Some Studies On The Solution Of Non - Newtonian Fluid Equation

We study two different class of compressible non-Newtonian fluids in this paper. In the first part of this paper, we proved the existence and uniqueness of a 1-D model. Moreover, we give the blow-up criterion of the solution. In the second part, since the existence of the solution has been proved, our work is to prove that the perturbations of density and velocity tend to zero in appropriate norms along a chosen time sequence.The motion of a compressible non-Newtonian fluid in 1-D can be described as follow with the initial and boundary conditions Here, Lu=-((|ux|2+μ)p-2/2ux+λ(ρ)ux)x, π≡π(ρ)≡ Apγ λ(ρ)= pα. 0<α<1, μ>0, p>2 are constants. We denote the unknown variables p, u and π the density, velocity and pressure, respectively. The pressure π is a smooth function of density p and the constant 7> 1 represents the adiabatic gas expo-nent. Beside that, the constant A> 0 in the pressure π(p) doesn’t play any role in the following analysis, without losing generality, we take A=1. Moreover, we denote (0,1) x (0, T)= ΩT is the space-time domain for the evolution of the fluid, where T is a positive number and (0,1) is a 1-D interval.The third chapter is concerned with a compressible non-newtonian fluid in the following form: with the initial-boundary conditions: u|(?)Ω×(0,T)= 0, ρ|t=0=ρo, ρu|t=0= m0, (0.10) where A=A(Du)=|Du|p-2Du+υI|divu|p-2divu, (0.11) π(ρ)= ργ, (0.12) ρ, u and d denote the unknown density, velocity and dimension respectively. p≥3, υ≥0,γ>d/2, and p> d. I is the unit matrix. Suppose that f∈(L∞(QT))d, ρ0 ∈L1(Ω).Define ρ0= 0 on m0=0, m0 ∈ (L1(Ω))d. Consider problem 0.9 on QT=Ω× (0, T), where Ω is the Lipschitz domain and Rd(d≥ 3).The models can be regarded as the simplified form of non-Newtonian fluids. The study of non-Newtonian fluids is very important since they can be found in many fields, such as in chemistry, biology, geology and glaciology. Many substances which are capable of flowing but which exhibit flow characteristics, cannot be adequately described by the classical linearly viscous fluid model. In order to describe some of the departures from Newtonian behavior (rheological properties, elastic features such as yield stress, stress relaxation and nonzero normal stress differences), many idealized material models have been suggested.This paper is divided into four chapter. In chapter 2, our main idea is to establish the existence and uniqueness of solutions for a class of compressible non-Newtonian fluids whose viscous term depends on density with p> 2. Moreover, it is also proved that the solutions are to blow up and the maximum norm of velocity gradients controls the possible break down of the strong solutions. In chapter 3, the existence of a compressible non-newtonian fluid equation has been proved. Our work is to prove that the perturbations of density and velocity tend to zero in appropriate norms along a chosen time sequence.