A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

<p>In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.</p>

Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

<p>In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.</p>

One component regularity criteria for the axially symmetric MHD-Boussinesq system

<p style='text-indent:20px;'>In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time <inline-formula><tex-math id="M1">\begin{document}$ T_\ast $\end{document}</tex-math></inline-formula> if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field or the temperature fluctuation is imposed.</p>