Similarity solutions of a class of unsteady laminar boundary layer

The study of laminar unsteady boundary layer flows is essential for understanding the transition from laminar to turbulent flow, as well as the origins of turbulence. However, finding solutions to this phenomenon poses significant challenges. In this study, we introduce a novel method that employs a similarity transformation to convert the two-dimensional unsteady laminar boundary layer equations into a single partial differential equation with constant coefficients. By applying this transformation, we successfully derive similarity solutions for flat plate boundary layer flow, expressed in terms of Kummer functions. For convergent boundary layer flow, we derive an approximate analytical solution that includes both shock wave and soliton wave solutions. The superposition of these solutions provides evidence for the existence of solitons or soliton-like coherent structures (SCS) within boundary layers. Additionally, this paper explores two- and three-dimensional laminar flows, as well as three-dimensional turbulent flow equations, revealing that they all incorporate third-order derivatives with respect to spatial coordinates. This finding suggests that all viscous fluid motions have the potential to exhibit solitons/like coherent structures (SCS).