Analysis Of Solutions Of Two Kinds Of Partial Differential Equations With Carreau Type Viscosity

Non-Newtonian fluid with Carreau type viscous is one of the most important fluids in industrial production.Especially,high polymer polymers such as polyethylene and polypropylene in chemical production are typical Carreau type non-Newtonian fluids.This is a kind of shear thinning fluid.Due to the non-linearity of the viscosity coefficient of the fluid velocity gradient,the analysis of the solution of the partial differential equations of the fluid becomes more difficult.For this point,we have done the following research:Firstly,based on the Navier-Stokes equations,we discuss the steady problem of Carreau-type non-Newtonian fluid flow with inflow and outflow in the straight pipe,and the corresponding boundary condition is a kind of mixed boundary condition with inflow and outflow boundary.For this problem,we first construct the approximate solution of the corresponding nonlinear equations by the Galerkin method,and then through the energy estimation and monotone operator method combined with the fixed point theorem,we prove the existence of the smooth solution of the system in the thin pipe And some new energy inequalities are obtained,which can overcome the difficulties of the mixed boundary and the non-linearity of the viscous,and get the new results and have practical application value.In addition,we through the Polyflow software,made the corresponding numerical simulation,the results consistent with the analysis results.Secondly,we discuss the asymptotic stability of the shock wave solution of the Cauchy problem for the nonconvex conservation laws of non-Newtonian fluid with Carreau type viscous.In the case of small perturbation,the monotone operator theory and the weighted energy estimation method are used to prove that the viscous shock wave solution of the viscous nonconvex law is asymptotically stable.There is no restriction on the strength of the shock wave.