【作者】 刘燕；

【导师】 袁洪君；

【作者基本信息】 吉林大学 ， 基础数学， 2010， 硕士

【摘要】 本文是一篇综述类文章,主要介绍非牛顿流体动力学中的Navier-stokes方程：其中Ω是R3中有界光滑区域,x∈Ω,I是二阶单位张量；F(t,x)是流体所受的外力,u=(u1,u2,u3)是速度向量；p是密度；p为压力；这里的p不仅包括表面的正应力,还包括表面上的切应力；它们都是时间t和x=(χ1,χ2,χ3)的函数.非牛顿流体是指剪应力与剪切变形速率之间不满足线性关系的流体,非牛顿流体力学在各工业领域应用广泛Navier-Stokes方程就被认为是描述一些非牛顿流体的很好的模型,如上的非牛顿流体的Navier-Stokes方程包括质量守恒方程和动量守恒方程.本文分为两个部分,第一部分主要介绍了一些不可压缩的非牛顿流体动力学方程的解的存在性和唯一性；第二部分主要介绍了可压缩的非牛顿流体动力学方程的解的存在性和唯一性.更多还原

【Abstract】 This paper is a review of the article,a brief introduction to the equations of Non-Newtonian fluids dynamics.In this paper, we consider a class of Navier-Stokes Equations with non-Newtonian flu-ids: whereΩ. is a smooth bounded domain in R3, x∈Ω,I denotes a second-order unit ten-sor; F(t,x) denotes the external force,u= (u1,u2,u3) denotes velocity vector; p denotes density;p is pressure,p includes not only the surface of normal stress, but also the shear stress on the surface of it;they are the function of time t and x= (x1, x2, x).When the density of the fluid does not change with pressure and temperature,we can put such a fluid as incompressible fluid,the density of the incompressible fluid is a con-stant,Therefore,the equations of conservation of mass has changed to:Sometimes fluids dynamics equations also include another equation,the equations of conservation of energy: If the relation between shear stress and the rate of strain is not linear,the fluids are called non-Newtonian.non-Newtonian fluids dynamics widely are used in various industrial fields.the Navier-Stokes equations are generally used as the governing equations for non-Newtonian fluids.the Navier-Stokes equations include the equations of conservation of mass and momentum.The actual flow is different from the ideal fluid,because of its viscosity and thermal conductivity,This is determined by intrinsic molecular structure of fluid,this fluids include gases and liquids. Fluids dynamics equations reflects the basic laws of viscous fluid.please prefer to [2]. Chen Wenfang and Fan Chun introduce that non-Newtonian fluids don’t obey Newtonian Law in [5].There are two parties in the paper.As follows:In the first part introducing existence and uniqueness of solutions of the equations of in-compressible Non-Newtonian fluids dynamics:firstly introducing the existence of solutions of the equations of incompressible Non-Newtonian fluids dynamics.Introducing existence of Young measure-valued solutions in the case of different parameters separately in [1]and[8]; Namely: andIntroducing existence of solutions of two kinds of unsteady incompressible non-Newtonian fluids’ with shear rate dependent viscosity separately in[12]and[16]; the questions can be for-mulated in the following form: andIntroducing existence of weak solutions of two kinds of incompressible non-Newtonian fluids with in [31], Namely:Introducing existence of solutions of a incompressible non-Newtonian system in two-dimensional unbounded domains in [29], incompressible non-Newtonian fluids undergoing isothermal processes can be formulated in the following form:Secondly introducing uniqueness of solutions of the equations of incompressible Non-Newtonian fluids dynamics.Maria Carmela Lombardo proved uniqueness of solutions of the time dependent incompressible Navier-Stokes equations on an half plane using the Abstract Cauchy-Kovalevskaya Theorem in Banach spaces in [20].In a 3D half space for the veloc-ity field u and v being the components of the velocity tangent and orthogonal respectively.Namely:Introducing Frederick Bloom and Wenge Hao proved uniqueness of a class of bipolar viscous fluids in [19].Bipolar viscous fluid flow in an infinite strip of R2.the questions can be formulated in the following form:In the second part introducing existence and uniqueness of solutions of the equations of compressible Non-Newtonian fluids dynamics:firstly introducing the existence of solutions of the equations of incompressible Non-Newtonian fluids dynamics.S.Matusu Necasova and M.Medvidova Lukacova proved existence of a class of isentropic non-Newtonian bipolar fluids in [21];introducing existence of measure-valued solutions to an initial and boundary value problem for a flow of compressible gases undergoing isothermal processes in bounded domains in [1]. Introducing the existence of the global solution to a class of compressible Navier-Stokes equations with Non-Newtonian potential.Secondly introducing uniqueness of solutions of the equations of compressible Non-Newtonian fluids dynamics.Introducing Xiaojing Xu and Hong jun Yuan proved the unique-ness of solutions for a class of Non-Newtonian fluids with vacuum in 1D bounded intervals in [26].Introducing global uniqueness of solution for a class of Non-Newtonian fluids with vacuum in ID bounded intervals in [27].更多还原