Random Non-newtonian Flow Solution Of Appropriate Qualitative And Its Power System

When differentiates the Newtonian fluids and non-Newtonian fluids, the key is whether the constitutive relation between stress tensor and the velocity gradi-ent can be characterized linearly. Non-Newtonian fluids are commonly found in the natural world and the real lives, there is a wide range of applications in chem-ical industry, petroleum industry, biomechanics, glaciology, geology, hemorheol-ogy, and so on. Therefore, the research of non-Newtonian fluids has a very im-portant practical significance. This dissertation mainly discusses the isothermal nonlinear incompressible bipolar viscosity model proposed by Bellout, Bloom and Necas, and the well-posedness and long-time behavior of solution to non-Newtonian fluids evolutionary equation driven by deterministic and stochastic force is considered.The dissertation consists of five chapters.In Chapter 1, we pay attention to introduce the physical background, some already known results and recently development of non-Newtonian fluids, and present some research status of stochastic differential equations.In Chapter 2, we mainly study the convergence of solution for non-Newtonian fluid to the solution for Navier-Stokes equation under the L2, H1-norm as the viscositiesμ0,μ1→0, and estimate the convergence rate. The key of proof is some norm estimates, some are uniform estimates independent onμ0,μ1,the others may be dependent onμ0,μ1, but whenμ0,μ1→0, these estimates are still bounded. At the same time, because these estimates lie on p, we analysis shear thinning and shear thickening case according to the value of p.In Chapter 3, we prove the existence, uniqueness, and decay of weak so-lution to fractal Boussinesq Approximation. First, we are concerned with the periodic boundary problem, and obtain some important a prior estimates by the commutator estimate, which are independent on domainΩ, the existence and uniqueness of the weak solution is obtained by Galerkin method. Then, we let│Ω│→∞, the existence and uniqueness of the weak solution for Cauchy prob- lem is easily established. Finally, we make use of the Fourier splitting method to prove the decay of weak solution. Owing to the appearance of the diffusion operator, the decay estimate of solution lies onα. Thus, we discuss it in three cases of 0<α<1/2,α=1/2,1/2<α<1 respectively.In Chapter 4, the existence of random attractors for stochastic non-Newtonian fluid with additive white noise is proved. Firstly, translating the stochastic differ-ential equation to a one with random coefficients by Ornstein-Uhlenbeck process. Secondly, applying the method proposed by Crauel, Debussche and Flandoli to prove the compact attracting set at time zero, then we can conclude the existence of random attractor. In the end, we verify regularity of the random attractors by showing the equivalence of attractor lying in different energy space, which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.In Chapter 5, we discuss the stochastic non-Newtonian fluid driven by mul-tiplicative noise, and adopt a method developed by Flandoli and Gatarek to con-struct martingale solutions (namely, the weak solution in the stochastic sense), this method is a stochastic generalization of compactness methods for deter-ministic partial differential equations. First, existence of martingale solution to approximating finite-dimensional problem whose nonlinear terms satisfy linear growth and continuity condition is obtained. Second, the family of laws is tight, because we obtain some norm estimates with expectation of solution to approx-imating finite-dimensional problem by Ito formula. Third, taking the limit of approximating finite-dimensional problem, the martingale solution of equations can be obtained with the aid of representation theorem for martingale. If the uniqueness of weak solution can be obtained, then the existence of invariant measure is also proved.