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:
()
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 =
ψy and
v = −
ψx. 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]):
()
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)
ψy and
v = (−1/
x)
ψx. 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:
()
Consider the infinitesimal operator:
()
The invariant surface condition is
()
The nonclassical symmetries determining equations are
()
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:
()
The solution determining equations in (
2.5) yield all classical symmetries (see [
5]) and the following infinite many nonclassical symmetry generators:
()
where
h(
x,
y) and
g(
x) satisfy
()
Equation (
2.7) yields
()
and thus the nonclassical symmetry generators in (
2.6) take the following form:
()
Now,
ψ =
ϕ(
x,
y) is group invariant solution of (
2.1) if
()
where the operator
X is given in (
2.9). The solution of (
2.10) for
ψ =
ϕ(
x,
y) is of the form
()
Substitution of (
2.11) in (
2.1) yields
()
and thus
()
The invariant solution (
2.11) with the help of (
2.13) takes the following form:
()
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:
()
Case 1 (ξ1 ≠ 0). In this case we set ξ1 = 1 and using SADE we have following six determining equations:
()
The solution determining equations in (
3.2) yield all classical symmetries and the following infinite many nonclassical symmetry generators:
()
where
h(
x,
y) and
g(
x) satisfy
()
The nonclassical symmetry generators (3.3) finally become
()
and we have used
()
Now,
ψ =
ϕ(
x,
y) is group invariant solution of (
3.1) if
()
where the operator
X is given in (
3.5). The solution of (
3.7) for
ψ =
ϕ(
x,
y) is of the form:
()
Substitution of (
3.8) in (
3.1) yields
()
and thus
()
Finally, we have following form invariant solution:
()
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.
