Abstract
Conserved quantities play a central role in the solution of jet flow problems. A systematic way of deriving conserved quantities for the radial jets with swirl is presented. The multiplier approach is used to derive the conservation laws for the system of three boundary layer equations for the velocity components and for the system of two partial differential equations for the stream function. When the swirl is zero or at a large distance from the orifice (at infinity), the boundary layer equations for the radial jets with swirl reduce to those of the purely radial jets. The conserved quantities for the radial liquid, free and wall jets with swirl are derived by integrating the conservation laws across the jets.
Highlights
It is of great interest to study the three dimensional boundary layer flows, for example the radial jets with swirl due to their extensive range of applications
Riley in [4] discussed the two-dimensional boundary layer flow of purely radial liquid, free and wall jets. In all these jet flow problems, the boundary conditions are homogeneous and the unknown exponent in the similarity solution cannot be obtained from the boundary conditions
The boundary layer equations for the velocity components as well as for the stream function formulation are given in Table 1 for both the radial jets with swirl and for the purely radial jets
Summary
It is of great interest to study the three dimensional boundary layer flows, for example the radial jets with swirl due to their extensive range of applications. Riley in [4] discussed the two-dimensional boundary layer flow of purely radial liquid, free and wall jets. In all these jet flow problems, the boundary conditions are homogeneous and the unknown exponent in the similarity solution cannot be obtained from the boundary conditions. By integrating the momentum equation in the swirl direction and using the boundary conditions and the continuity equation, one conserved quantity for the radial free jet with swirl and one for the radial wall jet with swirl were derived. The second conserved quantity for the radial free and wall jets with swirl were established by integrating the momentum equation in the radial direction and requiring that swirl is zero at x = ∞. The boundary conditions determine which conserved vector is associated with which jet
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