O.A.OIeinik’s Linearization Method And Its Application

In this paper, we study the two-dimensional boundary layer problem associated withincompressible fluid. Since the fluid is influenced by the viscous, the speed of normal directionchanges rapid in a thin layer, this layer is formed in the vicinity of the surface of the fluid, whichis called the boundary layer. The classical prandtl system reveals the nature of the equationsgoverning 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’sproperties on the characteristics of the boundary layer. This theory is not able to provide a fulland complete explanation for the flows concerning the interaction between the solid wall and thewater flows, which are often involved in engineering practice, especially the flow of micro-fluidproblem. Therefore, Prandtl boundary layer theory is incomplete. Considering the strongabsorbing effect of the surface of solid on water molecules, experiments show that: the boundarylayer system embodies the micro-fluid characteristic. We can use the micro-fluid boundary layerequations to describe the above system[1]. In the paper we will use Oleinik’s linear method,which was used to deal with Prandtl boundary layer theory, to discuss the micro-fluid boundarylayer equation. We prove the existence and the uniqueness of classical solution for themicro-fluid boundary layer equation with initial and boundary value condition.This paper is divided into two chapters. The first chapter discusses the existence anduniqueness of classical solution for the two-dimensional nonstationary micro-fluid boundarylayer equation. We first translate the problem into the initial and boundary value problem of aquasilinear parabolic equation by Crocco transformation. Then, using the oleinik’s the linearmethod, we can get a series of linear parabolic equations and we can get the solution of linearparabolic equations by the elliptic’s theory. At last, we use the inverse Crocco transformation andget the solution of the original equation. In the second part, we discuss the two-dimensionalnonstationary micro-fluid boundary layer continuation problem by two kinds of differencemethods. The result act the theoretical basis for the numercial solution of the mirco-fluidboundary layer.