Analysis Of Well-posedness For Inextensible Immersed Interface Problem And Numerical Methods Of The Navier-stokes-nernst-planck-poisson Equations

In recent year,the coupling models related to the fluid(Navier-Stokes/Stokes flow)system are widely applied in scientific research and practice,covering physics,biology,chemistry,materials science and related industries.In this paper,the main work is focused on theoretical and numerical analysis of two coupling fluid model problems.The first model we investigate is an incompressible fluid with an immersed interface.This kind of interface models has been used in the areas such as composite materials,multi-phase flow,bio-membrane,microbiology,etc.We are particularly interested in the problems with the inextensibility constraint on the interface.These models are commonly used to simulate the red blood cells and drug-carrying capsules.There exist several numerical methods for the numerical solution of the related problems without constraint.However,there is few result in the theoretical and numerical analysis of the problems with constraint.The main contribution of the first part of this paper includes:1)a Stokes flow with an inextensible immersed interface is introduced,and the inextensibility constraint on the interface is proposed in detail;2)a new functional space is introduced,which is proved to be a Banach space;3)a steady immersed interface problem is given,and it is proved that if the solution for the velocity field is in the new Banach space,then the problem admits a unique solution in the weak sense;4)a spectral discrete method is proposed and analyzed for a simplified model,together with a proof of the well-posedness of the discrete problem.In the second part of the thesis,we consider the Navier-Stokes-Nernst-Planck-Poisson(NSNPP)equation.This equation has been frequently used to model the behavior of charged colloidal particles,electro-hydro,and micro-/nano-fluidic dynamics.It can also been found in the research related to the fuel cell.This practical application background has stimulated the development of the new more robust numerical methods for the NSNPP equation.The first part of this paper aims at developing and analyzing efficient methods for the numerical solution of the above mentioned problem.The main contribution of this work includes:1)establish the weak form of the system and derive the stability inequality for the weak solution;2)propose a time stepping scheme and prove the non-negativity of the discrete concentration solution,which is well-known for its continuous counterpart.A stability condition for the scheme is also provided;3)design an efficient pressure-correction method to decouple the velocity and pressure in the fluid field.Once again it is proved that This new scheme preserves the non-negativity of the discrete concentration and the same stability condition as the previous scheme;4)propose a stabilized time stepping scheme,which is more stable than the non stabilized one;5)construct a full discrete method for the NSNPP system.This method combines a spectral method for the spatial discretization and semi-implicit second order difference scheme for the temporal discretization.The obtained numerical results show that our method is spectral accurate in space and second-order convergent in time.