-One of the big advantages of the standard finite element method FEM, is its efficiency in treating complicated

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The Marangoni effect is a very important phenomenon happening at an interface between two immiscible fluids creating a source of convection. This effect is very important in two phase flow problems. Unfortunately, the Marangoni effect is neglected by many studies in two phase fluid flow and is still considered a challenging problem.A mathematical model has been developed in this paper showing the Marangoni effect in the case of two immiscible fluids in Navier-Stokes equation. The mathematical translation of the convection term at the interface is developed in detail from the starting point of physical parameters using powerful mathematical tools.Citation: Mekhlouf R, Baggag A. Mathematical aspect of the Marangoni effect at the interface between two immiscible fluids. Fluid Mech Res Int. 2018;2(2):55-58.

One of the big advantages of the standard finite element method is its efficiency in treating complicated geometries and imposing the associated boundary conditions. However in some cases, such as handling the Dirichlet-type boundary conditions, the stability and the accuracy of FEM are seriously compromised.In this work, Nitsche's method is introduced, as an efficient way of expressing the Dirichlet boundary conditions in the weak formulation. It is shown that Nitsche's method preserves the rate of convergence and gives more accuracy than the classical approach. The method is implemented for the simplest case of Poisson equation, for Stokes flow and Navier-Stokes equations, with slip and noslip boundary conditions, in the case of viscous Newtonian incompressible flows. Error norms are calculated on different meshes in terms of size, topology and adaptivity, for a fair assessment of the proposed Nitsche's method.