Lie Group Analysis of Unsteady Flow and Heat Transfer over a Porous Surface for a Viscous Fluid

Abstract

The problem of a two-dimensional, unsteady flow and a heat transfer of a viscous fluid past a surface in the presence of variable suction/injection is analyzed. The unsteadiness is due to the time dependent free stream flow. The governing equations are derived with the usual boundary layer approximation. Using Lie group theory, a group classification of the equations with respect to the variable free stream flow and suction/injection velocity is performed. Restrictions imposed by the boundary conditions on the symmetries are discussed. Adopting the obtained symmetry groups, governing partial differential equations are converted into ordinary differential equations and then solved numerically. Effects of the dimensionless problem parameters on the velocity and temperature profiles are outlined in the figures.

1. Introduction

Boundary layer flow and heat transfer for an incompressible viscous fluid past a heated porous surface have attracted a great interest because of the importance in engineering applications such as cooling of nuclear reactors and turbine blades, food processing, crystal growth, flow control on airfoils, and electronic cooling. Blasius [1] first presented a similarity solution for velocity distribution within the boundary layer for a viscous incompressible fluid over a flat plate. Khaled and Vafai [2] studied hydromagnetic effects on dynamical and thermal boundary layer characteristics. This analysis is concerned with a family of squeezed flows. They displayed effects of the magnetic parameter, Prandtl number, suction/injection parameter, and squeezing parameter on velocity and temperature profiles graphically. Elbashbeshy and Bazid [3] studied the heat transfer over an unsteady stretching surface. They obtained a new similarity solution for the temperature field. They investigated the effect of the Prandtl number and the unsteadiness parameter which determine the velocity and temperature profiles and heat transfer coefficient. Mahmood et al. [4] studied flow and heat transfer over a permeable sensor surface placed in a squeezing channel. They displayed effects of the squeezing parameter on values of velocity and temperature and the variation of skin friction coefficient and heat transfer coefficient against the transpiration parameter with the graphics. Bataller [5] investigated the flow influenced by nonlinearly stretching of the sheet with heat transfer. They analyzed the effects of various physical parameters on heat transfer phenomena in a viscous flow over a nonlinearly stretching sheet. Mukhopadhyay [6] investigated the effects of thermal radiation on unsteady boundary layer mixed convection heat transfer problem from a vertical porous stretching surface embedded in a porous medium. Their study indicated that the flow and temperature field are significantly influenced by the unsteadiness parameter, buoyancy force, and suction parameter in both porous and nonporous media. Aziz [7] studied thermal boundary layer flow over a flat plate considering convective surface heat flux at the lower surface of the plate and determined the condition which the convection heat transfer coefficient must meet for a similarity solution to exist. Aziz [8] studied the boundary layer flow over a flat plate with slip flow and constant heat flux surface condition velocity and shear stress distributions were presented for a range of values of the parameter characterizing the slip flow. Ishak [9] considered the problem of steady laminar boundary layer flow and heat transfer over a stationary flat plate immersed in a uniform free stream with convective boundary condition. They found that suction increases the surface shear stress and as a consequence increases the heat transfer rate at the surface. Yao et al. [10] investigated heat transfer flow over a stretching/shrinking sheet with convective boundary conditions. They found that temperature at the surface of the plate increases with the increase of the surface convection parameter. Rahman [11] studied locally similar solutions for the hydromagnetic thermal boundary layer flow of a viscous incompressible electrically conducting fluid over a flat plate with a partial slip at the surface of boundary in the presence of the convective surface boundary condition numerically and displayed results for the velocity, temperature, and Prandtl number within the boundary layer delineating the effect of various parameters characterizing the flow.

In most of the published studies in the literature [12–16], the partial differential equations were converted to ordinary differential equations via a similarity transformation found by ad hoc methods. The resulting ordinary differential equations were then solved by numerical methods. Lie group methods are performed to investigate similarity transformation of the boundary layer equations in only a few studies [17–19]. However, a detailed investigation of the complete symmetries for unsteady boundary layer flow and heat transfer equations are lacking in the literature.

In this paper, Lie group analysis of unsteady flow and heat transfer over a porous surface in the presence of suction/injection velocity is analyzed. Group classification of the governing equations with respect to the variable free stream flow and suction/injection velocity is performed. The infinitesimal generators are obtained. The restrictions imposed by the boundary conditions on the generators are discussed. Using the resulting generators, the governing equations are transformed into ordinary differential equations and solved numerically. Numerical results are presented for the selected dimensionless problem parameters and discussed in detail. Present study reveals that injection decreases temperature gradient whereas suction increases the temperature gradient. The increase in the Prandtl number causes increase in the temperature gradient with increasing the slope.

2. Mathematical Formulation

Consider the unsteady two-dimensional, incompressible laminar flow over a porous surface. Under the boundary layer approximation, the continuity, momentum, and energy equations can be written as 𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦𝜕=0,𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝜕𝑈𝜕𝑡+𝑈𝜕𝑈𝜕𝑥𝜕+𝜈²𝑢𝜕𝑦²,𝜕𝑇𝜕𝑡+𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦𝜕=𝛼²𝑇𝜕𝑦², (2.1) where 𝑥 is the coordinate along the surface and 𝑦 is the coordinate vertical to 𝑥. 𝑡 is time, 𝑢 and 𝑣 are the velocity components in 𝑥 and 𝑦 coordinates. 𝑈 is the free stream velocity outside the boundary layer, 𝑇 is the temperature, 𝜈 and 𝛼 are the kinematic viscosity and thermal diffusivity of the fluid, respectively.

The boundary conditions are 𝑢𝑥,0,𝑡=0,𝑣𝑥,0,𝑡=𝑉𝑥,𝑡,𝑇𝑥,0,𝑡=𝑇_𝑠,𝑢𝑥,∞,𝑡=𝑈𝑥,𝑡,𝑇𝑥,∞,𝑡=𝑇_∞, (2.2) where 𝑉(𝑥,𝑡) is the suction/injection velocity of the permeable surface. 𝑇_𝑠 is the constant surface temperature and 𝑇_∞ is the temperature of the ambient fluid.

The nondimensional form of (2.1) and boundary conditions are as follows: 𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦=0,𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢()+1𝜕𝑦=𝐹𝑥,𝑡𝜕Re²𝑢𝜕𝑦²,𝜕𝑇𝜕𝑡+𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=1𝜕𝑦𝜕RePr²𝑇𝜕𝑦²,𝑢()()()()𝑢()()()𝑥,0,𝑡=0,𝑣𝑥,0,𝑡=𝑉𝑥,𝑡,𝑇𝑥,0,𝑡=1,𝑥,∞,𝑡=𝑈𝑥,𝑡,𝑇𝑥,∞,𝑡=0. (2.3)(2.4)(2.5)(2.6) The nondimensional parameters are related to the dimensional ones through the following relations: 𝑢=𝑢𝑈₀,𝑣=𝑣𝑈₀,𝑈=𝑈𝑈₀,𝑇=𝑇−𝑇_∞𝑇_𝑠−𝑇_∞,𝑥=𝑥𝐿,𝑦=𝑦𝐿𝑈,𝑡=⁰𝑡𝐿,𝑈Re=⁰𝐿𝜈𝜈,Pr=𝛼,𝐹(𝑥,𝑡)=𝜕𝑈𝜕𝑡+𝑈𝜕𝑈,𝜕𝑥 (2.7) where 𝑈₀, 𝐿, Re, and Pr are the average free stream velocity, length of the horizontal surface, Reynolds number, and Prandtl number, respectively. 𝐹(𝑥,𝑡) is used for simplicity in the equations.

3. Symmetry Analysis

The symmetry groups of (2.3)–(2.5) are calculated by using classical Lie group approach. Details of the theory can be found in Bluman and Kumei [20] and Stephani [21]. The one-parameter infinitesimals Lie group of transformations are defined as 𝑥^∗=𝑥+𝜀𝜉₁,𝑦𝑥,𝑦,𝑡,𝑢,𝑣,𝑇_∗=𝑦+𝜀𝜉₂,𝑡𝑥,𝑦,𝑡,𝑢,𝑣,𝑇_∗=𝑡+𝜀𝜉₃,𝑢𝑥,𝑦,𝑡,𝑢,𝑣,𝑇_∗=𝑢+𝜀𝜂₁,𝑣𝑥,𝑦,𝑡,𝑢,𝑣,𝑇_∗=𝑣+𝜀𝜂₂,𝑇𝑥,𝑦,𝑡,𝑢,𝑣,𝑇_∗=𝑇+𝜀𝜂₃𝑥,𝑦,𝑡,𝑢,𝑣,𝑇. (3.1) The prolonged infinitesimal generator which includes higher order derivatives is 𝑋=𝜉₁𝜕𝜕𝑥+𝜉₂𝜕𝜕𝑦+𝜉₃𝜕𝜕𝑡+𝜂₁𝜕𝜕𝑢+𝜂₂𝜕𝜕𝑣+𝜂₃𝜕𝜕𝑇+𝜂_1𝑥𝜕𝜕𝑢_𝑥+𝜂_1𝑦𝜕𝜕𝑢_𝑦+𝜂_1𝑡𝜕𝜕𝑢_𝑡+𝜂_2𝑦𝜕𝜕𝑣_𝑦+𝜂_3𝑥𝜕𝜕𝑇_𝑥+𝜂_3𝑦𝜕𝜕𝑇_𝑦+𝜂_3𝑡𝜕𝜕𝑇_𝑡+𝜂_1𝑦𝑦𝜕𝜕𝑢_𝑦𝑦+𝜂_3𝑦𝑦𝜕𝜕𝑇_𝑦𝑦. (3.2) The invariance conditions for (2.3)–(2.5) are obtained by applying the generator to the equations: 𝜂_1𝑥+𝜂_2𝑦𝜂=0,_1𝑡+𝜂₁𝑢_𝑥+𝑢𝜂_1𝑥+𝜂₂𝑢_𝑦+𝑣𝜂_1𝑦=𝜉₁𝐹_𝑥+𝜉₃𝐹_𝑡+1𝜂Re^1𝑦𝑦,𝜂_3𝑡+𝜂₁𝑇_𝑥+𝑢𝜂_3𝑥+𝜂₂𝑇_𝑦+𝑣𝜂_3𝑦=1𝜂RePr^3𝑦𝑦. (3.3) MathLie program [22] is used to calculate higher order infinitesimals and then equations are separated with respect to higher order variables. The initial restrictions on the infinitesimals are 𝜉₁=𝜉₁(𝑥,𝑡),𝜉₂=𝜉₂𝑥,𝑦,𝑡,𝜉₃=𝜉₃(𝑡),𝜂₁=𝜂₁𝑥,𝑦,𝑡,𝑢,𝜂₂=𝜂₂𝑥,𝑦,𝑡,𝑢,𝑣,𝜂₃=𝜂₃𝑥,𝑦,𝑡,𝑇, (3.4) and the remaining determining equations are 𝜕²𝜂₁𝜕𝑢²𝜕=0,₂𝜂₃𝜕𝑇²=0,−2𝜕𝜉₂+𝜕𝑦𝜕𝜉₃𝜕𝑡=0,𝜕𝜂₁+𝜕𝑥𝜕𝜂₂−𝜕𝑦=0,𝜕𝜉₂+𝜕𝑥𝜕𝜂₂𝜂𝜕𝑢=0,₁−𝜕𝜉₁𝜕𝑡−𝑢𝜕𝜉₁𝜕𝑥+2𝑢𝜕𝜉₂−𝜕𝑦=0,𝜕𝜉₁+𝜕𝑥𝜕𝜉₂+𝜕𝑦𝜕𝜂₁−𝜕𝑢𝜕𝜂₂𝜕𝑣=0,PrRe𝜂₂−PrRe𝜕𝜉₂𝜕𝑡−PrRe𝑢𝜕𝜉₂𝜕𝑥+PrRe𝑣𝜕𝜉₂+𝜕𝜕𝑦²𝜉₂𝜕𝑦²𝜕−2²𝜂₃𝜕𝑦𝜕𝑇=0,Re𝜂₂−Re𝜕𝜉₂𝜕𝑡−Re𝑢𝜕𝜉₂𝜕𝑥+Re𝑣𝜕𝜉₂+𝜕𝜕𝑦²𝜉₂𝜕𝑦²𝜕−2²𝜂₁𝜕𝑦𝜕𝑢=0,PrRe𝜕𝜂₃𝜕𝑡+PrRe𝑢𝜕𝜂₃𝜕𝑥+PrRe𝑣𝜕𝜂₃−𝜕𝜕𝑦²𝜂₃𝜕𝑦²=0,−Re𝜉₁𝜕𝐹𝜕𝑥−Re𝜉₃𝜕𝐹𝜕𝑡−2𝐹Re𝜕𝜉₂𝜕𝑦+Re𝜕𝜂₁𝜕𝑡+𝐹Re𝜕𝜂₁𝜕𝑢+Re𝑢𝜕𝜂₁𝜕𝑥+Re𝑣𝜕𝜂₁−𝜕𝜕𝑦²𝜂₁𝜕𝑦²=0. (3.5) Equations (3.5) are an overdetermined partial differential system. Solving the system, the infinitesimals are finally obtained: 𝜉₁(𝑡),𝜉=𝑎𝑥+𝑏₂=𝑐2(),𝜉𝑦+𝑑𝑥,𝑡₃𝜂=𝑐𝑡+𝑒,₁=(𝑎−𝑐)𝑢+𝑑𝑏,𝜂𝑑𝑡₂𝑐=−2𝑣+𝜕𝑑𝜕𝑡+𝑢𝜕𝑑,𝜂𝜕𝑥₃=𝑔𝑇+ℎ. (3.6)𝐹(𝑥,𝑡) is to be determined from the following equation: 𝑑²𝑏𝑑𝑡²+𝐹(𝑎−2𝑐)−𝜉₁𝜕𝐹𝜕𝑥−𝜉₃𝜕𝐹𝜕𝑡=0. (3.7) The generator is applied to the boundary conditions which should also be invariant under the transformation leading to restrictions in (3.6): 𝜉₁𝜉=𝑎𝑥+𝑏,₂=𝑐2𝜉𝑦,₃𝜂=𝑐𝑡+𝑒,₁=()𝜂𝑎−𝑐𝑢,₂𝑐=−2𝜂𝑣,₃𝐹()=0,𝑎−2𝑐−𝜉₁𝜕𝐹𝜕𝑥−𝜉₃𝜕𝐹𝜕𝑡=0 (3.8) or 𝐹=(𝑐𝑡+𝑒)^{(𝑎−2𝑐)/𝑐}𝑓𝜇(),𝜇=𝑎𝑥+𝑏^1/𝑎(𝑐𝑡+𝑒)^1/𝑐. (3.9)𝑉(𝑥,𝑡) and 𝑈(𝑥,𝑡) functions should satisfy the following equations: −𝑐2𝑉=(𝑎𝑥+𝑏)𝜕𝑉+()𝜕𝑥𝑐𝑡+𝑒𝜕𝑉,()()𝜕𝑡𝑎−𝑐𝑈=𝑎𝑥+𝑏𝜕𝑈+()𝜕𝑥𝑐𝑡+𝑒𝜕𝑈.𝜕𝑡 (3.10) Different functions can be obtained from (3.10) for 𝑉(𝑥,𝑡) and 𝑈(𝑥,𝑡) via setting the parameters as follows.

(I) Parameter a: selecting parameter 𝑎=1 without a loss of generality and all other parameters zero yields 𝑥𝜕𝑉(𝑡),𝜕𝑥=0⇒𝑉=𝑠𝑈=𝑥𝜕𝑈(𝑡).𝜕𝑥⇒𝑈=𝑥𝑓 (3.11) In this case, suction or injection velocity is an arbitrary function of time. Solving (3.11) for 𝑈(𝑥,𝑡) yields 𝑈=𝑥𝑓(𝑡), where 𝑓(𝑡) is an arbitrary function of time for this special choice.

(II) Parameter c: selecting parameter 𝑐=1 and all other parameters zero yields −𝑉2=𝑡𝜕𝑉𝑠(𝑥)𝜕𝑡=0⇒𝑉=√𝑡,−𝑈=𝑡𝜕𝑈1𝜕𝑡⇒𝑈=𝑡𝑓(𝑥). (3.12) For this choice, 𝑉(𝑥,𝑡) and 𝑈(𝑥,𝑡) yield 𝑉=𝑠(𝑥)/√𝑡 and 𝑈=(1/𝑡)𝑓(𝑥) from (3.10).

(III) Parameter b: selecting parameter 𝑏=1 and all other parameters zero yields 𝜕𝑉(𝑡),𝜕𝑥=0⇒𝑉=𝑠𝜕𝑈(𝑡).𝜕𝑥=0⇒𝑈=𝑓 (3.13) In this case, suction or injection velocity and free stream velocity are arbitrary functions of time.

(IV) Parameter e: selecting parameter 𝑒=1 and all other parameters zero yields 𝜕𝑉(𝑥),𝜕𝑡=0⇒𝑉=𝑠𝜕𝑈(𝑥).𝜕𝑡=0⇒𝑈=𝑓 (3.14) In this case, suction or injection velocity and free stream velocity are arbitrary functions of the spatial variable 𝑥.

4. Similarity Transformations and Reductions

Similarity transformations and reductions of the partial differential system to an ordinary differential system will be considered.

4.1. A Similarity Transformation

In this section, we choose the first case (case I) for obtaining similarity transformations as follows: 𝑑𝑥𝑥=𝑑𝑦0=𝑑𝑡0=𝑑𝑢𝑢=𝑑𝑣0=𝑑𝑇0. (4.1) Similarity variables and functions are obtained from (4.1): 𝜇₁=𝑦,𝜇₂=𝑡,𝑢=𝑥𝐹₁𝜇₁,𝜇₂,𝑣=𝐹₂𝜇₁,𝜇₂,𝑇=𝐹₃𝜇₁,𝜇₂,𝑉=𝑠𝜇₂,𝑈=𝑥𝑓𝜇₂. (4.2) The substitution of these variables and functions into the original equations and boundary conditions yields 𝐹₁+𝜕𝐹₂𝜕𝜇₁=0,𝜕𝐹₁𝜕𝜇₂+𝐹²₁+𝐹₂𝜕𝐹₁𝜕𝜇₁=𝑑𝑓𝜇₂𝑑𝜇₂+𝑓𝜇₂²+1𝜕Re²𝐹₁𝜕𝜇₁²,𝜕𝐹₃𝜕𝜇₂+𝐹₂𝜕𝐹₃𝜕𝜇₁=1𝜕RePr²𝐹₃𝜕𝜇₁²,𝐹₁0,𝜇₂=0,𝐹₂0,𝜇₂=𝑠𝜇₂,𝐹₃0,𝜇₂=1,𝐹₁∞,𝜇₂=𝑓𝜇₂,𝐹₃∞,𝜇₂=0. (4.3)(4.4) The partial differential system with three independent variables is reduced to a system with two independent variables.

4.2. The Second Reduction

To reduce (4.3) and (4.4) into an ordinary differential system, one-parameter infinitesimal Lie group of transformations is defined as 𝜇₁^∗=𝜇₁+𝜀𝛾₁𝜇₁,𝜇₂,𝐹₁,𝐹₂,𝐹₃,𝜇₂^∗=𝜇₂+𝜀𝛾₂𝜇₁,𝜇₂,𝐹₁,𝐹₂,𝐹₃,𝐹₁^∗=𝐹₁+𝜀𝜑₁𝜇₁,𝜇₂,𝐹₁,𝐹₂,𝐹₃,𝐹₂^∗=𝐹₂+𝜀𝜑₂𝜇₁,𝜇₂,𝐹₁,𝐹₂,𝐹₃,𝐹₃^∗=𝐹₃+𝜀𝜑₃𝜇₁,𝜇₂,𝐹₁,𝐹₂,𝐹₃. (4.5) The prolonged infinitesimal generator is 𝑋=𝛾₁𝜕𝜕𝜇₁+𝛾₂𝜕𝜕𝜇₂+𝜑₁𝜕𝜕𝐹₁+𝜑₂𝜕𝜕𝐹₂+𝜑₃𝜕𝜕𝐹₃+𝜑_1𝜇1𝜕𝜕𝐹_1𝜇1+𝜑_2𝜇1𝜕𝜕𝐹_2𝜇1+𝜑_3𝜇1𝜕𝜕𝐹_3𝜇1+𝜑_1𝜇2𝜕𝜕𝐹_1𝜇2+𝜑_3𝜇2𝜕𝜕𝐹_3𝜇2+𝜑_1𝜇1𝜇1𝜕𝜕𝐹_1𝜇1𝜇1+𝜑_3𝜇1𝜇1𝜕𝜕𝐹_3𝜇1𝜇1. (4.6) The invariance conditions for (4.3) are 𝜑₁+𝜑_2𝜇1𝜑=0,_1𝜇2+2𝐹₁𝜑₁+𝜑₂𝐹_1𝜇1+𝐹₂𝜑_1𝜇1=𝛾₂𝑑²𝑓𝑑𝜇²₂+2𝛾₂𝑓𝑑𝑓𝑑𝜇₂+1𝜑Re^1𝜇1𝜇1,𝜑_3𝜇2+𝜑₂𝐹_3𝜇1+𝐹₂𝜑_3𝜇1=1𝜑RePr^3𝜇1𝜇1. (4.7)(4.8)(4.9) Substituting the higher order infinitesimals, separating the equations with respect to higher order variables yields 𝜕²𝜑₁𝜕𝐹₁²𝜕=0,₂𝜑₃𝜕𝐹₃²=0,−2𝜕𝛾₁𝜕𝜇₁+𝜕𝛾₂𝜕𝜇₂𝜑=0,₂−𝜕𝛾₁𝜕𝜇₂+1𝜕Re²𝛾₁𝜕𝜇²₁1−2𝜕Re²𝜑₁𝜕𝜇₁𝜕𝐹₁+𝐹₂𝜕𝛾₁𝜕𝜇₁𝜑=0,₂−𝜕𝛾₁𝜕𝜇₂+1𝜕RePr²𝛾₁𝜕𝜇²₁1−2𝜕RePr²𝜑₃𝜕𝜇₁𝜕𝐹₃+𝐹₂𝜕𝛾₁𝜕𝜇₁1=0,Re𝜕𝜑₃𝜕𝜇₂+1𝑓RePr²𝜕𝜑₃𝜕𝐹₁+1𝑓RePr^′𝜕𝜑₃𝜕𝐹₁−1Re²𝜕Pr²𝜑₃𝜕𝜇₁²−1𝐹RePr¹²𝜕𝜑₃𝜕𝐹₁+1𝜕Re²𝜑₃𝜕𝜇₁𝜕𝐹₂=0,−2𝑓𝑓^′𝛾₂−2𝑓²𝜕𝛾₁𝜕𝜇₁−2𝑓^′𝜕𝛾₁𝜕𝜇₁+𝜕𝜑₁𝜕𝜇₂+𝑓²𝜕𝜑₁𝜕𝐹₁+𝑓^′𝜕𝜑₁𝜕𝐹₁−𝑓^′′𝛾₂−1𝜕Re²𝜑₁𝜕𝜇₁²+2𝐹₁𝜑₁+2𝐹²₁𝜕𝛾₁𝜕𝜇₁−𝐹²₁𝜕𝜑₁𝜕𝐹₁+𝐹₂𝜕𝜑₁𝜕𝜇₁−2=0,Re²𝜕Pr²𝜑₃𝜕𝜇₁𝜕𝐹₁−1𝐹RePr²𝜕𝜑₃𝜕𝐹₁+1𝐹Re²𝜕𝜑₃𝜕𝐹₁−2=0,Re²𝜕Pr²𝜑₁𝜕𝜇₁𝜕𝐹₃+1𝐹RePr²𝜕𝜑₁𝜕𝐹₃−1𝐹Re²𝜕𝜑₁𝜕𝐹₃𝜑=0,₁+𝜕𝜑₂𝜕𝜇₁+𝐹₁𝜕𝛾₁𝜕𝜇₁−𝐹₁𝜕𝜑₂𝜕𝐹₂=0, (4.10) from which the above equations are solved: 𝛾₁𝑐=−¹2𝜇₁+𝑐₂𝜇₂,𝛾₂=−𝑐₁𝜇₂+𝑐₃,𝜑₁=𝑐₁𝐹₁,𝜑₂=𝑐₁2𝐹₂+𝑑𝑐₂𝑑𝜇₂,𝜑₃=𝑐₄𝐹₃+𝑐₅. (4.11) The determining equation for 𝑓(𝜇₂) which is selected to 𝑈(𝑥,𝑡), outer velocity, is obtained as follows: −2𝑓𝑑𝑓𝑑𝜇₂𝛾₂−𝑑²𝑓𝑑𝜇²₂𝛾₂−𝑓²𝛾^′₂−𝑑𝑓𝑑𝜇₂𝛾^′₂+𝑓²+𝑑𝑓𝑑𝜇₂𝜕𝜑₁𝜕𝐹₁=0. (4.12) The boundaries and boundary conditions should be invariant under the transformations also (Bluman and Kumei [20]): 𝛾₁𝑐=−¹2𝜇₁,𝛾₂=−𝑐₁𝜇₂+𝑐₃,𝜑₁=𝑐₁𝐹₁,𝜑₂=𝑐₁2𝐹₂,𝜑₃=0,−2𝑓𝑑𝑓𝑑𝜇₂𝛾₂−𝑑²𝑓𝑑𝜇²₂𝛾₂−𝑓²𝛾^′₂−𝑑𝑓𝑑𝜇₂𝛾^′₂+𝑓²+𝑑𝑓𝑑𝜇₂𝜕𝜑₁𝜕𝐹₁=0. (4.13) Two parameters are involved in Lie group Transformations.

𝑠(𝜇₂) and 𝑓(𝜇₂) functions should satisfy the following equations: 𝑐₁2𝑠=−𝑐₁𝜇₂+𝑐₃𝑑𝑠𝑑𝜇₂,𝑐₁𝑓=−𝑐₁𝜇₂+𝑐₃𝑑𝑓𝑑𝜇₂. (4.14) Different functions can be obtained from the above equations for 𝑠(𝜇₂) and 𝑓(𝜇₂) via setting the parameters. The general case in which both parameters are involved will be considered.

Solving (4.14) results 𝜆𝑠=¹√𝑐₁𝜇₂−𝑐₃1,𝑓=𝑐₁𝜇₂−𝑐₃. (4.15) The equations determining the similarity transformations are 𝑑𝜇₁−𝑐₁/2𝜇₁=𝑑𝜇₂−𝑐₁𝜇₂+𝑐₃=𝑑𝐹₁𝑐₁𝐹₁=𝑑𝐹₂𝑐₁/2𝐹₂=𝑑𝐹₃0, (4.16) from which the similarity variables and functions are derived: 𝛽=𝜇₁𝑐₁𝜇₂−𝑐₃^−1/2,𝜃₁𝛽=𝑐₁𝜇₂−𝑐₃𝐹₁,𝜃₂^∗𝛽=𝑐₁𝜇₂−𝑐₃^1/2𝐹₂,𝜃₃𝛽=𝐹₃. (4.17) Inserting these variables and functions into (4.3) and (4.4) yields the following ordinary differential system: 𝜃₁+𝜃^∗_2𝛽𝜃=0,₂₁−𝑐₁𝜃₁+𝜃^∗₂𝜃_1𝛽−𝑐₁𝛽2𝜃_1𝛽=1−𝑐₁+1𝜃Re^1𝛽𝛽−𝑐₁𝛽2𝜃_3𝛽+𝜃₂𝜃_3𝛽=1𝜃RePr^3𝛽𝛽. (4.18) Further, we scale variables 𝜃^∗₂ and 𝛽 as 𝜃^∗₂=1√𝜃Re²1,𝛽=√Re𝑌. (4.19) The substitution of these variables into (4.18) yields 𝜃²_2𝑌+𝑐₁𝜃_2𝑌−𝜃₂𝜃_2𝑌𝑌+𝑐₁𝑌2𝜃_2𝑌𝑌=1−𝑐₁−𝜃_{2𝑌𝑌𝑌},−𝑐₁𝑌𝜃_3𝑌+𝜃₂𝜃_3𝑌=1𝜃Pr^3𝑌𝑌. (4.20) The associated boundary conditions are 𝜃₁(0)=0,𝜃₂(0)=𝜆₁,𝜃₃(0)=1,𝜃₁(∞)=1,𝜃₃(∞)=0, (4.21) where 𝜆₁ is the suction/injection parameter; 𝑐₁ and 𝑐₃ are the unsteadiness parameters.

5. Result and Discussion

The nonlinear reduced ordinary differential equation system (4.20) with the boundary conditions in (4.21) is solved numerically by applying a finite difference code that implements the three-stage Lobatto IIIA formula that provides continuous solutions of fourth order accuracy in the problem domain. Description of the numerical method is given by Shampine et al. [23] and Kierzenka and Shampine [24]. Results are displayed graphically for various values of problem parameters to see developments in the velocity and temperature fields.

The variation of the dimensionless 𝑥-component of velocity is shown for various values of unsteadiness parameter 𝑐₁ in the absence of suction/injection in Figure 1. It is clear that an increase in the 𝑐₁ causes a reduction in the velocity until a certain point. After this point, reverse effect in velocity is observed for increasing values of 𝑐₁.

Figure 1 

Variation of 𝜃1 for different values of unsteadiness parameter 𝑐1 (Pr=10, 𝜆1=0).

The effect of suction/injection parameter on the velocity profile is presented in Figure 2. It is noted that suction decreases the boundary layer thickness whereas injection increases.

Figure 2 

Variation of 𝜃1 for different values of suction/injection (Pr=10, 𝑐1=0.5).

Effects of Prandtl number on the temperature profiles are given in Figure 3. An increase in the Prandtl number decreases the temperature profiles. Effects of unsteadiness parameter 𝑐₁ on the temperature profiles are depicted in Figure 4. An increase in 𝑐₁ decreases the temperature profiles. Thermal boundary layer is narrower for higher unsteadiness parameter. In Figure 5 effects of suction-injection parameter on the temperature profiles are depicted. 𝜆₁=0 corresponds to no suction/injection case. Suction decreases the thickness of thermal boundary layer whereas injection increases the thickness of thermal boundary layer.

Figure 3 

Variation of 𝜃3 for different values of the Prandtl number (𝜆1=0, 𝑐1=0.5).

Figure 4 

Variation of 𝜃3 for different values of 𝑐1 (Pr=10, 𝜆1=0).

Figure 5 

Variation of 𝜃3 for different values of suction/injection (Pr=10, 𝑐1=0.5).

In Figure 6, we displayed the velocity gradient against suction/injection parameter for the case of flow along a horizontal porous surface for different values of the unsteadiness parameter, 𝑐₁. Injection decreases the velocity gradient whereas suction increases the velocity gradient. The reduction in 𝑐₁ causes an increase in velocity gradient with decreasing the slope.

Figure 6 

Variation of the velocity gradient with 𝑐1 (Pr=1).

In Figure 7, we show the temperature gradient against suction/injection parameter for different 𝑐₁-unsteadiness parameter. Injection decreases the temperature gradient whereas suction increases the temperature gradient. In Figure 8, we displayed the temperature gradient against unsteadiness parameter for different Prandtl numbers. The increase in the Prandtl number causes increase in temperature gradient.

Figure 7 

Variation of the temperature gradient with 𝜆1 (Pr=1).

Figure 8 

Variation of the temperature gradient with 𝑐1 (𝜆1=0).

6. Conclusions

In this paper, we have studied the problem of two-dimensional, unsteady flow and heat transfer of a viscous fluid past a surface in the presence of variable suction/injection. The unsteadiness is due to time dependent free stream flow.

The governing equations are derived with the usual boundary layer approximation. Using Lie group theory, a group classification of the equations with respect to the variable free stream flow and suction/injection velocity is performed. Restrictions imposed by the boundary conditions on the symmetries are discussed. Adopting the obtained symmetry groups, governing partial differential equations are converted into ordinary differential equations and then solved numerically.

Effects of dimensionless problem parameters on the velocity and temperature profiles are outlined in the figures. From the present investigation the following results are obtained.(i)An increase in the unsteadiness parameter 𝑐₁ causes a reduction in the velocity.(ii)Suction decreases the boundary layer thickness whereas injection increases.(iii)An increase in the Prandtl number and unsteadiness parameter decreases the temperature profiles.(iv)Suction decreases the thickness of thermal boundary layer whereas injection increases the thickness of boundary layer. (v)Injection decreases the velocity gradient whereas suction increases. Moreover the reduction of 𝑐₁ causes an increase in the velocity gradient.(vi)Injection decreases the temperature gradient whereas suction increases the temperature gradient. The increase in the Prandtl number causes increase in the temperature gradient with increasing the slope.