Two-dimensional thin-film flow of an incompressible inhomogeneous fluid in a channel

We study the 2D flow of an inhomogeneous incompressible fluid in a channel of small aspect ratio. The constitutive equation of the fluid consists of the classical incompressible Navier-Stokes equation plus an extra term depending on the density and its spatial gradient. This type of fluid is particularly suitable for describing the flow of incompressible granular materials. We formulate the mathematical problem and we exploit the smallness of the aspect ratio of the channel to obtain a simplified version (leading order of the lubrication expansion) that can be solved numerically. Choosing a particular set of boundary data we also determine a particular solution that can be compared to the numerical one to prove the consistency of the numerical scheme. The comparison shows good agreement. Finally we extend the model to the case of a channel with variable thickness and we compare our simulations with the ones obtained in Massoudi et al. (2012) [7].