Nonclassical Symmetry Analysis of Heated Two-Dimensional Flow Problems

1 Introduction

Prandtl [1] derived the boundary layer equations by simplifying the Navier–Stokes equations. Schlichting [2, 3] developed boundary layer equations for two-dimensional flow. The problem of the radial laminar heated free jet was studied by Schwarz [4] and later by Naz et al. [5]. The similarity solutions and conserved quantities were established. The boundary layer equations are usually solved subject to certain initial and boundary conditions to describe flow in jets, films, fractures, and plates. Each jet flow problem has a conserved quantity, and that is a measure of the strength of the jet. Naz et al. [6] developed a new method to establish conserved quantities for different types of jet flow problems. Three different types of jets widely studied in literature are the liquid jet, free jet and wall jet. The boundary conditions and conserved quantities for each jet flow problem are different, but the governing partial differential equation is the same for all these jets.

The exact solutions of nonlinear PDEs are derived by classical and nonclassical methods. Bluman and Cole developed nonclassical method to construct nonclassical symmetries for PDEs and assumed that the required analytical solution is invariant under symmetry transformations preserving both the form of the differential equation and the invariant solution condition [7]. This is an elegant approach and yields more symmetries than the Lie classical approach. The direct method from Clarkson and Kruskal [8] is adopted by many researchers to construct nonclassical symmetries and new solutions for PDEs. For a scalar equation, the direct method and the nonclassical method yield the same number of reductions (see Olver [9]). Bila and Niesen [10] proposed a new procedure for finding nonclassical symmetries, but this method is applicable to a specific class of PDEs. Bruzon and Gandarias [11] modified this new procedure to find nonclassical symmetries and reported a new nonclassical symmetry for the shallow water wave equation. One can obtain determining equations for nonclassical symmetries by some powerful computer packages (see e.g. [12, 13]). Filho and Figueiredo [13] developed a computer package SADE on Maple for calculating the determining equations and nonclassical symmetries. SADE is based on the nonclassical method developed by Bluman and Cole [7]. The method adopted in this article is the nonclassical method. The Maple-based computer package SADE is utilised to derive the determining equations for nonclassical symmetries, and the solution of determining equations yield a number of nonclassical symmetries. The group invariant solutions of boundary layer equations for two-dimensional heated flow problems are established by these nonclassical symmetries.

The Lie classical symmetry analysis for jet flow problems is studied in the literature up to this point. The group invariant solutions for boundary layer equations are constructed by the Lie classical symmetries. Naz et al. [14] studied the nonclassical symmetries of boundary layer equations for two-dimensional and radial flows. A new group invariant solution was obtained by utilising the nonclassical symmetries. In Ludlow et al. [15], the similarity solutions of the unsteady incompressible boundary-layer equations were established by direct method. The similarity reductions of the steady-state boundary layer equations were studied by Brude [16]. The similarity solutions of the steady two-dimensional boundary-layer equations in the flat and axisymmetric case are indeed invariant solutions under the action of nonclassical symmetries by Saccomandi [17] . In this article, we consider nonclassical symmetry analysis for two-dimensional heated flows and obtain a number of new solutions. This artile adds to the literature in three ways. First, we derive the nonclassical symmetries for the two-dimensional heated flows. This analysis will help to explore new problems in jets, films, fractures, and plates by imposing suitable conditions and physical conserved quantities. We obtain the solution of determining equations for nonclassical symmetries directly without using SADE. It is a tedious job to compute nonclassical symmetries. Second, we have derived a number of group invariant solutions for boundary layer equations for two-dimensional heated flows via nonclassical symmetries. Third, we have successfully formulated a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plat and, with suitable boundary conditions, we solved this problem.

This article is organised in the following manner. In Section 2 the nonclassical symmetries of boundary layer equations for two-dimensional flows is presented. The group invariant solutions of boundary layer equations for two-dimensional flows are given in Section 3. In Section 4, heat transfer analysis for fluid flow over a linearly stretching porous plate is studied via nonclassical symmetry analysis. Finally, conclusions are summarised in Section 5.

2 Nonclassical Symmetries of Boundary Layer Equations for Two-Dimensional Heated Flows

The two-dimensional heated flow problems are represented by the following boundary layer equations:

Here (x, y) denotes the usual orthogonal cartesian coordinates parallel and perpendicular to the boundary y=0, T is temperature, and ψ the stream function. The velocity components in the x and y directions are u(x, y) and v(x, y), respectively. The velocity components are related to stream function ψ as

Consider the infinitesimal operator

The system (1) can be rewritten as

The invariant surface conditions are

The nonclassical symmetries determining equations are (see e.g. [7])

where X^[1] and X^[3] are the usual first and third prolongations of operator X and can be computed from

where ζi1, i2, …, isα can be determined from

ζiα = Di(ηα) − ujαDi(ξj)ζi1, i2, …, isα = Dis(ζi1, i2, …, is−1α) − uji1, i2, …, is−1αDis(ξj),  s > 1,

in which

is the total derivative operator with respect to xⁱ. The standard procedure for deriving nonclassical symmetries is to consider following two scenarios: Scenario 1: ξ¹≠0 and Scenario 2: ξ¹=0, ξ²≠0.

3 Group Invariant Solutions of Boundary Layer Equations for Two-Dimensional Heated Flows

In this section we construct the group invariant solutions of boundary layer equations for two-dimensional heated flows for both Scenarios 1 and 2. When ξ¹≠0, we fix ξ¹=1 and obtain six nonclassical symmetry generators X₁, …, X₆. For the case ξ¹=0, ξ²=1, we derive three nonclassical symmetry generators X₇, X₈, and X₉. We set F(x, y, ψ, T)=e^x+y and G(x, ψ, T)=T in X₉ in order to derive group invariant solutions using symmetries X₇, X₈, and X₉. The similarity variables, reduced forms, and group invariant solutions of system (1) for both scenarios are presented in Table 1. The complete details of the method of how to construct a group invariant solution is provided in next section.

Now the question arises, what is the significance of derived group invariant solutions? It would be of interest to identify what type of physical phenomena can be associated with the solutions derived via nonclassical symmetry analysis. This analysis could help to explore new problems in jets, films, fractures, and plates by imposing suitable conditions and physical conserved quantities. With suitable boundary conditions the solution of system (1) derived via nonclassical symmetry X₃ can be utilised to describe flow over a heated porous plate.

4 Heat Transfer Analysis for Fluid Flow Over a Linearly Stretching Porous Plate

Consider an incompressible flow of viscous two dimensional flow over a porous plate. Assume that plate is along x-axis and y-axis is perpendicular to the plate. The plate is moving with stretching velocity u=ax. Moreover the plate is also porous having velocity v=b (or we can take v=−b) and fluid above the plate is at rest. A constant temperature T₀ is applied to the moving plate while the temperature of the fluid is taken as T_∞. The governing equations for this two dimensional fluid flow in terms of stream function ψ and temperature T are given by (1). The fluid flow over a linearly stretching porous plate is shown in Figure 1.

Figure 1:

Flow over a linearly stretching porous plate.

The boundary conditions for this problem are given by

Now, ψ=ϕ(x, y) and T=γ(x, y) are group invariant solutions of the system (1) if

where the operator X is given in (16) . The solution of (31) for u=U(x, y) is of the form

Now, substitution of (32) into system (1) yields system of following two ODEs:

The solution of (33) gives

Substituting A(r) and B(r) from (34) into (32), one can obtain solution as given in Table 1. Now, we find a solution subject to boundary conditions (30). The boundary conditions (30) yield p(x) = p0, b = 6p0, 6a + b2 = 0 and following conditions are obtained:

The boundary conditions (35) provides

The group invariant solution (32) with the aid of (34) and (36) takes following form:

where p0 = 6b and the solution is valid under restriction 6a+b²=0. The velocity components are given by

According to the best of our knowledge, the solution (37) is new and not reported in literature before.

5 Conclusions

The nonclassical symmetries of boundary layer equations for two-dimensional heated flows were computed by computer package SADE. First we derived the nonclassical symmetry determining equations by SADE. We solved these equations directly to obtain nonclassical symmetries. The nonclassical symmetries were computed for two different scenarios, ξ¹≠0 and ξ¹=0, ξ²≠0. We obtained six nonclassical symmetry generators for Scenario 1. The nonclassical symmetry generators X₁ and X₂ contain arbitrary functions. For Scenario 2, ξ¹=0, ξ²≠0, five nonclassical symmetries involving arbitrary functions were established. We explicitly mentioned three nonclassical symmetries, X₇, X₈, and X₉. The two additional nonclassical symmetries in this case were not explicitly given, as the arbitrary functions are too complex. Furthermore, the group invariant solutions by utilising the nonclassical symmetries were derived. We introduced similarity variables associated with each nonclassical symmetry. The variables reduced the system of PDEs to a system of ODEs in terms of similarity variables. The reduced ODEs were solved to obtain group invariant solution for boundary layer equations governing flow in two-dimensional heated flow problems. It is of interest to identify what type of physical phenomena can be associated with the solutions derived via nonclassical symmetry. We successfully formulated a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plate.