Of Lie Group Theory In Fluid Mechanics Applications

Although objects of nature world exhibit some astonishing variety of shapes, it's the truth that they universally own certain form of symmetry. Symmetry is universal and of immense practical importance. Our human beings have a perception of every kind of symmetry that lies at core of our life. Symmetries provide cues that help us relate to our environment and guide our movements through the world. The Lie group is a sort of symmetry which is useful in both mathematics and physics. This paper is trying to present the applications of Lie groups from both the Geometry and algebraic aspects. For the aspect of geometry, we will briefly discuss some symmetries of topological structure in flows and show how to describe them by assistants of Lie group. As to the aspect of algebra, we follow the classic Lie symmetry method that is usually used for dealing differential equations.The first chapter in this paper is to introduce some basic knowledge of the differentiable manifolds, tangent space, the symmetries of differential equations and so forth.The second chapter will present systematical discuss about the symmetries of topological structure in flows. We first give an incisive conclusion when ignoring the boundary conditions. At the meantime, as the Boltzmann equation is widely applied into fluid dynamics, we will introduce Lie group to analyze this equation. The reason why we do this work is that Lattice Boltzmann method is still not good enough for compressible fluid flow as well as turbulence. We will give the simplification of Boltzmann equation on S3 (1).After introduce the basic concept of Lie group method at the beginning of chapter three, we will first show the classic result of Navier-Stokes equation. These symmetries were calculated by S. P. Lloyd in 1979. We will then transform Navier-Stokes equation to a scalar equation. This operation helps us to avoid dealing with coupled equations, which will be more convenient for using Lie group method. Furthermore, we will also introduce some kind of modern soft wares which are used for calculating determine equations and symmetries.A major obstacle in the application of Lie group theory to solving differential equations is that usually a large number of tedious calculations is involved. It is not easy to guarantee the validity of results from handy calculating hundreds of partial differential equations. Therefore, we will investigate some popular available soft wares in the forth chapter. These programs will assist us to calculate the symmetry groups of differential equation systems.