Binary-fluid–solid interaction based on the Navier–Stokes–Cahn–Hilliard Equations

We consider amodel for binary-fluid-solid interaction based on a diffuse-interface model for the binary fluid and a hyperelastic-materialmodel for the solid. The diffuse-interface binary-fluidmodel is described by the quasi-incompressible Navier-Stokes-Cahn-Hilliard equations with preferential-wetting boundary conditions at the fluid-solid interface. The fluid traction on the interface includes a capillary-stress contribution in addition to the regular viscous-stress and pressure contributions. The dynamic interface condition comprises the traction exerted by the nonuniform solid-fluid surface tension in accordance with the Young-Laplace law for the solid-fluid interface. The solid is modeled as a hyperelastic material. We present a weak formulation of the aggregated binary-fluid-solid interaction problem, based on an arbitrary Lagrangian-Eulerian formulation of theNavier-Stokes-Cahn-Hilliard equations and a properweak evaluation of the binary-fluid traction and of the solid-fluid surface tension. We also present an analysis of the essential properties of the binary-fluid-solid interaction problem, including a dissipation relation for the complete fluid-solid interaction problem. To validate the presented binary-fluid-solid interactionmodel, we consider numerical simulations for the elasto-capillary interaction of a droplet with a soft solid substrate and present a comparison to corresponding experimental data.