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

本研究已在1998-1999年期间交付，以便订购和分析带有自由界面的流体流动的有限元素方案，在自然和繁琐的工业过程中经常找到有限元素方案。在1998年，我们考虑了两个流体流动的低密度比率的有限元素方案。我们认为这两个流体都是可见的、不复杂的和不相容的。作为这个问题的数学模型，我们使用了一个流体模型，并将Boussinesq近似于Navier-Stokes equations。该接口被认为是一个0级伪密度函数集合，其中是传输方程的解决方案。我们为这些等式提出了一个混合的有限元素方案，P1 iso P2/P1元素。特别是，我们还提出了使用双重良好潜力的运输等式的再初始化技术。在1999年，我们考虑了两个流体流动的有限元素方法，其中包括自由界面的表面张力效应。这里，我们使用一个真正的双流体流动系统来处理密度和能见度的高比率。在一般情况下，表面张力效应由界面上的线积分表示。在我们的研究中，表面张力效应被解释为一个身体力量跨越一个狭窄的界面区域，在另一个方面，作为两个流体流动的有限元素方案的数学分析，我们将考虑到近似界面的融合。在事实上，估计L-D1 p-D1（Ω）-伪密度函数的正值之间的差异的规范，以及在有限元素解决方案的融合下的近似值，我们已经证明了这一点，这里我们已经使用了重型操作员。对总有限元素方案的数学分析，无论如何，是我们未来的主题。

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.