Correct Statement, Analysis and Numerical Solution of Singular Nonlinear Problems for Self-Similar Solutions to the Boundary Layer Equations with Zero Pressure Gradient

TL;DR

This paper develops a new approach to analyze and numerically solve singular nonlinear boundary layer problems for self-similar solutions in fluid dynamics, providing insights into particle trajectories and related physical phenomena.

Contribution

A novel method for analyzing and numerically solving singular nonlinear boundary layer problems with applications to self-similar flow solutions.

Findings

Efficient numerical methods for singular nonlinear problems

Numerical particle trajectories in flow plane obtained

Connections established with physical jet problems

Abstract

For the problems indicated in the title, a further development of a new approach (different from those applied before) is given. A basic problem under consideration arises in viscous incompressible fluid dynamics and describes self-similar solutions to the boundary layer equation for a stream function with zero pressure gradient (connected with the plane-parallel laminar flow in a mixing layer). Some previous results concerning singular nonlinear Cauchy problems, smooth stable initial manifolds, and parametric exponential Lyapunov series are used to state correctly and analyze the singular "initial-boundary-value" problem for a third-order nonlinear ordinary differential equation defined on the entire real axis. The detailed analysis of this singular nonlinear problem leads, in particular, to efficient methods for solving it approximately and gives a possibility to obtain numerically the particle trajectories in the plane of flow. Some results of the numerical experiments are displayed and their physical interpretation is discussed. A connection of this basic problem with some known physical and mathematical problems, arising for self-similar solutions to the boundary layer equations with zero pressure gradient, is described, namely the "flooded jet", the plane "semi-jet" and the "near-wall jet" problems are considered which are of interest by themselves.