On The Solution Of Micro-fluid Boundary Layer Equation

The Prandtl system reveals the nature of the equations governing the flow of fluids within the boundary layer with small viscosity. Nevertheless, Prandtl boundary theory obviously does not give any consideration to the influence of wall's properties on the characteristics of the boundary layer. This theory is not able to provide a full and complete explanation for the flows concerning the interaction between the solid wall and the water flows, which are often involved in engineering practice, especially the flow of micro-flow problem. It sometimes may lead to contradictory or erroneous conclusions. Therefore, Prandtl boundary layer theory is incomplete. Given the much stronger absorbing effect of the surface of solid on water molecules, experiments show that: the boundary layer system embodies the micro-fluid characteristic. We can use the following micro-fluid boundary layer equations to describe the above system [1].Oleinik has proved the existence of classical local solutions to the Prandtl system under some assumptions [2]. It's quite natural to consider the following question: under what initial and boundary circumstances, the uniqueness of classical solution for the micro-fluid boundary layer equation are true. Here we borrow the Oleinik's line method, which was used to deal with Prandtl boundary layer theory.This paper is divided into two chapters. The first chapter discusses the local existence and uniqueness of classical solution to the micro-fluid boundary layer equation (X is fixed, t being sufficiently small). We first translate the initial and boundary value problem of the micro-fluid boundary layer equation to the initial and boundary value problem for a degenerate parabolic equation by the Crocco transformation. Then using the Prandtl's line method, we reduce the PDE problem to the system of ordinary differential equations and prove some prior estimates to the solution of the corresponding ODE system. Then, we linearly extend the solutions of ODE to PDE's. Finally, we prove the existence and uniqueness of the local classical solution for the micro-fluid boundary layer equation. This idea comes from Oleinik; However, with emergence of low-order nonlinear derivative items in micro-flow boundary layer discussed in this article, so there exists far more difficulties than the Prandtl system. The discretization method, which is used in dealing with the nonlinear terms, is accomplished.In the second part, we get the existence and uniqueness of classical solution for the micro-fluid boundary layer equation with random t>0 and X being sufficiently small. The proving process is similar to the first part. But the specific calculation process, including the construction of the corresponding function, is different.