Study on the finite element scheme for fluid flows with free interface

This study has been carried out during 1998-1999 in order to develop and analyze the finite元素scheme for fluid flows with free interface，which are often found in nature and manay工业process . in 1998我们已经考虑了一个finite element scheme for two-fluid flows with low ratio of densities. weassume that two fluids are both viscousincompressible and immiscible. As the mathematical model for this problem we use the one-fluid modelassuming thr Boussinesq approximation to the Navier-Stokes equations. the接口是相关的as the 0-level Set of the pseudo-density function which is the Solution of the transport equation。我们提供了一个mixed finite element scheme与P1 iso P2/P1 element for these equations。especially,我们也提供了re-initialization technique in the finite element scheme for the transport1999年我们已经考虑了最终的元素方法two-fluid flows with free interface including surface tension effect. Here we use the true two-fluidflow system in order to deal the problems with the high ratios of density and viscosity. ingeneral,surface tension effect is represented by the line integral on the interface. In our studysurface tension effect is interpreted as a body force spread across the interfacial region with afinite thickness.On the other hand，as mathematical analysis of the finite element scheme for two-fluid flows we have considered theconvergence of the approximate interface. In fact，estimating the L - 1p - D1（Ω） norm of difference between the measure of positive value of thepseudo-density function and its approximation under the convergence of the finite element solution我们在这里使用了Heaviside operator. mathematical analysis of totalfinite element scheme, however, is our theme in the near future。

This study has been carried out during 1998-1999 in order to develop and analyze the finite element scheme for fluid flows with free interface, which are often found in nature and manay industrial processes.In 1998, we have considered a finite element scheme for two-fluid flows with low ratio of densities. We assume that two fluids are both viscous, incompressible and immiscible. As the mathematical model for this problem we use the one-fluid model assuming thr Boussinesq approximation to the Navier-Stokes equations. The interface is considered as the 0-level Set of the pseudo-density function which is the Solution of the transport equation. We have proposed a mixed finite element scheme with P1 iso P2/P1 element for these equations. Especially, we have also proposed the re-initialization technique in the finite element scheme for the transport equation by using a double well potential.In 1999 we have considered the finite element method for two-fluid flows with free interface including surface tension effect. Here we use the true two-fluid flow system in order to deal the problems with the high ratios of density and viscosity. In general, the surface tension effect is represented by the line integral on the interface. In our study the surface tension effect is interpreted as a body force spread across the interfacial region with a finite thickness.On the other hand, as the mathematical analysis of the finite element scheme for two-fluid flows we have considered the convergence of the approximate interface. In fact, estimating the LィイD1pィエD1(Ω)-norm of difference between the measure of positive value of the pseudo-density function and its approximation under the convergence of the finite element solution, we have prove it. Here we have used the Heaviside operator. The mathematical analysis of the total finite element scheme, however, is our theme in the near future.