Compressible Non-Newtonian Fluids With Gravitational Potential And Particles

Fluid mechanics is the study of the laws of mechanical motion in fluids,and due to the need of industrial development,the integration of fluid mechanics and other disciplines gave rise to many branches.Nowadays non-Newtonian fluid mechanics has developed as part of the basic scientific system,and widely used in petroleum,chemical industry,medicine,biology,food and other fields at same time,therefore,it is significant to study the properties of non-Newtonian flows.In this thesis,two non-Newtonian equations with gravitational potential and particle interactions are studied,and the well-posedness of the solution of corresponding initial boundary value problem is discussed under the condition that the initial density vacuum is allowed.And the main content is as follows:(1)A mathematical model for the evolution of particles in compressible non-Newtonian fluids is studied,and it is proved that the initial boundary value problem with vacuum has a unique local strong solution.Since the problem is vacuum,it is difficult to prove the existence and uniqueness of the solution directly,first consider the absence of a vacuum in p >2,q >2,the conclusion with vacuum condition is discussed by using the conclusion without vacuum condition.By estimating each term of the auxiliary function,a uniform estimation is obtained,which overcomes the difficulty caused by the strong non-linearity of the system and the interaction between multiple equations,and thus proves that the limit of the solution is the solution of the original boundary value problem.Finally,the existence and uniqueness theorem of the solution of this problem is proved.(2)A shear thickening flow model describing particle interaction in viscous compressible non-Newtonian fluid is studied.Based on conservation of mass and conservation of energy,considering the absence of a vacuum in p >2,1<q <2,the conclusion with vacuum condition is given by using the conclusion without vacuum condition.The approximate solution is constructed by using iteration method and compatibility condition,and the convergence of approximate solutions is discussed by combining the functional analysis and space theory of partial differential equations.Finally,the existence and uniqueness theorem of the solution of this problem is proved.