Analysis And Calculation Of Mathematical Model Of Fluid Phase Transition

Van der Waals fluid with viscosity is an important fluid in fluid mechanics.However,it is difficult to solve the nonlinear partial differential equations composed of the flow and heat conduction of Van der Waals Newtonian fluid.In the numerical simulation,the numerical solution of the Van der Waals fluid model is very different from the actual problem.The difference is mainly due to the instability of the problem in physics and the discomfort in mathematics.In order to solve the oscillation phenomenon in mathematical simulation,this model has been improved in previous studies.In other words,the constant artificial viscosity coefficient greater than zero is introduced.Because the artificial viscosity is added to solve the oscillation phenomenon of the mathematical model,and the artificial viscosity only needs to play a role in the unstable ellipse.When the artificial viscosity is constant,the error in the non-elliptic region is increased.Aiming at this problem,this paper improves the artificial viscosity,i.e.?1=?1(v)or ?1=?1(?)Let the value of artificial viscosity be as small as possible in the non-elliptic region;And the value is larger in the unstable ellipse region to play a role in mitigating the shock.Therefore,this paper makes the following research:First,based on the Van der Waals fluid equations,we discuss the steady-state solution of the equations and the relationship between the steady-state solution and the viscosity coefficient when the artificial viscosity coefficient is a function of the specific volume(v)and the value is small in the non-phase-change region.Then,for Language coordinates,the uniqueness and multiplicity of solutions under periodic boundary conditions of steady state are analyzed,sufficient conditions are given for the existence of only trivial solutions and nontrivial solutions,and relevant computational verification is given.Finally,in Euler coordinates,various boundary value problems of one-dimensional Van der Waals fluid mathematical model are calculated,and the results of numerical simulation are analyzed and compared.For the above study,the steady state form of the equation is used.By constructing the corresponding functional and certain spatial conditions,the problem of finding the solution of the system of equations is transformed into the problem of solving the extreme value of the functional,and then the relationship between the existence of the solution of the system of equations and the coefficient of viscosity is proved.Then,MATLAB software was used to simulate the Van der Waals fluid.