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

Authors: W. Sinkala; M. Chaisi
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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 x-axis as shown in Figure 1. The free stream velocity is denoted by u∞ (m/s), CA0 is the initial concentration of molecules of species A (mol/m3), CAi is the concentration of molecules of species A at the plate interface (mol/m3), δ is the momentum (or hydrodynamic) boundary layer thickness (m), and δc is the concentration boundary layer thickness (m). Molecules of species A from the solid slab diffuse along the y-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 A over an infinitesimal element based on the Cartesian coordinates leads to the PDE
(1.1)xCyy=αyCx,
for the concentration C of molecules of species A, with the associated boundary conditions
(1.2)BC1:C(x,y)=CA0wheny=δ(x)BC2:C(x,0)=CAi,
where δ(x)=x/β, and α and β are constants. Precisely
(1.3)α=3u∞1.59.28Dv0.5,β=2D3u∞α,
where D is the diffusion coefficient (m2/s) and v 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 speciesA from the solid slab diffuse along the y-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 k,
(2.1)F(x,u,u(1),u(2),…,u(k))=0,
with n independent variables, x=(x1,x2,…,xn), and one dependent variable u, where u(j) represents all the jth order partial derivatives of u with respect to x. An element of the set represented by u(j) is denoted by ui1i2,…,ij=∂u/∂xi1∂xi2,…,∂xij, ij=1,2,…,n for j=1,2,…,k. We want to define invariance of (2.1) under a one-parameter Lie group of transformations in the parameter ε(2.2)x~i=Xi(x,u;ε)=εξi(x,u)+O(ε2),u~=U(x,u;ε)=εη(x,u)+O(ε2),i=1,2,…,n, acting on the (x,u)-space. The group (2.2) has as its infinitesimal generator
(2.3)X=ξi(x,u)∂xi+η(x,u)∂u,
where
(2.4)ξi(x,u)=∂Xi∂ε(x,u;ε)|ε=0,i=1,2,…,n,η(x,u)=∂Ui∂ε(x,u;ε)|ε=0,
and the kth extension given by
(2.5)u~i1i2…x~i=Xi(x,u;ε)=εξi(x,u)+O(ε2),u~i1i2…u~=U(x,u;ε)=εη(x,u)+O(ε2)u~i1i2…iku~u~⋮u~i1i2…ik=Ui1i2⋯ik(x,u,u(1),u(2),…,u(k);ε)u~i1i2…ik=ui1i2⋯ik+εηi1i2⋯ik(k)(x,u,u(1),u(2),…,u(k))+O(ε2),
where i=1,2,…,n and ij=1,2,…,n for j=1,2,…,k. The kth extension of the group (2.2) is generated by the following (kth extended) infinitesimal generator:
(2.6)X(k)=ξi(x,u)∂xi+η(x,u)∂u+ηi(1)(x,u,u(1))∂ui+⋯+ηi1i2⋯ik(k)∂i1i2⋯ik,
with the explicit formulas for the extended infinitesimals given recursively by
(2.7)ηi(1)=Diη-(Diξj)uj,i=1,2,…,n,ηi1i2…ik-1(k)=Dikηi1i2…ik(k-1)-(Dikξj)ui1i2…ik-1j,ij=1,2,…,n for j=1,2,…,k with k≥2, where Di is the total derivative operator defined by
(2.8)Di=DDxi=∂∂xi+ui∂∂u+uij∂∂uj+⋯+uii1i2⋯in∂∂ui1i2⋯in+⋯,
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 x~=(x~1,x~2,…,x~n) and u~, that is,
(2.9)F(x~,u~,u~(1),u~(2),…,u~(k))=0.
When this is the case we sometimes (loosely) simply say that X (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
(2.10)X(k)F(x,u,u(1),u(2),…,u(k))=0whenF(x,u,u(1),u(2),…,u(k))=0.
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
(2.11)F(x,u,u(1),u(2),…,u(k))=0,(2.12)u|s=h(x),
where s is a given manifold. Suppose that (2.11) admits m one-parameter symmetries
(2.13)Xi=ξj(x,u)∂xj+η(x,u)∂u,i=1,…,m.
We say that (2.11)-(2.12) is invariant under a symmetry
(2.14)X=∑i=1mεiXi,
for some constants εi, provided that(i)
the manifolds is invariant under X;(ii)
the boundary (initial) conditionu|s=h(x) is invariant under X restricted to s.We now state the invariance principle [15, 18]: if the BIVP (2.11)-(2.12) admits one-parameter symmetries Xμ, one should seek the solution of the problem among the functions invariant under Xμ. 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:
(3.1)X1=(1x)∂x,X2=y∂y+2x∂x,X3=∂C,X4=yx3/2∂y+x5/2∂x-C(αy34+x3/2)∂C,(3.2)Xϕ=ϕ(x,y)∂C,
where ϕ(x,y) 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,
(3.3)X=∑i=14εiXi,
of the symmetries in (3.1), with the constants εi’s suitably chosen so that the boundary conditions (1.2) are invariant under (3.3), that is, we require that
(3.4)X(C-CA0)=0wheny=δ(x),X(C-CAi)=0wheny=0.From (3.4) we obtain that
(3.5)ε3-ε4(1+α4β3)x3/2=0,
from which it follows that ε3=ε4=0. Therefore (3.3) reduces to
(3.6)Γ=(2x+κx)∂x+y∂y,
the general form of the symmetry admitted by both the PDE (1.1) and the  boundary conditions (1.2), where κ is an arbitrary constant. We can now assume the existence of an invariant solution C=ϕ(x,y) 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),
(3.7)dx2x+κ/x=dyy=dC0,
yield the integrals
(3.8)J1=y32x3/2+κ,J2=C,
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
(3.9)J2=ϕ(J1),
or
(3.10)C(x,y)=ϕ(y32x3/2+κ),
where ϕ is an arbitrary function. Upon substituting (3.10) into (1.1) we obtain the following second-order linear ODE:
(3.11)3zϕ′′(z)+(2+αz)ϕ′(z)=0.
The solution of (3.11) is
(3.12)ϕ(z)=K1+K2Γ(13,αz3),
where K1 and K2 are arbitrary constants and Γ(a,z) is the incomplete gamma function defined by
(3.13)Γ(a,z)=∫z∞ta-1e-tdt.
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 K1 and K2 in (3.12) are determined from the  boundary conditions (1.2):
(3.14)C(x,y)|y=δ(x)=CA0⇒K1+K2Γ(13,φ(x))=CA0,C(x,0)=CAi⇒K1+K2Γ(13)=CAi,
where
(3.15)φ(x)=αx3/23β3(2x3/2+κ),Γ(z)=∫0∞tz-1e-tdt.
We now solve the equations (3.14) simultaneously for K1 and K2. We obtain that K1 and K2 are constants only if we set
(3.16)κ=0,
in which case
(3.17)K1=CA0Γ(1/3)-CAiΓ(1/3,α/6β3)Γ(1/3)-Γ(1/3,α/6β3),K2=CA0-CAiΓ(1/3,α/6β3)-Γ(1/3).
Finally from (3.10), (3.12), and (3.17) we have that the BVP (1.1)-(1.2) is solved by
(3.18)C(x,y)=CA0Γ(1/3)+(CAi-CA0)Γ(1/3,αy3/6x3/2)-CAiΓ(1/3,α/6β3)Γ(1/3)-Γ(1/3,α/6β3).


## 4. Concluding Remarks

When a PDE is richly endowed with Lie point symmetries (i.e., when the PDE admits the Lie symmetry algebrassl(2,ℝ)⨁sA1 or sl(2,ℝ)⨁sW, where W 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:
(4.1)C(x,y)-CAiCA0-CAi=λβ(w0yx-0.5-α24w1y4x-2+α2504w2y7x-3.5-α312960w3y10x-5λβ+α4404352w1y13x-6.5-α514929920w5y16x-8+⋯),
where wi’s and λ 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
(4.2)C(x,y)=ψ(ynxm)
for suitable numbers m and n and some function  ψ. This approach also leads to the exact solution
(4.3)C(x,y)-CAiCA0-CAi=α/33Γ(4/3)(yx-0.5-αy4x-23×4+α2y7x-3.59×2!×7-α3y10x-527×3!×10α/33Γ(4/3)+α4y13x-6.581×4!×13-α5y16x-8243×5!×16+⋯).
We remark here that Fatoorehchi and Abolghasemi [1] only determine the constants wi’s and λ by comparing (4.1) with the exact solution (4.3). These constants are determined to be
(4.4)λ=4.64/9.283Γ(4/3)Sc1/3,wi=2i,
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 m=-1/2 and n=1 [1]. This makes (3.10) and (4.2) equivalent in the light of (3.16).



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