Numerical simulation of the absorption of a droplet in a porous medium

In this paper we study the behavior of a droplet printed on a porous substrate by an inkjet printer. An extended model for absorption of a droplet in a porous substrate is proposed. The model is based on a model proposed by Alleborn et al.[1][2], with the extension of the dynamics of the wetting front due to capillary forces and radial velocity. As a basic assumption, an initially spherically shaped droplet is considered such that the model can be simplified as an axially symmetric problem. The droplet dynamics is driven by pressure that acts on the droplet, and consists of Laplace pressure, disjoining pressure and gravity. Hence, fluid flow in the droplet is modeled by the Navier-Stokes equation and continuity equation, where the lubrication approximation is taken into account. The mass loss due to sorption is modeled by the Darcy equation. Here, we extend our model by considering the radial velocity of the fluid inside the porous medium. The extension implies the spreading of the fluid inside the substrate in radial direction. Hence the dynamics of the wetting front due to the radial velocity is modeled using the mass balance principle. For the condition in the surface of the substrate, we use continuity of velocity and pressure. In the wetting front, a discontinuous pressure is assumed, in order to distinguish between the saturated and unsaturated porous medium. The numerical method has good stability properties. Numerical results agree well with simulations from a commercial CFD package where the full Navier-Stokes equation is solved numerically.