Solutions of the multi-face jet flow problems via symmetries of the nonlinear diffusion equation
Exact solutions to the multi-face jet flow problems are obtained via symmetries of the nonlinear diffusion equation. All those problems are characterized by that the internal boundaries separating different fluids are not known in advance and should be determined as a part of solution of a whole problem. Finding solutions to the fluid dynamics equations satisfying boundary conditions at the boundaries of unknown shapes, in general, represents a very difficult task. The method we used includes the following steps: (i) Transformation of the boundary layer equations to another coordinate system with one of the coordinates representing a stream function of the °ow. For example, transformation of the steady boundary layer equations to the von Mises variables which results in a nonlinear diffusion equation. In such coordinates, interfaces between moving fluids will be coordinate surfaces. (ii) Application of the symmetry group methods to the transformed equations and adjusting the solutions to the transformed boundary conditions. (iii) Transformation of the solution back to the original variables to obtain the solution of the complex flow problem. The problems considered include: the planar or radial jet discharge into a quiescent immiscible fluid; the multi-fluid jet flows in the presence of rigid boundaries; weakly swirling free fan jets penetrating into the quiescent domain of immiscible fluids.
This is joint work with Georgy Burde