Lie Group Analysis of Mixed Convection Flow with Mass Transfer over a Stretching Surface with Suction or Injection

Abstract

The mixed convection flow with mass transfer over a stretching surface with suction or injection is examined. By using Lie group analysis, the symmetries of the equations are calculated. A four-finite parameter and one infinite parameter Lie group transformations are obtained. Two different cases are discussed, one for the scaling symmetry and the other for spiral symmetry. The governing partial differential equations are transformed into ordinary differential equations using these symmetries. It has been noted that the similarity variables and functions available in the literature become special cases of the similarity variables and functions discussed in this paper.

1. Introduction

The study of continuously stretching sheets has many applications in manufacturing industries. Application of stretching sheets can be found in the areas like paper production, hot rolling, glass blowing, continuous casting of metals, and wire drawing. First of all Sakiadis [1, 2] investigated the boundary layer behavior on stretching surfaces and presented numerical solution for the sheet having constant speed. Extension to this problem where velocity is proportional to the distance from the slit was given by Crane [3]. Flow and heat transfer in the boundary layer on stretching surface was studied by Tsou et al. [4]. Fox et al. [5] presented different methods (analytical or numerical) for solving problems of stretching sheet with suction and injection. Heat and mass transfer on stretched surface with suction and injection was introduced by Erickson et al. [6]. P. S. Gupta and A. S. Gupta [7] studied the same problem for linearly stretching sheet. Heat transfer past a moving continuous plate with variable temperature was studied by Soundalgekar and Murty [8] and Grubka et al. [9]. Ali [10] presented similarity solutions for stretched surface with suction and injection. Hayat et al. [11] investigated the effect of heat and mass transfer for Soret and Dufour's effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid.

Lie group analysis is a classical method discovered by Norwegian mathematician Sophus Lie for finding invariant and similarity solutions [12–15]. Yürüsoy and Pakdemirli [16] presented exact solution of boundary layer equations of a special non-Newtonian fluid over a stretching sheet by the method of Lie group analysis. They extended their work to creeping flow of second-grade fluid [17]. Sivasankaran et al. [18, 19] studied the problem of natural convection heat and mass transfer flow past an inclined plate for various parameters using Lie group analysis without and with heat generation.

In this paper, Lie group analysis of mixed convection flow with mass transfer over a stretching surface with suction or injection is studied. Here the application of Lie group analysis for the fluid flow problem is discussed, with the help of examples from the literature, in such a way that one can easily understand its importance for finding similarity variables and functions. There are two important features of this work that make it different from the other. First, instead of taking a specific form of the boundary conditions (i.e., stretching velocity of the surface, wall temperature, and concentration), these are taken as arbitrary functions. Second, symmetry operator is applied not only to the equations of the problem but also to the boundary conditions of the problem in such a way that with infinite parameter Lie group transformations those finite parameters from infinitesimals are also discarded under which boundary conditions fail to be invariant. It has been wiliness that the number of symmetries reduce in order to comply with the invariance of boundary conditions. It is shown that mixed convection flow with mass transfer is possible only for two types of vertical stretching, the polynomial stretching and the exponential stretching. From examples it is shown that the similarities of the problems discussed by Ali and Al-Yousef [20], Yih [21], and Chen [22] become special subcases of the case of polynomial stretching and the problem discussed by Partha et al. [23] becomes a special subcase of the exponential stretching case. In the case of polynomial stretching a new example is included.

2. Equations of Motion

Consider a two-dimensional laminar flow of a steady incompressible viscous fluid over a permeable surface stretching with velocity 𝑢_𝑤(𝑥). The temperature distribution and concentration at the surface are 𝑇_𝑤(𝑥) and 𝐶_𝑤(𝑥), respectively. The surface is moving through a quiescent ambient fluid with constant temperature 𝑇_∞ and constant concentration 𝐶_∞. 𝑣_𝑤(𝑥) is the suction or injection velocity through the surface. Here we do not assume a specific form of velocity, temperature, and concentration variations for the stretching surface. Using Boussinesq approximations, the governing equations for the boundary layer flows over a stretching surface with suction or injection are expressed as𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦𝑢𝜕𝑢=0,𝜕𝑥+𝑣𝜕𝑢𝜕𝑦𝜕=𝜈²𝑢𝜕𝑦²+𝑔𝛽(𝑇−𝑇_∞)+𝑔𝛽^∗(𝐶−𝐶_∞𝑢𝜕𝑇),𝜕𝑥+𝑣𝜕𝑇𝜕𝑦𝜕=𝛼²𝑇𝜕𝑦²,𝑢𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝑦𝜕=𝐷²𝐶𝜕𝑦², (2.1)(1)(2)(3) along with the following boundary conditions:𝑢=𝑈₀𝑢_𝑤(𝑥),𝑇−𝑇_∞=𝑇₀𝑇_𝑤(𝑥),𝑣=𝑉₀𝑣_𝑤(𝑥),𝐶−𝐶_∞=𝐶₀𝐶_𝑤(𝑇𝑥)at𝑦=0,𝑢→0,𝑇→_∞,𝐶→𝐶_∞as𝑦→∞, (2.2) where 𝑢 and 𝑣 are the components of velocity in 𝑥 and 𝑦 directions, 𝜈 is the fluid kinematic viscosity, 𝑔 is the gravitational acceleration, 𝛽 is the coefficient of thermal expansion, 𝛽^∗ is the coefficient of concentration expansion, 𝛼 is the thermal diffusivity, and 𝐷 is the mass diffusivity.𝑈₀ and 𝑉₀ are the reference velocities, whereas 𝑇₀ and 𝐶₀ are the reference temperature and mass concentration, respectively.

Introducing the nondimensional parameters𝑥𝑥=𝐿𝑦,𝑦=𝐿(𝑈₀𝐿𝜈)^1/2𝑢,𝑢=𝑈₀,𝑣𝑣=𝑈₀(𝑈₀𝐿𝜈)^1/2(𝑇,𝑇=𝑇−_∞)𝑇₀(𝐶,𝐶=𝐶−_∞)𝐶₀, (2.3) in (2.1), one gets𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+v𝜕𝑢=𝜕𝜕𝑦²𝑢𝜕𝑦²+𝑔𝛽𝑇₀𝐿𝑈²₀𝑇+𝑔𝛽^∗𝐶₀𝐿𝑈²₀𝑢𝐶,𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=1𝜕𝑦𝜕Pr²𝑇𝜕𝑦²𝑢,𝜕𝐶𝜕𝑥+𝑣𝜕𝐶=1𝜕𝑦𝜕Sc²𝐶𝜕𝑦², (2.4)(2.5)(2.6)(2.7) where Pr=𝜈/𝛼 is the Prandtl number and Sc=𝜈/𝐷 is the Schmidt number. Boundary conditions for the above equations are𝑦=0;𝑢=𝑢_𝑤𝑉(𝑥),𝑣=⁰𝑣_𝑤(𝑥)𝑈₀(𝑈₀𝐿𝜈)^1/2,𝑇=𝑇_𝑤(𝑥),𝐶=𝐶_𝑤(𝑥),𝑦→∞;𝑢=0,𝑇=0,𝐶=0. (2.8)

3. Symmetries of the Problem

By applying Lie group method [12] to (2.4)–(2.7), the infinitesimal generator for the problem can be written as𝑋=𝜉₁𝜕𝜕𝑥+𝜉₂𝜕𝜕𝑦+𝜙₁𝜕𝜕𝑢+𝜙₂𝜕𝜕𝑣+𝜙₃𝜕𝜕𝑇+𝜙₄𝜕.𝜕𝐶 (3.1)

One requires that the equations remain invariant under the infinitesimal Lie point transformations𝑥^∗=𝑥+𝜖𝜉₁(𝑥,𝑦,𝑢,𝑣,𝑇,𝐶)+𝑂(𝜖²𝑦),_∗=𝑦+𝜖𝜉₂(𝑥,𝑦,𝑢,𝑣,𝑇,𝐶)+𝑂(𝜖²𝑢),_∗=𝑢+𝜖𝜙₁(𝑥,𝑦,𝑢,𝑣,𝑇,𝐶)+𝑂(𝜖²𝑣),_∗=𝑣+𝜖𝜙₂(𝑥,𝑦,𝑢,𝑣,𝑇,𝐶)+𝑂(𝜖²𝑇),_∗=𝑇+𝜖𝜙₃(𝑥,𝑦,𝑢,𝑣,𝑇,𝐶)+𝑂(𝜖²𝐶),_∗=𝐶+𝜖𝜙₄(𝑥,𝑦,𝑢,𝑣,𝑇,𝐶)+𝑂(𝜖²). (3.2)

By employing a tedious and straightforward algebra, the form of the infinitesimals is𝜉₁=𝑐+𝑑𝑥,𝜉₂=𝑏𝑦+𝛾(𝑥),𝜙₁𝜙=(−2𝑏+𝑑)𝑢,₂=−𝑏𝑣+𝑢𝛾^′(𝑥),𝜙₃𝜙=−𝑒+(−4𝑏+𝑑)𝑇,₄=𝛽𝑇₀𝛽^∗𝐶₀𝑒+(−4𝑏+𝑑)𝐶. (3.3)

Therefore, the equations admit four finite parameter Lie group transformations. 𝛾(𝑥) is infinite parameter Lie group transformation. Parameter 𝑐 corresponds to the translation in the variable 𝑥, and parameter 𝑒 corresponds to the translation in the variables 𝑇 and 𝐶. Parameter 𝑏 corresponds to scaling in the variables 𝑦, 𝑢, 𝑣, 𝑇, and 𝐶. Parameter 𝑑 corresponds to the scaling in the variables 𝑥, 𝑢, 𝑇, and 𝐶. In the following sections, similarity variables and functions corresponding to the above symmetries will be derived.

Before finding similarity variables and functions for all of the symmetries of the problem, the symmetry operator 𝑋 will be applied to the boundary conditions and then only those symmetries will be discussed that leave the boundary conditions invariant.

4. Invariance of the Boundary Conditions

By applying the symmetry operator 𝑋, that is defined in (3.1), on boundary conditions, we get differential equations𝜕𝑢_𝑤−𝜕𝑥(−2𝑏+𝑑)𝑢(𝑐+𝑑𝑥)^𝑤=0,𝜕𝑣_𝑤+𝑏𝜕𝑥𝑣(𝑐+𝑑𝑥)^𝑤=0,𝜕𝑇_𝑤−𝜕𝑥(−4𝑏+𝑑)𝑇(𝑐+𝑑𝑥)^𝑤=0,𝜕𝐶_𝑤−𝜕𝑥(−4𝑏+𝑑)𝐶(𝑐+𝑑𝑥)^𝑤=0, (4.1) with some restrictions 𝑒=𝛾(𝑥)=0.

Therefore, the possible boundary conditions for the current problem are as follows: for 𝑑≠0,𝑢_𝑤(𝑥)=𝐴₁(𝑐+𝑑𝑥)^{(−2𝑏+𝑑)/𝑑},𝑣_𝑤(𝑥)=𝐴₂(𝑐+𝑑𝑥)^−𝑏/𝑑,𝑇_𝑤(𝑥)=A₃(𝑐+𝑑𝑥)^{(−4𝑏+𝑑)/𝑑},𝐶_𝑤(𝑥)=𝐴₄(𝑐+𝑑𝑥)^{(−4𝑏+𝑑)/𝑑}. (4.2) For 𝑑=0 and 𝑐≠0,𝑢_𝑤(𝑥)=𝐴₅exp(−2𝑏𝑐𝑥),𝑣_𝑤(𝑥)=𝐴₆𝑏exp(−𝑐𝑇𝑥),_𝑤(𝑥)=𝐴₇exp(−4𝑏𝑐𝑥),𝐶_𝑤(𝑥)=𝐴₈exp(−4𝑏𝑐𝑥), (4.3) where 𝐴_𝑖 for 𝑖=1,2,3,…,8 are constants of integration.

5. Determining Similarity Variables and Functions

From (4.2) and (4.3) it is shown that only two types of vertical stretching are possible: polynomial stretching for 𝑑≠0 and exponential stretching for 𝑑=0. In the following work each case will be described with examples.

5.1. Case  1: For 𝑑≠0

In this case, the parameters 𝑏, 𝑐, and 𝑑 are taken as arbitrary and 𝑒=𝛾(𝑥)=0 is considered. This case is combination of scaling and translational symmetry. For 𝑐=0 it becomes a scaling symmetry.

The characteristic equations for obtaining the similarity transformations would then be𝑑𝑥=(𝑐+𝑑𝑥)𝑑𝑦=𝑏𝑦𝑑𝑢=(−2𝑏+𝑑)𝑢𝑑𝑣=−𝑏𝑣𝑑𝑇=(−4𝑏+𝑑)𝑇𝑑𝐶.(−4𝑏+𝑑)𝐶 (5.1) The resulting similarity variable is𝑦𝜂=𝐾₁^1/2(𝑐+𝑑𝑥)^𝑏/𝑑, (5.2) and the similarity functions are𝑢=(𝑐+𝑑𝑥)^{(−2𝑏+𝑑)/𝑑}𝐹^′(𝜂),𝑣=(𝑐+𝑑𝑥)^−𝑏/𝑑𝐻(𝜂),𝑇=(𝑐+𝑑𝑥)^{(−4𝑏+𝑑)/𝑑}𝜃(𝜂),𝐶=(𝑐+𝑑𝑥)^{(−4𝑏+𝑑)/𝑑}𝜑(𝜂). (5.3) From (4.2) and (5.3), we have for 𝐴_𝑖=1  (𝑖=1,2,3,4)𝑢=𝑢_𝑤(𝑥)𝐹^′(𝜂),𝑣=−𝑣_𝑤(𝑥)𝐻(𝜂),𝑇=𝑇_𝑤(𝑥)𝜃(𝜂),𝐶=𝐶_𝑤(𝑥)𝜑(𝜂). (5.4)

Using the continuity equation we can define stream function 𝜓 such that𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓.𝜕𝑥 (5.5) Integrating (5.5) by using (5.3), we obtain𝜓=𝐾₁^1/2(𝑐+𝑑𝑥)^{(−𝑏+𝑑)/𝑑}𝐹(𝜂). (5.6) By putting value of 𝜓 from (5.6) into (5.5), then comparing with (5.3) the form of 𝐻(𝜂) becomes𝐻(𝜂)=𝐾₁^1/2[(−𝑏+𝑑𝑑)𝐹−𝑏𝜂𝐹^′]. (5.7)

Substituting the above new variables in (2.5)–(2.7), we get the following system of ordinary differential equations:𝐹^′′′+𝐾₁(−𝑏+𝑑)𝑑𝐹𝐹^′′−𝐾₁(−2𝑏+𝑑)𝑑𝐹^′2+𝐾₁GrRe²𝜃+𝐾₁𝐺𝑐Re²1𝜑=0,𝜃Pr^′′+𝐾₁(−𝑏+𝑑)𝑑𝐹𝜃^′−𝐾₁(−4𝑏+𝑑)𝑑𝐹^′1𝜃=0,𝜑Sc^′′+𝐾₁(−𝑏+𝑑)𝑑𝐹𝜑^′−𝐾₁(−4𝑏+𝑑)𝑑𝐹^′𝜑=0, (5.8)(5.9)(5.10) and boundary conditions are transformed to𝐹^′(0)=1,𝐹(0)=−𝑑^∗1𝐾₁^1/2((−𝑏+𝑑)/𝑑),𝜃(0)=1,𝜑(0)=1,𝐹^′(∞)=0,𝜃(∞)=0,𝜑(∞)=0, (5.11) where Gr=𝑔𝛽𝑇₀𝑇_𝑤(𝑥)𝐿³(𝑐+𝑑𝑥)³/𝜈² is the thermal Grashof number, Re=𝑈₀𝑢_𝑤(𝑥)𝐿(𝑐+𝑑𝑥)/𝜈is the Reynold number, and 𝐺𝑐=𝑔𝛽𝑇₀𝑇_𝑤(𝑥)𝐿³(𝑐+𝑑𝑥)³/𝜈² is the solutal Grashof number.

In  (5.11)𝑑^∗=𝑣_𝑤(𝑥)𝑈₀𝑢_𝑤(𝑈(𝑥)⁰𝑢_𝑤(𝑥)𝐿(𝑐+𝑑𝑥)𝜈)^1/2. (5.12) By using values of 𝑢_𝑤(𝑥) and 𝑣_𝑤(𝑥) in (5.12), we get𝑑^∗=𝑉₀𝑈₀(𝑈₀𝐿𝜈)^1/2, (5.13) which is a nondimensional constant.

5.1.1. Example  1

For 𝑐=0, 𝑏=(1−𝑚)/2, 𝑑=1, 𝐾₁=2/(𝑚+1), and −4𝑏+𝑑=𝑛, the similarity variable and similarity function are𝜂=𝑦(𝑚+12)^1/2(𝑥)^(𝑚−1)/22,𝜓=()𝑚+1^1/2(𝑥)^(𝑚+1)/2𝐹(𝜂). (5.14) Then 𝑢_𝑤(𝑥), 𝑇_𝑤(𝑥), and 𝐶_𝑤(𝑥) take the following form:𝑢_𝑤(𝑥)=𝑥^𝑚,𝑇_𝑤(𝑥)=𝑥^𝑛,𝐶_𝑤(𝑥)=𝑥^𝑛. (5.15) For these variables (5.8)–(5.10) become𝐹^′′′+𝐹𝐹^′′−2𝑚𝑚+1𝐹+2𝜆𝑚+1𝜃+2𝜆^∗1𝑚+1𝜑=0,𝜃Pr^′′+𝐹𝜃^′−2𝑛𝐹𝑚+1^′1𝜃=0,𝜑Sc^′′+𝐹𝜑^′−2𝑛𝐹𝑚+1^′𝜑=0, (5.16) subject to boundary conditions𝐹^′(0)=1,𝐹(0)=−𝑑^∗2𝑚+1,𝜃(0)=1,𝜑(0)=1,𝐹^′(∞)=0,𝜃(∞)=0,𝜑(∞)=0, (5.17) where 𝜆=Gr/Re² and 𝜆^∗=𝐺𝑐/Re². In this case from (5.3) and (5.7) we have2𝑣=−()𝑚+1^1/2𝑥^(𝑚−1)/2[(𝑚+12)𝐹+(𝑚−12)𝜂𝐹^′]. (5.18) This is all the same as that taken by Ali and Al-Yousef [20] for the problem of mixed convection flow over a stretching surface without considering mass transfer.

5.1.2. Example  2

Consider the case for 𝑐=0, 𝑏=0, 𝑑=1, and 𝐾₁=1. In this case the form of similarity variable and similarity function is𝜂=𝑦,𝜓=𝑥𝐹(𝜂). (5.19) Therefore, we have𝑢_𝑤(𝑥)=𝑥,𝑇_𝑤(𝑥)=𝑥,𝐶_𝑤(𝑥)=𝑥, (5.20) in consequence of these, (5.8)–(5.10) can be written as𝐹^′′′+𝐹𝐹^′′−𝐹^′2+𝜆𝜃+𝜆^∗1𝜑=0,𝜃Pr^′′+𝐹𝜃^′−𝐹^′1𝜃=0,𝜑Sc^′′+𝐹𝜑^′−𝐹^′𝜑=0, (5.21) subject to boundary conditions𝐹^′(0)=1,𝐹(0)=𝐹_𝑤𝐹,𝜃(0)=1,𝜑(0)=1,_′(∞)=0,𝜃(∞)=0,𝜑(∞)=0, (5.22) where 𝐹_𝑤=−𝑑^∗.

This problem exactly match to the work done by Yih [21] if we neglect MHD effects in his work.

5.1.3. Example  3

In the absence of suction or injection 𝑣_𝑤(𝑥)=0at𝑦=0. (5.23) In such case for 𝑐=0, 𝑏=(1−𝑚)/2, 𝑑=1, and 𝐾₁=1, the similarity variable and similarity function are obtained as𝜂=𝑦(𝑥)^(𝑚−1)/2,𝜓=(𝑥)^(𝑚+1)/2𝐹(𝜂). (5.24) We then have𝑢_𝑤(𝑥)=𝑥^𝑚,𝑇_𝑤(𝑥)=𝑥^2𝑚−1,𝐶_𝑤(𝑥)=𝑥^2𝑚−1, (5.25) and (5.8)–(5.10) take the form𝐹^′′′+(𝑚+12)𝐹𝐹^′′−𝑚𝐹^′2+𝜆𝜃+𝜆^∗1𝜑=0,𝜃Pr^′′+(𝑚+12)𝐹𝜃^′−(2𝑚−1)𝐹^′1𝜃=0,𝜑Sc^′′+(𝑚+12)𝐹𝜑^′−(2𝑚−1)𝐹^′𝜑=0, (5.26) together with the boundary conditions𝐹^′𝐹(0)=1,𝐹(0)=0,𝜃(0)=1,𝜑(0)=1,_′(∞)=0,𝜃(∞)=0,𝜑(∞)=0. (5.27) This in fact is the problem considered by Chen [22] neglecting mass transfer.

5.1.4. Example  4

For 𝑏=−2, 𝑐=1, 𝑑=2, and 𝐾₁=2. The physical quantities 𝑢_𝑤(𝑥), 𝑇_𝑤(𝑥), and 𝐶_𝑤(𝑥) take the following form𝑢_𝑤(𝑥)=(1+2𝑥)³,𝑇_𝑤(𝑥)=(1+2𝑥)⁵,𝐶_𝑤(𝑥)=(1+2𝑥)⁵, (5.28) and the similarity variable and functions are𝜂=𝑦(1+2𝑥)√2,𝜓=√2(1+2𝑥)²𝐹(𝜂),𝑇=𝑇_𝑤(𝑥)𝜃(𝜂),𝐶=𝐶_𝑤(𝑥)𝜑(𝜂), (5.29) as a result (5.8)–(5.10) become𝐹^′′′+4𝐹𝐹^′′−6𝐹^′2+2𝜆𝜃+2𝜆^∗1𝜑=0,𝜃Pr^′′+4𝐹𝜃^′−10𝐹^′1𝜃=0,𝜑Sc^′′+4𝐹𝜑^′−10𝐹^′𝜑=0, (5.30) subject to the boundary conditions𝐹^′(0)=1,𝐹(0)=−𝑑^∗,𝜃(0)=1,𝜑(0)=1,𝐹^′(∞)=0,𝜃(∞)=0,𝜑(∞)=0, (5.31) where 𝑑^∗ is the same as given in (5.13).

Such types of stretching velocity, wall temperature, and concentration have not been discussed so far.

5.2. Case  2: For 𝑑=0

In this section, the parameters 𝑏 and 𝑐 are taken to be arbitrary and 𝑒=𝑑=𝛾(𝑥)=0 is considered. This becomes a spiral symmetry where we have considered translation in the variable 𝑥 and scaling in the variables 𝑦, 𝑢, 𝑣,𝑇, and 𝐶. Setting 𝑏=0 results in a translational symmetry.

The characteristic equations then are𝑑𝑥𝑐=𝑑𝑦=𝑏𝑦𝑑𝑢=−2𝑏𝑢𝑑𝑣=−𝑏𝑣𝑑𝑇=−4𝑏𝑇𝑑𝐶.−4𝑏𝐶 (5.32) The similarity variable is 𝑦𝜂=𝐾₂^1/2exp((𝑏/𝑐)𝑥), (5.33) and the similarity functions are𝑢=exp(−2𝑏𝑐𝑥)𝐹^′𝑏(𝜂),𝑣=exp(−𝑐𝑥)𝐻(𝜂),𝑇=exp(−4𝑏𝑐𝑥)𝜃(𝜂),𝐶=exp(−4𝑏𝑐𝑥)𝜑(𝜂). (5.34) From (4.3) and (5.34) we have for 𝐴_𝑖=1(𝑖=5,6,7,8)𝑢=𝑢_𝑤(𝑥)𝐹^′(𝜂),𝑣=−𝑣_𝑤(𝑥)𝐻(𝜂),𝑇=𝑇_𝑤(𝑥)𝜃(𝜂),𝐶=𝐶_𝑤(𝑥)𝜑(𝜂). (5.35) Following the same procedure as done in case 1, we have𝜓=𝐾₂^1/2𝑏exp(−𝑐𝑥)𝐹(𝜂). (5.36) Note that the variables will be balanced if𝑏𝐻(𝜂)=𝑐𝐾₂^1/2[𝐹+𝜂𝐹^′]. (5.37) In this case, (2.5)–(2.7) will take the following form𝐹^′′′−𝐾₂𝑏𝑐𝐹𝐹^′′+𝐾₂𝑏𝑐𝐹^′2+𝐾₂GrRe²𝜃+𝐾₂𝐺𝑐Re²1𝜑=0,𝜃Pr^′′−𝐾₂𝑏𝑐𝐹𝜃^′+4𝐾₂𝑏𝑐𝐹^′1𝜃=0,𝜑Sc^′′−𝐾₂𝑏𝑐𝐹𝜑^′+4𝐾₂𝑏𝑐𝐹^′𝜑=0, (5.38) subject to the boundary conditions𝐹^′(0)=1,𝐹(0)=−𝑑^∗1𝐾₂^1/2𝐹(𝑏/𝑐),𝜃(0)=1,𝜑(0)=1,_′(∞)=0,𝜃(∞)=0,𝜑(∞)=0, (5.39) where the nondimensional physical parameters areGr=𝑔𝛽𝑇₀𝑇_𝑤(𝑥)𝐿³𝜈²𝑈,Re=⁰𝑢_𝑤(𝑥)𝐿𝜈,𝐺𝑐=𝑔𝛽^∗𝑇₀𝑇_𝑤(𝑥)𝐿³𝜈²,𝑑^∗=𝑣_𝑤(𝑥)𝑈₀𝑢_𝑤(𝑈(𝑥)⁰𝑢_𝑤(𝑥)𝐿𝜈)^1/2. (5.40)(5.41) By incorporating (5.34) in (5.41), we get𝑑^∗=𝑉₀𝑈₀(𝑈₀𝐿𝜈)^1/2. (5.42) Again 𝑑^∗ is a nondimensional constant.

5.2.1. Example  1

In the case of rigid surface𝑣_𝑤(𝑥)=0at𝑦=0. (5.43) For 𝑏/𝑐=−1/2 and 𝐾₂=2 the similarity variable and function are𝜂=𝑦[exp(𝑥)2]^1/2,𝜓=[2exp(𝑥)]^1/2𝐹(𝜂), (5.44) and the surface velocity, temperature, and mass concentration are𝑢_𝑤(𝑥)=exp(𝑥),𝑇_𝑤(𝑥)=exp(2𝑥),𝐶_𝑤(𝑥)=exp(2𝑥). (5.45) With the above functions (5.38) become𝐹^′′′+𝐹𝐹^′′−2𝐹^′2+2𝜆𝜃+2𝜆^∗1𝜑=0,𝜃Pr^′′+𝐹𝜃^′−4𝐹^′1𝜃=0,𝜑Sc^′′+𝐹𝜑^′−4𝐹^′𝜑=0, (5.46) and boundary conditions are 𝐹^′𝐹(0)=1,𝐹(0)=0,𝜃(0)=1,𝜑(0)=1,_′(∞)=0,𝜃(∞)=0,𝜑(∞)=0, (5.47) where 𝜆=Gr/Re² and 𝜆^∗=𝐺𝑐/Re².

Which is the same as that taken by Partha et al. [23] for mixed convection flow along exponentially stretching surface without considering the mass transfer. In that problem effects of viscous dissipation were examined that are not considered in this paper.

6. Conclusion

Mixed convection flow with mass transfer over a stretching surface with suction or injection is studied. Through Lie group analysis symmetries of equations are determined. By applying these similarity variables and functions, original PDEs are converted into ODEs. For specific values of parameters of Lie group transformations, the similarity variables and functions which are available in the literature become special case of our problem. Following the same procedure, now one can easily find the similarity variable and function of his own choice for the particular problem. Specially for stretching problem, use of integral power in boundary conditions is not necessary. Therefore it is a breakthrough in the study of similarity solutions in the sense that only those symmetries that accept restrictions from the boundary conditions are discussed and similarity variables and functions obtained in this way have physical importance. The methodology will help directly to obtain similarity variables and functions of the boundary value problems for any specific boundary value problem at hand.