BVP (1.1)-(1.2)

Using LIE [8, 13] we obtain that (1.1) admits an infinite dimensional Lie symmetry algebra spanned by the following operators:

where is an arbitrary solution of (1.1).

In the light of the invariance principle, the starting point in the “search” for the solution of (1.1)-(1.2) is the determination of the subalgebra of the Lie algebra spanned by the symmetries in (3.1) that leaves the boundary conditions (1.2) invariant. To do this we construct a special linear combination,

of the symmetries in (3.1), with the constants ’s suitably chosen so that the boundary conditions (1.2) are invariant under (3.3), that is, we require that

From (3.4) we obtain that

from which it follows that . Therefore (3.3) reduces to

the general form of the symmetry admitted by both the PDE (1.1) and the boundary conditions (1.2), where is an arbitrary constant. We can now assume the existence of an invariant solution of the BVP (1.1)-(1.2), the form of which is determined by functions invariant under the group generated by (3.6). The characteristic equations of (3.6),

yield the integrals

from the first and second equations, respectively. These integrals are the similarity variables of the BVP and the invariant solution is expressed in terms of them in the form

or

where is an arbitrary function. Upon substituting (3.10) into (1.1) we obtain the following second-order linear ODE:

The solution of (3.11) is

where and are arbitrary constants and is the incomplete gamma function defined by

We now have that all invariant solutions of (1.1) arising from (3.6) are embedded in the family of solutions (3.10). The values of and in (3.12) are determined from the boundary conditions (1.2):

where

We now solve the equations (3.14) simultaneously for and . We obtain that and are constants only if we set

in which case

Finally from (3.10), (3.12), and (3.17) we have that the BVP (1.1)-(1.2) is solved by

4. Concluding Remarks

When a PDE is richly endowed with Lie point symmetries (i.e., when the PDE admits the Lie symmetry algebras or , where is the Heisenberg-Weyl [19]) the symmetry analysis approach usually provides a neat solution (when it exists) to the associated BIVP. Algorithms for obtaining admitted Lie point symmetries and for using the symmetries to construct invariant solutions are well developed and quite routine. It is instructive to compare the solution obtained in this paper, (3.18), with the series-form solution obtained by Fatoorehchi and Abolghasemi [1] via Adomian decomposition:

where ’s and are “suitable” constants. As reported in [1] one can also solve (1.1)-(1.2) by the combination of variables method, the starting point of which is the assumption that

for suitable numbers and and some function . This approach also leads to the exact solution

We remark here that Fatoorehchi and Abolghasemi [1] only determine the constants ’s and by comparing (4.1) with the exact solution (4.3). These constants are determined to be

where Sc is the Schmidt number.

Finally, it is noteworthy that in the combination of variables approach to the solution of (1.1)-(1.2) it turns out that and [1]. This makes (3.10) and (4.2) equivalent in the light of (3.16).

Acknowledgments

The authors thank the Directorate of Research Development at Walter Sisulu University for its continued support and the anonymous referees for useful comments that helped to improve the paper.

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