Several Research On Compressible Non-newtonian Fluids

Fluid mechanics is a branch of physics which studies the phenomenon and behaviorof fluids in the associate field. So far, Fluid mechanics has formed many branches betweenother sciences, many physicists and mathematicians have devoted in this field.In classical newtonian mechanics, in parallel flow, shear stress is proportional to theshear rate. The coefcient of proportionality is called viscous coefcient. On this basis,one can get the famous Navier-Stokes equation.In recent years, with the growing awareness of the important characteristics of nonNewtonian fluid, one found that, in daily life, there are a lot of fluids do not obey theconstant viscosity of Newtonian law, namely the non Newtonian fluid. Non-Newtonianfluid is universal in nature. For this type of fluid, under stress, will continue to changeits state of motion, its constitutive relationship with Newton viscosity law have obviousdiferences. However, there are few results about this type of fluids. So it is necessary tostudy the non Newtonian fluids.This thesis is devoted to study several classes of compressible non-Newtonian fluid models. In chapter2, we consider the following system in one-dimensional bounded intervalwith the initial and boundary conditionswhere ΩT=I x (0,T),I=(0,1), the initial density p0≥0,4/3＜p, q<2are given constants.The unknown variables p,u,P,Φ denote the density, velocity, pressure and the non-Newtonian gravitational potential, respectively.Since the second equation (13) has singularity, and the vacuum may appear, We first regularized the viscous term, then by using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition. We obtain the following theorem:Theorem0.0.1. If (p0,u0, f) satisfies the following conditions and if there is a function g E L2(I), such that the initial data satisfy the following compatibility condition: Then there exist a time T*∈(0,+∞) and a unique strong solution (p, u,Φ) to (13)-(14) such thatIn chapter3,we consider the following system in one-dimensional bounded intervalwith the initial and boundary conditionswhere p,u,δη,P(p)=apγ denotes the fluid density, velocity, the density of particle in the mixture and pressure respectively, a>0,γ>1,μ0>0,p>2, the given function Φ(χ) denotes the external potential (typically incorporating gravity and buoyancy). In particular, if Φ is the gravitational potential, then Φ=cx, where c is the gravitational constant.γ>0is the viscosity coefficient and β≠0is a constant.We obtain the local existence and uniqueness of strong solutions based on some compatibility condition. Our main result is stated in the following theoremTheorem0.0.2. Let μ0>0be a positive constant and Φ∈C2(Ω)，and assume that the initial data(p0,u0,η0) satisfy the following conditions and the compatibility condition ((u0x2+μ0)(p-2)/2u0χ)χ-(p(ρ0)+η0)χ-η0Φχ=ρ0(g+βΦχ),(19) for some g∈L2(Ω).Then there exist a T*∈(0,+∞) and unique strong solution (ρ,u,η)to(17)-(18such thanIn chapter4,we consider the following system in one-dimensional bounded interval with the initial and boundary conditionswhere p,u,η,P(ρ)=aργ denotes the fluid density,velocity,the density of particle in the mixture and pressure respectively,a>0,γ>1,4/3<p<2,the given function Φ(χ)denotes the external potential(typically incorporating gravity and buoyancy).In particular,if Φ is the gravitational potential,then Φ=cχ,where c is the gravitational constant.λ>0is the viscosity coefficient and β≠0is a constant.Similar to the method in chapter2,Firstly we regularized the viscous term,then by using the iterative method, the local existence and uniqueness of strong solutions was proved.we have the following theorem:Theorem0.0.3. LetΦ∈C2(Ω),and assume that the initial data (po,uo,ηo) satisfy the following conditions and the compatibility condition for some g∈L2(Ω). Then there exist a time T*∈(0,+∞) and a unique strong solution (p,u,η) to (21)-(22)such that...