Introduction Of Research Advance For The Equations Of Non-Newtonian Fluids Dynamics

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].