Nonclassical Symmetry Analysis of Boundary Layer Equations

Abstract

The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

1. Introduction

Prandt [1] derived the boundary layer equations by simplifying the Navier-Stokes equations. Schlichting [2, 3] showed that the two-dimensional flow is represented by the boundary layer equation: 𝜓_𝑦𝜓_𝑥𝑦−𝜓_𝑥𝜓_𝑦𝑦−𝜓_𝑦𝑦𝑦=0. (1.1) Here (𝑥,𝑦) denote the usual orthogonal cartesian coordinates parallel and perpendicular to the boundary 𝑦=0, 𝜓 denotes the stream function. The velocity components in the 𝑥 and 𝑦 directions, 𝑢(𝑥,𝑦) and 𝑣(𝑥,𝑦), are related with 𝜓 as 𝑢=𝜓_𝑦 and 𝑣=−𝜓_𝑥. The boundary-layer equations are usually solved subject to certain conditions to describe flow in jets, films, and plates. In jet flow problems due to homogeneous boundary conditions a further condition known as conserved quantity is required. The conserved quantity is a measure of the strength of the jet. A new method of deriving conserved quantities for different types of jet flow problems was discussed by Naz et al. [4]. The liquid jet, the free jet, and the wall jet satisfy the same partial differential equations (1.1), but the boundary conditions and conserved quantities for each jet flow problem are different. The boundary-layer equations were solved subject to certain boundary conditions and conserved quantity for two-dimensional free, wall, and liquid jets in [2, 3, 5–8].

The radial flow is represented by the boundary layer equation (see, e.g., Squire [9]): 1𝑥𝜓_𝑦𝜓_𝑥𝑦−1𝑥²𝜓²_𝑦−1𝑥𝜓_𝑥𝜓_𝑦𝑦−𝜓_𝑦𝑦𝑦=0. (1.2) Cylindrical polar coordinates (𝑥,𝜃,𝑦) are used. The radial coordinate is 𝑥, the axis of symmetry is 𝑥=0, and all quantities are independent of 𝜃. The velocity components in the 𝑥 and 𝑦 directions, 𝑢(𝑥,𝑦) and 𝑣(𝑥,𝑦), are related to 𝜓 as 𝑢=(1/𝑥)𝜓_𝑦 and 𝑣=(−1/𝑥)𝜓_𝑥. The boundary layer equations were solved subject to certain boundary conditions and conserved quantity for radial-free jet in works [6, 9–11], wall jet by Glauert [7], and liquid jet in [8, 10].

The classical and nonclassical symmetry methods play a vital role in deriving the exact solutions to nonlinear partial differential equations. The nonclassical method due to Bluman and Cole [12] and the direct method due to Clarkson and Kruskal [13] have been successfully applied for constructing the nonclassical symmetries and new solutions for partial differential equations. Olver [14] has shown that for a scalar equation, every reduction obtainable using the direct method is also obtainable using the nonclassical method. An algorithm for calculating the determining equations associated with the nonclassical method was introduced by Clarkson and Mansfield [15]. In the nonclassical method the invariant surface condition is augmented by the invariant surface condition. A new procedure for finding nonclassical symmetries is given in [16, 17], but this is restricted to a specific class of PDEs. Recently Filho and Figueiredo [18] developed a powerful computer package SADE for calculating the nonclassical symmetries by converting given PDE system to involutive form or without converting it to involutive form. We will use SADE to calculate the nonclassical symmetries and similarity reductions of boundary layer equations for two-dimensional as well as radial flows.

The paper is arranged in the following pattern: in Section 2 the nonclassical symmetries and similarity solution of boundary layer equations for two-dimensional flows are presented. The nonclassical symmetries and similarity solution of boundary layer equations for radial flows are given in Section 3. Finally, Conclusions are summarized in Section 4.

2. Nonclassical Symmetries and Similarity Solution of Boundary Layer Equations for Two-Dimensional Flows

The two-dimensional flow is represented by the boundary layer equation: Δ₁=𝜓_𝑦𝜓_𝑥𝑦−𝜓_𝑥𝜓_𝑦𝑦−𝜓_𝑦𝑦𝑦=0. (2.1) Consider the infinitesimal operator: 𝑋=𝜉¹𝜕𝑥,𝑦,𝜓𝜕𝑥+𝜉²𝜕𝑥,𝑦,𝜓𝜕𝜕𝑦+𝜂𝑥,𝑦,𝜓.𝜕𝜓 (2.2) The invariant surface condition is Δ₂=𝜉¹𝑥,𝑦,𝜓𝜓_𝑥+𝜉²𝑥,𝑦,𝜓𝜓_𝑦−𝜂𝑥,𝑦,𝜓=0. (2.3) The nonclassical symmetries determining equations are 𝑋^[3]Δ_1Δ1=0,Δ2=0=0,𝑋^[1]Δ_2Δ1=0,Δ2=0, (2.4) where 𝑋^[1] and 𝑋^[3] are the usual first and third prolongations of operator 𝑋. Two cases arise: Case 1  𝜉¹≠0 and Case 2  𝜉¹=0, 𝜉²≠0.

Case 1 (𝜉¹≠0). In this case we set 𝜉¹=1, the SADE package yields the following six determining equations: 𝜉²_𝑢𝜂=0,_𝑢𝑢𝜂=0,_𝑦𝑦𝑦−𝜂_𝑦²+𝜂𝜂_𝑦𝑦=0,3𝜉²_𝑦𝑦−3𝜂_𝑦𝑢−𝜂𝜉²_𝑦−𝜂_𝑢𝜂−𝜂_𝑥−𝜂_𝑦𝜉²𝜉=0,_{2𝑦𝑦𝑦}−3𝜂_𝑦𝑢+𝜂𝜉²_𝑦𝑦+𝜉²𝜂_𝑦𝑦+𝜂_𝑥𝑦−𝜂𝜂_𝑦𝑢+2𝜂_𝑢𝜂_𝑦𝜂=0,_𝑥𝑢−𝜉²_𝑦²−𝜉²_𝑥𝑦+𝜉²𝜂_𝑦𝑢−𝜉²𝜉²_𝑦𝑦+𝜂_𝑢²=0. (2.5) The solution determining equations in (2.5) yield all classical symmetries (see [5]) and the following infinite many nonclassical symmetry generators: 𝜕𝑋=(𝑥)𝜕𝜕𝑥+𝑔𝜕𝜕𝑦+ℎ𝑥,𝑦,𝜕𝑢 (2.6) where ℎ(𝑥,𝑦) and 𝑔(𝑥) satisfy ℎ_𝑥+𝑔(𝑥)ℎ_𝑦=0,ℎ_𝑦𝑦𝑦−ℎ²_𝑦+ℎℎ_𝑦𝑦=0. (2.7) Equation (2.7) yields ℎ=6𝑥,𝑦𝑦−𝐺(𝑥),𝐺^′(𝑥)=𝑔(𝑥), (2.8) and thus the nonclassical symmetry generators in (2.6) take the following form: 𝜕𝑋=𝜕𝑥+𝐺^′(𝑥)𝜕+6𝜕𝑦(𝑥)𝜕𝑦−𝐺.𝜕𝑢 (2.9) Now, 𝜓=𝜙(𝑥,𝑦) is group invariant solution of (2.1) if 𝑋𝜓−𝜙𝑥,𝑦_𝜓=𝜙=0, (2.10) where the operator 𝑋 is given in (2.9). The solution of (2.10) for 𝜓=𝜙(𝑥,𝑦) is of the form 𝜓𝑥,𝑦=6𝑥𝜒+𝑤𝜒,𝜒=𝑦−𝐺(𝑥). (2.11) Substitution of (2.11) in (2.1) yields 𝜒²𝑑³𝑤𝑑𝜒³𝑑+6𝜒²𝑤𝑑𝜒²+6𝑑𝑤𝑑𝜒=0 (2.12) and thus 𝑤𝜒=𝑐₁+𝑐₂𝜒+𝑐₃𝜒². (2.13) The invariant solution (2.11) with the help of (2.13) takes the following form: 𝜓𝑥,𝑦=6𝑥𝜒+𝑐₁+𝑐₂𝜒+𝑐₃𝜒²,𝜒=𝑦−𝐺(𝑥). (2.14) The invariant solution (2.14) is new solution for boundary layer equations for two-dimensional flows.

Case 2 (𝜉¹=0,𝜉²≠0). Results are in no-go case.

3. Nonclassical Symmetries and Similarity Solution of Boundary Layer Equations for Radial Flows

The radial flow is represented by the boundary layer equation: 1Δ=𝑥𝜓_𝑦𝜓_𝑥𝑦−1𝑥²𝜓²_𝑦−1𝑥𝜓_𝑥𝜓_𝑦𝑦−𝜓_𝑦𝑦𝑦=0. (3.1)

Case 1 (𝜉¹≠0). In this case we set 𝜉¹=1 and using SADE we have following six determining equations: 𝜉²_𝑢𝜂=0,_𝑢𝑢=0,𝑥𝜂_𝑦𝑦𝑦−𝜂_𝑦²+𝜂𝜂_𝑦𝑦=0,3𝑥²𝜉²_𝑦𝑦−3𝑥²𝜂_𝑦𝑢−𝑥𝜂𝜉²_𝑦−𝑥𝜂_𝑢𝜂−𝑥𝜂_𝑥−𝑥𝜂_𝑦𝜉²𝑥+𝜂=0,₂𝜉²_𝑦𝑦𝑦−3𝑥²𝜂_𝑦𝑦𝑢+𝑥𝜂𝜉²_𝑦𝑦+𝑥𝜉²𝜂_𝑦𝑦+𝑥𝜂_𝑥𝑦−𝑥𝜂𝜂_𝑦𝑢+2𝑥𝜂_𝑢𝜂_𝑦−3𝜂_𝑦𝑥=0,₂𝜂_𝑥𝑢−𝑥²𝜉²_𝑦²−𝑥²𝜉²_𝑥𝑦+𝑥²𝜉²𝜂_𝑦𝑢−𝑥²𝜉²𝜉²_𝑦𝑦+𝑥²𝜂_𝑢²−2𝑥𝜂_𝑢+2=0. (3.2) The solution determining equations in (3.2) yield all classical symmetries and the following infinite many nonclassical symmetry generators: 𝜕𝑋=+1𝜕𝑥𝑥(𝑥𝜕−𝑦+𝑥𝑔)𝜕𝜕𝑦+ℎ𝑥,𝑦,𝜕𝑢 (3.3) where ℎ(𝑥,𝑦) and 𝑔(𝑥) satisfy 𝑥ℎ_𝑥−𝑦ℎ_𝑦+𝑥𝑔(𝑥)ℎ_𝑦−2ℎ=0,𝑥ℎ_𝑦𝑦𝑦−ℎ²_𝑦+ℎℎ_𝑦𝑦=0. (3.4)
The nonclassical symmetry generators (3.3) finally become 𝜕𝑋=+1𝜕𝑥𝑥−𝑦+𝑥𝐺^′(𝑥)(𝑥𝜕+𝐺)+𝜕𝑦6𝑥(𝑥)𝜕𝑦−𝐺,𝜕𝑢 (3.5) and we have used ℎ𝑥,𝑦=6𝑥𝑦−𝐺(𝑥),𝑥𝐺^′(𝑥)+𝐺(𝑥)=𝑥𝑔(𝑥). (3.6) Now, 𝜓=𝜙(𝑥,𝑦) is group invariant solution of (3.1) if 𝑋𝜓−𝜙(𝑥,𝑦)_𝜓=𝜙=0 (3.7) where the operator 𝑋 is given in (3.5). The solution of (3.7) for 𝜓=𝜙(𝑥,𝑦) is of the form: 𝜓𝑥,𝑦=2𝑥³𝜒+𝑤𝜒,𝜒=𝑥𝑦−𝐺(𝑥). (3.8) Substitution of (3.8) in (3.1) yields 𝜒²𝑑³𝑤𝑑𝜒³𝑑+6𝜒²𝑤𝑑𝜒²+6𝑑𝑤𝑑𝜒=0 (3.9) and thus 𝑤𝜒=𝑐₁+𝑐₂𝜒+𝑐₃𝜒². (3.10) Finally, we have following form invariant solution: 𝜓𝑥,𝑦=2𝑥³𝜒+𝑐₁+𝑐₂𝜒+𝑐₃𝜒²,𝜒=𝑥𝑦−𝐺(𝑥), (3.11) and this is new solution not obtained in the literature.

Case 2 (𝜉¹=0,𝜉²≠0). Results are in no-go case for radial flow also.

4. Conclusions

The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows were computed by computer package SADE. A new similarity solution for two-dimensional flows was given in (2.14). For radial flows a new similarity solution (3.11) was derived. It would be of interest to identify what type of physical phenomena can be associated with the solutions derived in this paper.