Binary-fluid–solid interaction based on the Navier–Stokes–Korteweg equations

Abstract

We consider a computational model for binary-fluid–solid interaction based on an arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes–Korteweg equations, and we assess the predictive capabilities of this model. Due to the presence of two distinct fluid components, the stress tensor in the binary-fluid exhibits a capillary component in addition to the pressure and viscous-stress components. The distinct fluid–solid surface energies of the fluid components moreover lead to preferential wetting at the solid substrate. Compared to conventional FSI problems, the dynamic condition coupling the binary-fluid and solid subsystems incorporates an additional term associated with the binary-fluid–solid surface tension. We consider a formulation of the Navier–Stokes–Korteweg equations in which the free energy associated with the standard van-der Waals equation of state is replaced by a polynomial double-well function to provide better control over the diffuse-interface thickness and the surface tension. For the solid subsystem, we regard a standard hyperelastic model. We explore the main properties of the binary-fluid–solid interaction problem and establish a dissipation relation for the aggregated system. In addition, we present numerical results based on a fully monolithic approach to the complete nonlinear system. To validate the computational model, we consider the elasto-capillary interaction of a sessile droplet on a soft solid substrate and compare the numerical results with a corresponding solid model with fabricated fluid loads and with experimental data.