Existence Of Global Weak Solutions For A Class Of Non-Homogeneous Non-Newtonian Fluid Coupled With Vlasov Equations

Fluid mechanics is a branch of mechanics.It is the science of studying fluid phenomena and related mechanical behavior.Classical Newtonian fluid mechanics believes that in parallel flow,the stress tensor and the shear rate are linear,and its proportionality coefficient is called Is the viscosity coefficient.On this basis,the well-known Navier-Stokes equation can be obtained.Corresponding to the Newtonian fluid is another fluid whose stress tensor is not linearly related to the shear rate.People usually call it Non-Newtonian fluid.For this type of fluid,under the action of stress,it will continuously change its state of motion,and its constitutive relationship is significantly different from Newton's law of constant viscosity.Non-Newtonian fluids are ubiquitous in na-ture.They are widely used in aerospace,energy,ocean,chemistry,biomedicine,geology and other fields,which also makes people more and more interested in the research of non-Newtonian fluid systems.So far,fluids Mechanics and other disciplines are infiltrated and merged with each other,and many branches have been formed.The coupling system of fluid and particle is one of the hot research fields in the field of partial differential equation fluid mechanics.At present,there are few results on the coupling mod-el of particle equations and non-homogeneous non-Newtonian fluids,and most of the existing results are concentrated in the study of local solutions.There-fore,it is necessary to study the global solutions of non-Newtonian fluids.This article draws on the literature gives the existence of global weak solutions for a class of incompressible non-Newtonian fluid equations,and establishes a three-dimensional bounded region Vlasov equation and a class of inhomogeneous incompressible non-Newtonian fluid coupling equations.Among them The cou-pling is achieved through the resistance term in the non-Newtonian fluid equation and the acceleration term in the Newtonian equation.This paper aims at the initial-boundary value problem of a class of non-homogeneous non-Newtonian fluid and the coupling model of the Newtonian equation.By introducing an appropriate approximation system,a truncation is adopted.Techniques and monotone operator theory to establish the existence of the global weak solution of the initial-boundary value problem.In this paper,the exponent of the stress tensor of the non-Newtonian fluidsatisfies12/5.