Unsteady incompressible laminar boundary layer: time and space variable molecular viscosity

Abstract

The subject of this work is two approaches to describe the laminar unsteady flow of an incompressible fluid in the boundary layer. In the first approach, the viscosity of the fluid and the acceleration with which the plane is set in motion are considered constant. In essence, this is Rayleigh's problem. The solution obtained on the basis of these assumptions asymptotically converges to the well-known self-similar Stokes solution. It is important that the solutions of Stokes and Rayleigh asymptotically at large values of time correspond to the disappearance of shear stresses between the liquid and moving plane after acceleration. A paradox emerges the equations derived by Stokes to describe internal friction indicate the absence of the same friction between a moving body and fluid. Since research using the calculus of variation methods revealed that the molecular viscosity inside the stationary boundary layer should depend on the distance to the moving surface, the corresponding non-steady problem was considered. As a result, as before for the steady case, solutions describing both non-gradient and gradient flows of incompressible fluid in the boundary layer are obtained. The asymptotic analysis of the transition to steady flow testifies the consistency of these solutions. For the case of non-gradient flow, a comparison of the classical solution with the solution corresponding to the extreme fluid flow rate carried by the moving surface is made. It is shown that according to the solution obtained on the basis of the calculus of variation approach, the shear stress on the surface does not disappear anywhere after the motion becomes steady but, as expected, acquires a constant value. The research methods are purely theoretical and the results are analyzed by comparison with available theoretical and experimental data and compliance with the fundamental laws of physics, in particular the law of conservation of energy. These methods are based on the construction of analytical mathematical models, which are differential equations in partial derivatives supplemented with appropriate physical initial and boundary conditions. In addition, Euler's differential equations for the extreme of functional theory are used (in this paper, this is the extreme of fluid flow rate across the cross-section of the boundary layer). When solving these equations, the well-known Fourier method of variable separation is used. Arbitrary functions of time arising during partial integration (by one of the variables – the spatial coordinate) are determined from the conditions of asymptotic convergence of the solutions of non-steady problems to the corresponding solutions of steady problems. Conclusions. The presented results are of fundamental importance for understanding the physics of the flow around aircraft parts, as they indicate the contradiction of the existing idea of the reversibility of direct and inverse problems: the motion of a body in a still fluid and the flow of a fluid around an immobile body.