Symmetry Methods of Flow and Heat Transfer between Slowly Expanding or Contracting Walls

Abstract

An analysis has been carried out for the flow and heat transfer of an incompressible laminar and viscous fluid in a rectangular domain bounded by two moving porous walls which enable the fluid to enter or exit during successive expansions or contractions. The basic equations governing the flow are reduced to the ordinary differential equations using Lie-group analysis. Effects of the permeation Reynolds number 𝑅_𝑒, porosity 𝑅, and the dimensionless wall dilation rate 𝛼 on the self-axial velocity are studied both analytically and numerically. The solutions are represented graphically. The analytical procedure is based on double perturbation in the permeation Reynolds number 𝑅_𝑒 and the wall expansion ratio 𝛼, whereas the numerical solution is obtained using Runge-Kutta method with shooting technique. Results are correlated and compared for some values of the physical parameters. Lastly, we look at the temperature distribution.

1. Introduction

The studies on the boundary layer flow and heat transfer over a stretching surface have become more and more prominent in a number of engineering applications. For instance, during extrusion of a polymer sheet, the reduction of both thickness and width takes place in a cooling bath. The quality of the final product depends upon the heat transfer rate at the stretching surface. In the past, many experimental and theoretical attempts on this topic have been made. Such studies have been presented under the various assumptions of small Reynolds number 𝑅_𝑒, intermediate 𝑅_𝑒, large 𝑅_𝑒, and arbitrary 𝑅_𝑒. The steady flow in a channel with stationary walls and small 𝑅_𝑒 has been studied by Berman [1]. Dauenhauer and Majdalani [2] numerically discussed the two-dimensional viscous flow in a deformable channel when −50<𝑅_𝑒<200 and −100<𝛼<100 (𝛼 denotes the wall expansion ratio). Majdalani et al. [3] analyzed the channel flow of slowly expanding-contracting walls which leads to the transport of biological fluids. They first derived the analytic solution for small 𝑅_𝑒 and 𝛼 and then compared it with the numerical solution.

The flow problem given in study [3] has been analytically solved by Boutros et al. [4] when 𝑅_𝑒 and 𝛼 vary in the ranges −5<𝑅_𝑒<5 and −1<𝛼<1. They used the Lie-group method in this study. Mahmood et al. [5] discussed the homotopy perturbation and numerical solutions for viscous flow in a deformable channel with porous medium. Asghar et al. [6] computed exact solution for the flow of viscous fluid through expanding-contracting channels. They used symmetry methods and conservation laws.

The flow and heat transfer in square domain have been studied by Noor et al. [7]. Our main aim is to study the heat transfer in a rectangular domain. In this study, symmetry methods are applied to a natural convection boundary layer problem. The main advantage of such a method is that it can successfully be applied to a nonlinear differential equation. The symmetries of differential equations are those groups of transformation under which the differential equation remains invariant, that is, a symmetry group maps any solution to any other solutions. The symmetry solutions are quite popular because they result in reductions of independent variable of the problem.

The purpose of this paper is to generalize the flow analysis and heat distribution of [4]. The salient features have been taken into account when the fluid saturates the porous medium. Like in [4], the analytic solution for the arising nonlinear flow problem is given by employing the Lie-group method (with 𝑅_𝑒 and 𝛼 as the perturbation quantities). Finally, the graphs of velocity and temperature are plotted and discussed.

2. Mathematical Formulation of the Problem

Let us consider a rectangular domain bounded by two walls of equal permeability that enable the fluid to enter or exit during successive expansions or contractions. The walls expand or contract uniformly at the time-dependent rate ℎ̇. The continuous sheet aligned with the 𝑥-axis at 𝑦=0 means that the wall is impulsively stretched with the velocity 𝑈_𝑤 along the 𝑥-axis and 𝑇_𝑤(𝑥,𝑡) as our surface temperature. At 𝑦=ℎ(𝑡), it is assumed that the fluid inflow velocity 𝑉_𝑤 is independent of the position. A thin fluid film with uniform thickness ℎ(𝑡) rests on the horizontal wall. The governing time-dependent equations for mass, momentum, and energy are given by 𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦𝜕=0,𝑢+𝜕𝑡𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦1=−𝜌𝜕𝑃𝜕𝑥𝜕+𝜈²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²−𝜈𝜙𝑘𝜕𝑢,𝑣+𝜕𝑡𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑣𝜕𝑦1=−𝜌𝜕𝑃𝜕𝑦𝜕+𝜈²𝑣𝜕𝑥²+𝜕²𝑣𝜕𝑦²−𝜈𝜙𝑘𝑣+𝑔𝛽𝑇−𝑇_𝑤,𝜕𝑇+𝜕𝑡𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦𝜕=𝛼²𝑇𝜕𝑥²+𝜕²𝑇𝜕𝑦², (1)(2)(3)(4) where 𝑢 and 𝑣 are the velocity components in the 𝑥 and 𝑦 directions, respectively, and 𝑇 is temperature. We assume that the fluid properties are constant. Here, 𝜌 is the fluid density, 𝜇 is the dynamic viscosity, and 𝑘 is the thermal conductivity of an incompressible fluid. Thus, the kinematic viscosity is 𝜈=𝜇/𝜌, 𝑔 is the acceleration due to gravity, 𝛽 is the coefficient of the thermal expansion, and the thermal diffusivity is 𝛼=𝑘/𝜌𝑐_𝑝, where 𝑐_𝑝 is the specific heat, 𝑃 is the pressure, 𝑡 is time, and 𝜙 and 𝑘 are the porosity and permeability of porous medium, respectively.

The appropriate boundary conditions are (i)𝑢=0,𝑣=−𝑉_𝑤,𝑇=𝑇_𝑤at(𝑡),()𝜕𝑦=ℎii𝑢𝜕𝑦=0,𝑣=0,𝜕𝑇𝜕𝑦=0at()𝑦=0,iii𝑢=0at𝑥=0, (5) where ℎ(𝑡) is the film thickness. The boundary condition reflects that the fluid motion within the liquid film is caused by the viscous shear arising from the stretching of the elastic wall.

Now we will express the axial velocity, normal velocity, and boundary conditions in terms of the stream function Ψ. From the continuity Equation (1), there exists a dimensional stream function Ψ(𝑥,𝑦,𝑡) such that 𝜕𝑢=Ψ𝜕𝑦,𝜕𝑣=−Ψ𝜕𝑥, (6) which satisfies (1) identically.

Introducing the dimensionless normal coordinate 𝑦=𝑦/ℎ(𝑡), (6) becomes 1𝑢=ℎ𝜕Ψ,𝜕𝑦𝜕𝑣=−Ψ𝜕𝑥. (7) Substituting (7) into (2)–(4), we obtain ℎ²Ψ_𝑦𝑡−ℎℎ̇𝑦Ψ_𝑦𝑦−ℎℎ̇Ψ_𝑦+ℎΨ_𝑦Ψ_𝑥𝑦−ℎΨ_𝑥Ψ_𝑦𝑦ℎ=−³𝜌𝑃_𝑥+𝜈ℎ²Ψ_𝑥𝑥𝑦+Ψ_𝑦𝑦𝑦−ℎ²𝜈𝜙𝑘Ψ_𝑦,−ℎ²Ψ_𝑥𝑡+ℎℎ̇𝑦Ψ_𝑥𝑦−ℎΨ_𝑦Ψ_𝑥𝑥+ℎΨ_𝑥Ψ_𝑥𝑦ℎ=−𝜌𝑃_𝑦+𝜈−ℎ²Ψ_𝑥𝑥𝑥−Ψ_𝑥𝑦𝑦+ℎ²𝜈𝜙𝑘Ψ_𝑥+𝑔𝛽𝑇−𝑇_𝑤ℎ²,𝜕𝑇+1𝜕𝑡ℎΨ_𝑦𝜕𝑇𝜕𝑥−Ψ_𝑥𝜕𝑇𝜕𝑦𝜕=𝛼²𝑇𝜕𝑥²+𝜕²𝑇𝜕𝑦², (8) where a dot denotes the derivative with respect to 𝑡.

The variables in (8) are dimensionless according to 𝑢=𝑢𝑉_𝑤,𝑣=𝑢𝑉_𝑤,𝑥=𝑥ℎ(𝑡),𝑦=𝑦ℎ(𝑡),Ψ=Ψℎ𝑉_𝑤,𝑃=𝑃𝜌𝑉²_𝑤,𝑡=𝑡ℎ𝑉_𝑤ℎℎ̇,𝛼=𝜈,𝜃=𝑇−𝑇_ℎ𝑇_𝑤−𝑇_ℎ,1𝑅=𝜈𝜙𝑎𝑘𝑉_𝑤. (9) Substituting (9) into (8), we have Ψ_𝑦𝑡+Ψ_𝑦Ψ_𝑥𝑦−Ψ_𝑥Ψ_𝑦𝑦+𝑃_𝑥−1𝑅_𝑒𝛼Ψ_𝑦+𝛼𝑦Ψ_𝑦𝑦+Ψ_𝑥𝑥𝑦+Ψ_𝑦𝑦𝑦−1𝑅Ψ_𝑦Ψ=0,_𝑥𝑡+Ψ_𝑦Ψ_𝑥𝑥−Ψ_𝑥Ψ_𝑥𝑦−𝑃_𝑦−1𝑅_𝑒𝛼𝑦Ψ_𝑥𝑦+Ψ_𝑥𝑦𝑦+Ψ_𝑥𝑥𝑥+1𝑅Ψ_𝑥+1ℎ²𝐺_𝑟𝜃=0,𝜕𝜃𝜕𝑡+Ψ_𝑦𝜕𝜃𝜕𝑥−Ψ_𝑥𝜕𝜃=1𝜕𝑦𝑃_𝑟𝑅_𝑒𝜕²𝜃𝜕𝑥²+𝜕²𝜃𝜕𝑦², (10)(11)(12) in which 𝑅_𝑒=ℎ𝑉_𝑤/𝜈 is the permeation Reynolds number, 𝐺_𝑟=𝑔𝛽(𝑇_𝑤−𝑇_ℎ)ℎ³/𝑣² is the Grashof number, 𝑃_𝑟=𝜈/𝛼 is the Prandtl number, and ℎ̇=𝛼𝜈/ℎ.

Through (7) and (9), we have 𝑢=𝜕Ψ𝜕𝑦,𝑣=−𝜕Ψ,𝜕𝑥 (13) and, thus, the boundary conditions take the following forms: Ψ_𝑦=0,Ψ_𝑥Ψ=1,𝜃=1at𝑦=1,_𝑦𝑦=0,Ψ_𝑥=0,𝜃_𝑦Ψ=0at𝑦=0,_𝑦=0at𝑥=0. (14a)(14b)(14c)

3. Solution of the Problem

This section derives the similarity solutions using Lie-group method under which (10)–(12) are invariant.

3.1. Lie Symmetry Analysis

We consider the one-parameter (𝜀) Lie group of infinitesimal transformation in (𝑥,𝑦,𝑡,Ψ,𝑃,𝜃) given by 𝑥^∗=𝑥+𝜀𝜙𝑥,𝑦,𝑡,Ψ,𝑃,𝜃+0𝜀²,𝑦^∗=𝑦+𝜀𝜁𝑥,𝑦,𝑡,Ψ,𝑃,𝜃+0𝜀²,𝑡^∗=𝑡+𝜀𝐹𝑥,𝑦,𝑡,Ψ,𝑃,𝜃+0𝜀²,Ψ^∗=Ψ+𝜀𝜂𝑥,𝑦,𝑡,Ψ,𝑃,𝜃+0𝜀²,𝑃^∗=𝑃+𝜀𝑔𝑥,𝑦,𝑡,Ψ,𝑃,𝜃+0𝜀²,𝜃^∗=𝜃+𝜀𝐻𝑥,𝑦,𝑡,Ψ,𝑃,𝜃+0𝜀², (15) with 𝜀 as a small parameter.

In view of Lie's algorithm, the vector field is 𝜕𝑋=𝜙𝜕𝜕𝑥+𝜁𝜕𝜕𝑦+𝐹𝜕𝑡𝜕+𝜂𝜕𝜕Ψ+𝑔𝜕𝜕𝑃+𝐻,𝜕𝜃 (16) if it is left variant by the transformation (𝑥,𝑦,𝑡,Ψ,𝑃,𝜃)→(𝑥^∗,𝑦^∗,𝑡^∗,Ψ^∗,𝑃^∗,𝜃^∗).

The solutions Ψ=Ψ(𝑥,𝑦,𝑡), 𝑃=𝑃(𝑥,𝑦,𝑡) and 𝜃=𝜃(𝑥,𝑦,𝑡) are invariant under the symmetry (16) if Φ_Ψ=𝑋Ψ−Ψ𝑥,𝑦,𝑡=0,whereΨ=Ψ𝑥,𝑦,𝑡,Φ_𝑃=𝑋𝑃−𝑃𝑥,𝑦,𝑡=0,where𝑃=𝑃𝑥,𝑦,𝑡,Φ_𝜃=𝑋𝜃−𝜃𝑥,𝑦,𝑡=0,where𝜃=𝜃𝑥,𝑦,𝑡. (17) We set Δ₁=Ψ_𝑦𝑡+Ψ_𝑦Ψ_𝑥𝑦−Ψ_𝑥Ψ_𝑦𝑦+𝑃_𝑥−1𝑅_𝑒𝛼Ψ_𝑦+𝛼𝑦Ψ_𝑦𝑦+Ψ_𝑥𝑥𝑦+Ψ_𝑦𝑦𝑦−1𝑅Ψ_𝑦,Δ₂=Ψ_𝑥𝑡+Ψ_𝑦Ψ_𝑥𝑥−Ψ_𝑥Ψ_𝑥𝑦−𝑃_𝑦−1𝑅_𝑒𝛼𝑦Ψ_𝑥𝑦+Ψ_𝑥𝑦𝑦+Ψ_𝑥𝑥𝑥+1𝑅Ψ_𝑥+1ℎ²𝐺_𝑟Δ𝜃,₃=𝜕𝜃𝜕𝑡+Ψ_𝑦𝜕𝜃𝜕𝑥−Ψ_𝑥𝜕𝜃−1𝜕𝑦𝑃_𝑟𝑅_𝑒𝜕²𝜃𝜕𝑥²+𝜕²𝜃𝜕𝑦². (18) The vector field 𝑋 given by (16) is a symmetry generator of (10)–(12) if and only if 𝑋^[3]Δ_𝑗Δ𝑗=0=0,𝑗=1,2,3, (19) in which 𝑋^[3]𝜕=𝜙𝜕𝜕𝑥+𝜁𝜕𝜕𝑦+𝐹𝜕𝑡𝜕+𝜂𝜕𝜕Ψ+𝑔𝜕𝜕𝑃+𝐻𝜕𝜃+𝜂^𝑥𝜕𝜕Ψ_𝑥+𝜂^𝑦𝜕𝜕Ψ_𝑦+𝑔^𝑥𝜕𝜕𝑃_𝑥+𝑔^𝑦𝜕𝜕𝑃_𝑦+𝐻^𝑥𝜕𝜕𝜃_𝑥+𝐻^𝑦𝜕𝜕𝜃_𝑦+𝐻^𝑡𝜕𝜕𝜃_𝑡+𝜂^𝑥𝑦𝜕𝜕Ψ_𝑥𝑦+𝜂^𝑥𝑡𝜕𝜕Ψ_𝑥𝑡+𝜂^𝑦𝑡𝜕𝜕Ψ_𝑦𝑡+𝜂^𝑥𝑥𝜕𝜕Ψ_𝑥𝑥+𝜂^𝑦𝑦𝜕𝜕Ψ_𝑦𝑦+𝐻^𝑥𝑥𝜕𝜕𝜃_𝑥𝑥+𝐻^𝑦𝑦𝜕𝜕𝜃_𝑦𝑦+𝜂^𝑥𝑥𝑦𝜕𝜕Ψ_𝑥𝑥𝑦+𝜂^𝑥𝑦𝑦𝜕𝜕Ψ_𝑥𝑦𝑦+𝜂^𝑥𝑥𝑥𝜕𝜕Ψ_𝑥𝑥𝑥+𝜂^𝑦𝑦𝑦𝜕𝜕Ψ_𝑦𝑦𝑦 (20) is the third prolongation of 𝑋.

We now introduce the total derivatives by differentiating (15) with respect to 𝑥, 𝑦, and 𝑡 and construct 𝐷_𝑥=𝜕_𝑥+Ψ_𝑥𝜕_Ψ+𝑃_𝑥𝜕_𝑝+𝜃_𝑥𝜕_𝜃+Ψ_𝑥𝑥𝜕_Ψ𝑥+𝑃_𝑥𝑥𝜕_𝑃𝑥+𝜃_𝑥𝑥𝜕_𝜃𝑥+Ψ_𝑥𝑦𝜕_Ψ𝑦+𝜃_𝑥𝑦𝜕_𝜃𝑦𝐷+⋯,_𝑦=𝜕_𝑦+Ψ_𝑦𝜕_Ψ+𝑃_𝑦𝜕_𝑃+𝜃_𝑦𝜕_𝜃+Ψ_𝑦𝑦𝜕_Ψ𝑦+𝑃_𝑦𝑦𝜕_𝑃𝑦+𝜃_𝑦𝑦𝜕_𝜃𝑦+Ψ_𝑥𝑦𝜕_Ψ𝑥+𝜃_𝑥𝑦𝜕_𝜃𝑥𝐷+⋯,_𝑡=𝜕_𝑡+Ψ_𝑡𝜕_Ψ+𝑃_𝑡𝜕_𝑃+𝜃_𝑡𝜕_𝜃+Ψ_𝑡𝑡𝜕_Ψ𝑡+𝑃_𝑡𝑡𝜕_𝑃𝑡+𝜃_𝑡𝑡𝜕_𝜃𝑡+Ψ_𝑥𝑡𝜕_Ψ𝑥+𝜃_𝑥𝑡𝜕_𝜃𝑥+⋯. (21) Choosing small 𝐺_𝑟 when 𝑇_ℎ≈𝑇_𝑤, the system of (10)–(12) has the six parameter Lie-group points of symmetries generated by 𝑋₁=𝜕𝜕𝑡,𝑋₂𝜕=𝜃𝜕𝑡,𝑋₃=𝜕𝜕𝜃,𝑋₄=𝜕,𝑋𝜕Ψ₅=𝐹₂𝑡𝜕𝜕𝑦,𝑋₆=𝐹₁𝑡𝜕.𝜕𝑥 (22)

3.2. Invariant Solution

When calculating invariant solutions under the group generators 𝑋₃ and 𝑋₄, we found that there are no invariant solutions. Then 𝑋₅ and 𝑋₆ give solutions of (1)–(3) and this contradicts the boundary conditions.

For 𝑋₁ and 𝑋₂, the characteristic Φ=(Φ_Ψ,Φ_𝑃,Φ_𝜃) has the following components: Φ_Ψ=−Ψ_𝑡,Φ_𝑃=−𝑃_𝑡,Φ_𝜃=−𝜃_𝑡. (23) The general solutions of invariant surface conditions (17) are given by Ψ=ℎ𝑦𝐻𝑥,y,𝑃=Γ𝑥,𝑦,𝜃=𝜏𝑥,𝑦. (24) Invoking (24) into (10), we have 𝑑−𝐾³ℎ𝑑𝑦³+−𝛼𝐾𝑦−ℎ𝐾₁−3𝐾𝐾₂𝑑²ℎ𝑑𝑦²+−𝛼𝐾−2𝛼𝐾𝑦𝐾₂−ℎ𝐾₃+ℎ𝐾₄−𝐾𝐾₅−3𝐾𝐾₆+1𝑅𝑑ℎ,𝐾𝑑𝑦₁𝑑ℎ𝑑𝑦²+−𝛼𝐾𝐾₂+1𝑅𝐾₂−𝛼𝐾𝐾₆𝑦−𝐾𝐾₉−𝐾𝐾₁₀ℎ+𝐾₇−𝐾₈ℎ²+1𝐻𝑑Γ,𝐾𝑑𝑥₁=𝐻_𝑥,𝐾₂=𝐻_𝑦𝐻,𝐾₃=𝐻_𝑥𝐻_𝑦𝐻,𝐾₄=𝐻_𝑥𝑦,𝐾₅=𝐻_𝑥𝑥𝐻,𝐾₆=𝐻_𝑦𝑦𝐻,𝐾₇=𝐻_𝑦𝐻_𝑥𝑦𝐻,𝐾₈=𝐻_𝑥𝐻_𝑦𝑦𝐻,𝐾₉=𝐻_𝑥𝑥𝑦𝐻,𝐾₁₀=𝐻_𝑦𝑦𝑦𝐻,𝐾=𝑅_𝑒. (25)(26) Integration of 𝐻_𝑥=𝐾₁ from (26) leads to the following expression: 𝐻𝑥,𝑦=𝑥𝐾₁𝑦+𝐾₁₁𝑦. (27) The above equation when used into Ψ=ℎ(𝑦)𝐻(𝑥,𝑦) (from (24)) gives Ψ=𝑥𝐾₁𝑦+𝐾₁₁𝑦ℎ𝑦, (28) which after differentiating with respect to 𝑦 and using (14c) yields 𝐾₁₁𝑦ℎ𝑦=𝐾₁₂, (29) where 𝐾₁₂ is a constant of integration and hence (28) reads Ψ=𝑥𝐺𝑦+𝐾₁₂ (30) with 𝐺(𝑦)=𝐾₁(𝑦)ℎ(𝑦).

Putting 𝑃=Γ(𝑥,𝑦) from (24) and (27) into the last term of (25) yields 𝐾₁₁=0. (31) With the help of (27) and (31), one obtains 𝐻𝑥,𝑦=𝑥𝐾₁𝑦, (32) while (29)–(31) yield Ψ=𝑥𝐺𝑦. (33) Due to (13) and (33), one may write 𝑢=𝑥𝑑𝐺𝑑𝑦,𝑣=−𝐺. (34) Using (33) in (11) and then differentiating with respect to 𝑥, one arrives at the following result: 𝑃_𝑥𝑦=1ℎ²𝐺_𝑟𝜃_𝑥. (35) Putting (33) into (10), differentiating with respect to 𝑦, and then using (35), we obtain 𝑑⁴𝐺𝑑𝑦⁴𝑦𝑑𝑥+𝛼³𝐺𝑑𝑦³𝑑+2²𝐺𝑑𝑦²𝑥+𝑅_𝑒𝐺𝑑³𝐺𝑑𝑦³𝑥−𝑅_𝑒𝑑𝐺𝑑𝑑𝑦²𝐺𝑑𝑦²𝑥−𝑅_𝑒𝑑²𝐺𝑑𝑦²1𝑅1𝑥+ℎ²𝐺_𝑟𝜃_𝑥=0. (36) Using (33) and 𝜃=𝜏(𝑥,𝑦) from (24) in (12), we can write 𝑥𝑑𝐺𝑑𝑦𝜕𝜏𝜕𝑥−𝐺𝜕𝜏−1𝜕𝑦𝑃_𝑟𝑅_𝑒𝜕²𝜏𝜕𝑥²+𝜕²𝜏𝜕𝑦²=0, (37) and the boundary conditions (14a), (14b), and (14c) become (i)𝑑𝐺(1)()𝐺(1)()𝑑𝑑𝑦=0,ii=1,iii²𝐺(0)𝑑𝑦²()𝐺(0)(v)𝜏()()𝜏()=0,iv=0,𝑥,1=0,vi𝑥,0=1. (38) Using 𝜃=𝜏(𝑥,𝑦) and equating-like powers of ℎ, (36) helps in writing the following equations: 𝑑⁴𝐺𝑑𝑦⁴𝑦𝑑+𝛼³𝐺𝑑𝑦³𝑑+2²𝐺𝑑𝑦²+𝑅_𝑒𝐺𝑑³𝐺𝑑𝑦³−𝑅_𝑒𝑑𝐺𝑑𝑑𝑦²𝐺𝑑𝑦²−𝑅_𝑒𝑑²𝐺𝑑𝑦²1𝑅𝐺=0,_𝑟𝜏_𝑥=0. (39) The above equation implies that 𝜏=𝐸(𝑦) and 𝐺_𝑟=0 which satisfy our assumption that 𝐺_𝑟 is very small. Now (36)–(38) yield 𝑑⁴𝐺𝑑𝑦⁴𝑦𝑑+𝛼³𝐺𝑑𝑦³𝑑+2²𝐺𝑑𝑦²+𝑅_𝑒𝐺𝑑³𝐺𝑑𝑦³−𝑅_𝑒𝑑𝐺𝑑𝑑𝑦²𝐺𝑑𝑦²−𝑅_𝑒𝑑²𝐺𝑑𝑦²1𝑅𝐺𝑦=0,𝜕𝐸+1𝜕𝑦𝑃_𝑟𝑅_𝑒𝜕²𝐸𝜕𝑦²(i)(1)=0,𝑑𝐺()𝐺(1)()𝑑𝑑𝑦=0,ii=1,iii²𝐺(0)𝑑𝑦²()𝐺(0)(v)𝐸(1)()𝐸=0,iv=0,=1,vi^′(0)=0. (40)(41)(42)

3.3. Analytical Solution

The aim of this section is to find the solutions of (40)–(42) using double perturbation [3, 4]. For small 𝑅_𝑒 and 𝛼, we expand 𝐺=𝐺₁+𝑅_𝑒𝐺₂+𝑂𝑅²_𝑒,𝐺₁=𝐺₁₀+𝛼𝐺₁₁+𝑂𝛼²,𝐺₂=𝐺₂₀+𝛼𝐺₂₁+𝑂𝛼². (43) Using (43) into (40)–(42) and then solving the resulting problems for small 𝑅_𝑒 and 𝛼, we obtain 𝐺₁₀𝑦1=−2𝑦³+32𝐺𝑦,₁₁𝑦=3𝑦40⁵−3𝑦20³+3𝐺40𝑦,₂₀𝑦=1𝑦280⁷−3𝑦280³+1𝑦+1140𝑅−1𝑦40⁵+1𝑦20³−1𝑦,𝐺40₂₁𝑦=−13𝑦20160⁹−9𝑦2800⁷+9𝑦5600⁵+227𝑦25200³+227𝑦+133600𝑅1𝑦210⁷−3𝑦200⁵+11𝑦700³−23𝑦,𝐺4200₁𝑦1=−2𝑦³+323𝑦+𝛼𝑦40⁵−3𝑦20³+3𝑦,𝐺40₂𝑦=1𝑦280⁷−3𝑦280³+1𝑦+1140𝑅−1𝑦40⁵+1𝑦20³−1𝑦−40+𝛼13𝑦20160⁹−9𝑦2800⁷+9𝑦5600⁵+227𝑦25200³−227𝑦+133600𝑅1𝑦210⁷−3𝑦200⁵+11𝑦700³−23𝑦,𝐺𝑦=−142002𝑦³+323𝑦+𝛼𝑦40⁵−3𝑦20³+3𝑦40+𝑅_𝑒1𝑦280⁷−3𝑦280³+1𝑦+1140𝑅−1𝑦40⁵+1𝑦20³−1𝑦−40+𝛼13𝑦20160⁹−9𝑦2800⁷+9𝑦5600⁵+227𝑦25200³−227𝑦+133600𝑅1𝑦210⁷−3𝑦200⁵+11𝑦700³−23𝑦.4200 (44)(45) It is noted that for 𝑅→∞, the expression of 𝐺(𝑦) in [4] is recovered.

Let 𝐸=𝐸₁+𝑅_𝑒𝐸₂+𝑂𝑅²_𝑒. (46) From (41), (45), and (46), we obtain 𝑑²𝐸₁𝑑𝑦²=0,𝐸₁(1)=1,𝐸^′₁(0)𝑃=0,_𝑟𝐺𝑦𝑑𝐸₁𝑦+𝑑𝑑𝑦²𝐸₂𝑦𝑑𝑦²,𝐸₂(1)=0,𝐸^′₂(0)=0. (47) Solving the above problems and using (46), one obtains 𝐸𝑦=1. (48)

3.4. Numerical Solution

Now the numerical solution of (40)–(42) has been obtained using shooting method with Runge-Kutta scheme.

4. Results and Discussion

Figures 1, 2, 3, and 4 illustrate the behaviour of self-axial velocity over a range of 𝑅 with 𝑅_𝑒 and 𝛼 fixed.

Figure 1 

Self-axial velocity profiles over a range of 𝑅, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=1 and 𝛼=0.5.

Figure 2 

Self-axial velocity profiles over a range of 𝑅, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=1 and 𝛼=−0.5.

Figure 3 

Self-axial velocity profiles over a range of 𝑅 where blue=1, pink=0.5, yellow=−0.5 and green=−1 at 𝑅𝑒=−1 and 𝛼=−0.5.

Figure 4 

Self-axial velocity profiles over a range of 𝑅, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=−1 and 𝛼=0.5.

Figures 1 and 2 illustrate the behaviour of self-axial velocity 𝑢/𝑥 for permeation Reynolds number 𝑅_𝑒=1 (injection) and 𝛼=0.5,−0.5 (expansion and contraction, resp.) over a range of porosity parameter 𝑅. For 𝑅>0, these figures show that the higher porosity 𝑅 leads to higher self-axial velocity near the center and lower near the wall. The results for 𝑅<0 are quite opposite to that of 𝑅>0. A comparative study of these figures further indicates that the self-axial velocity near the center in case of injection with expanding wall and high porosity is higher than injection with contracting wall and high porosity.

The plots of self-axial velocity 𝑢/𝑥 for permeation Reynolds number 𝑅_𝑒=−1 (suction) and 𝛼=0.5,−0.5 (expansion and contraction, resp.) over a range of 𝑅 have been displayed in Figures 3 and 4. In case of 𝑅>0, these graphs depict that the higher porosity 𝑅 leads to lower self-axial velocity near the center and higher near the wall. For 𝑅<0, these figures depict that the lower porosity 𝑅 leads to higher self-axial velocity near the center and lower near the wall. By comparing Figures 3 and 4, we note that the self-axial velocity near the center in case of suction with expanding wall and high porosity is higher than suction with contracting wall and high porosity.

The behaviour of the self-axial velocity 𝑢/𝑥 for wall dilation rate 𝛼=−0.5 (contraction) and 𝑅_𝑒=1,−1 (injection and suction) over a range of 𝑅 has been displayed in Figures 2 and 3. For 𝑅>0, Figure 3 shows that the higher the porosity 𝑅, the lower the self-axial velocity at the center and higher near the wall. Figure 2 shows that the higher the porosity 𝑅, the higher the self-axial velocity at the center and lower near the wall. When 𝑅<0, Figure 3 elucidates that the lower porosity 𝑅 gives a higher self-axial velocity near the center and a lower one near the wall. Figure 2 elucidates that the lower porosity 𝑅 gives a lower self-axial velocity near the center and a higher one near the wall. A comparative study of Figures 2 and 3 indicates that the self-axial velocity near the center in case of injection with contracting wall and high porosity is higher than suction with contracting wall and high porosity.

The variations of self-axial velocity 𝑢/𝑥 for wall dilation rate 𝛼=0.5 (expansion) and 𝑅_𝑒=1,−1 (injection and suction) over a range of 𝑅 have been plotted in Figures 1 and 4. When 𝑅>0, then Figure 1 shows that the higher the porosity 𝑅, the higher the self-axial velocity at the center and lower near the wall. Figure 4 shows that the higher the porosity 𝑅, the lower the self-axial velocity at the center and higher near the wall. When 𝑅<0, Figure 1 describes that the lower porosity 𝑅 gives lower self-axial velocity near the center and higher near the wall. Figure 4 provides that the lower porosity 𝑅 yields higher self-axial velocity near the center and lower near the wall. Comparison of Figures 1 and 4 leads to the conclusion that the self-axial velocity near the center for suction with expanding wall and high porosity is higher than injection with expanding wall and high porosity.

Tables 1, 2, 3, and 4 depict that the percentage error decreases when 𝑅 increases.

Table 1 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅_𝑒=1, 𝛼=0.5.

Table 2 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1. for 𝑅_𝑒=1, 𝛼=−0.5.

Table 3 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅_𝑒=−1, 𝛼=0.5.

Table 4 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅_𝑒=−1, 𝛼=−0.5.

Figures 5, 6, 7, and 8 plot the behaviour of self-axial velocity over a range of 𝛼 with fixed 𝑅_𝑒 and 𝑅.

Figure 5 

Self-axial velocity profiles over a range of 𝛼, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=1 and 𝑅=0.5.

Figure 6 

Self-axial velocity profiles over a range of 𝛼, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=−1 and 𝑅=0.5.

Figure 7 

Self-axial velocity profiles over a range of 𝛼, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=−1 and 𝑅=−0.5.

Figure 8 

Self-axial velocity profiles over a range of 𝛼, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝑅𝑒=1 and 𝑅=−0.5.

For 𝛼>0, Figures 5–8 witness that the greater 𝛼 leads to higher self-axial velocity at the center and lower near the wall. For 𝛼<0, these figures show that an increase in contraction ratio leads to lower self-axial velocity near the center and higher near the wall. By comparing Figures 5 and 6, we note that the self-axial velocity near the center in case of suction with expanding wall and high porosity is higher than injection with expanding wall and high porosity.

Comparison of Figures 5 and 8 shows that the self-axial velocity near the center in case of injection with expanding wall and low porosity is higher than injection with expanding wall and high porosity. Comparative study of Figures 6 and 7 reveals that the self-axial velocity near the center in case of suction with expanding wall and high porosity is higher than suction with expanding wall and low porosity. By comparing Figures 7 and 8, the self-axial velocity near the center in case of injection with expanding wall and low porosity is higher than suction with expanding wall and low porosity.

Tables 5, 6, 7, and 8 indicate that the percentage error is an increasing function of 𝛼.

Table 5 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=1, 𝑅_𝑒=1.

Table 6 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=1, 𝑅_𝑒=−1.

Table 7 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=−1, 𝑅_𝑒=−1.

Table 8 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=−1, 𝑅_𝑒=1.

Figures 9, 10, 11, and 12 illustrate the behaviour of self-axial velocity over a range of 𝑅_𝑒 with fixed 𝛼 and 𝑅.

Figure 9 

Self-axial velocity profiles over a range of 𝑅𝑒, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝛼=0.5 and 𝑅=0.5.

Figure 10 

Self-axial velocity profiles over a range of 𝑅𝑒, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝛼=−0.5 and 𝑅=0.5.

Figure 11 

Self-axial velocity profiles over a range of 𝑅𝑒, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝛼=−0.5 and 𝑅=−0.5.

Figure 12 

Self-axial velocity profiles over a range of 𝑅𝑒, where blue=1, pink=0.5, yellow=−0.5, and green=−1 at 𝛼=0.5 and 𝑅=−0.5.

The self-axial velocity 𝑢/𝑥 for porosity parameter 𝑅=0.5 (high porosity) and wall dilation rate 𝛼=0.5,−0.5 (expansion and contraction, resp.) over a range of 𝑅_𝑒 has been sketched in Figures 9 and 10. For 𝑅_𝑒>0, we found that increasing injection 𝑅_𝑒 leads to a lower self-axial velocity at the center and a higher one near the wall. When 𝑅_𝑒<0, Figures 9 and 10 indicate that increasing suction ratio leads to a higher self-axial velocity near the center and a lower one near the wall. Comparison of Figures 9 and 10 shows that the self-axial velocity near the center in case of injection with expanding wall and high porosity is higher than injection with contracting wall and high porosity.

Figures 11 and 12 provide the variation of self-axial velocity 𝑢/𝑥 for porosity parameter 𝑅=−0.5 (low porosity) and wall dilation rate 𝛼=0.5,−0.5 (expansion and contraction, resp.) over a range of 𝑅_𝑒. In case of 𝑅_𝑒>0, Figures 11 and 12 show that increasing injection leads to a higher self-axial velocity near the center and a lower one near the wall. For 𝑅_𝑒<0, Figures 11 and 12 show that increasing suction ratio leads to a lower self-axial velocity at the center and a higher one near the wall. A comparison between Figures 11 and 12 shows that the self-axial velocity near the center in case of injection with expanding wall and low porosity is higher than injection with contracting wall and low porosity.

The self-axial velocity 𝑢/𝑥 for porosity parameter 𝑅=−0.5,0.5 (low and high porosity, resp.) and wall dilation rate 𝛼=−0.5 (contraction) over a range of 𝑅_𝑒 has been explained in Figures 10 and 11. When 𝑅_𝑒>0, Figure 10 shows that increasing injection leads to a lower self-axial velocity near the center and a higher one near the wall. Figure 11 shows that increasing injection leads to a higher self-axial velocity near the center and a lower one near the wall. In case of 𝑅_𝑒<0, Figure 10 shows that increasing suction ratio leads to a higher self-axial velocity at the center and a lower one near the wall. Increasing suction ratio leads to a lower self-axial velocity at the center and a higher one near the wall (Figure 11). A comparison shows that the self-axial velocity near the center in case of injection with contracting wall and low porosity is higher than injection with contracting wall and high porosity (Figures 10 and 11).

Figures 9 and 12 indicate the behaviour of self-axial velocity 𝑢/𝑥 for porosity parameter 𝑅=−0.5,0.5 (low and high porosity, resp.) and wall dilation rate 𝛼=0.5 (expansion) over a range of 𝑅_𝑒. In case of 𝑅_𝑒>0, Figure 9 shows that increasing injection leads to a lower self-axial velocity near the center and a higher one near the wall. Figure 12 shows that increasing injection leads to a higher self-axial velocity near the center and a lower one near the wall. In case of 𝑅_𝑒<0, Figure 9 depicts that increasing suction ratio leads to a higher self-axial velocity at the center and a lower one near the wall. Figure 12 shows that increasing suction ratio leads to a lower self-axial velocity at the center and a higher one near the wall. By comparing Figures 9 and 12, the self-axial velocity near the center in case of injection with expansion wall and low porosity is higher than injection with expansion wall and large porosity.

Tables 9, 10, 11, and 12 show the percentage error decrease for a small 𝑅_𝑒.

Table 9 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=1, 𝛼=−0.5.

Table 10 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=1, 𝛼=0.5.

Table 11 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=−1, 𝛼=−0.5.

Table 12 

Comparison between analytical and numerical solutions for self-axial velocity 𝑢/𝑥 at 𝑦=0.1 for 𝑅=−1, 𝛼=0.5.

The plots in Figure 13 elucidate that the temperature distribution is constant throughout and it is independent of physical parameter. Numerical solution for temperature is similar to our analytical solution, and therefore, temperature distribution has no error.

Figure 13 

Temperature distribution profile.

5. Conclusions

In this paper, we have generalized the flow analysis of [4] with the influence of porous medium and heat transfer. Analytical solution for the arising nonlinear problem is obtained by using Lie symmetry technique in conjunction with a second-order double perturbation method. We have studied the effects of porous medium (𝑅), permeation Reynolds 𝑅_𝑒, and wall dilation rate 𝛼 on the self-axial velocity and temperature distribution within the fluid. We compared the analytical solution with the numerical solution for self-axial velocity for the different values of 𝑅, 𝑅_𝑒, and 𝛼.

It was found that the temperature distribution has no error since analytical solution is similar to numerical solution and both are equal to one. We also found that as 𝑅 increases, the percentage error decreases and that temperature distribution is constant throughout. Here, we have noticed that the obtained analytical results match quite well with the numerical results for a good range of these parameters. We also noticed that in all cases, the self-axial velocity has similar trend as in [4], that is, the self-axial velocity approaches a cosine profile. Finally, we observed that when 𝑅 approaches infinity, our problem reduces to the problem in [4] and our results (analytical and numerical) also reduce to the results in [4].