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.

An adaptive isogeometric analysis approach to elasto‐capillary fluid‐solid interaction

“…for certain c 0 , c 1 , generally distinct from c 0 , c 1 in (26). The part contained in L L ∪ L tr L−1 can again be suppressed, leading to tr .…”

“…Colors represent the pressure [Color figure can be viewed at wileyonlinelibrary.com] at the contact line reduces to a delta distribution, and the solid-fluid interface exhibits a sharp kink at the apex. 21,26 For the considered value of the interface-thickness parameter, = 1 m, the interface deformation at the apex exhibits a small but finite radius of curvature. From Figure 5 one can observe that the pressure in the droplet is virtually uniform at an elevated level of approximately 520 Pa relative to the ambient pressure.…”

“…The truncation consists in suppression of the part of N L−1 that is already contained in L L , that is, by setting the coefficient c 0 in (26) to zero:…”

“…For diffuse-interface binary-fluid models, this kink in fact exhibits a finite radius of curvature, which vanishes in the sharp-interface limit, that is, as the thickness of the diffuse-interface passes to zero. 26 The overall deformation of the solid-fluid interface including the wetting ridge is however insensitive to the thickness of the diffuse interface, provided that the diffuse-interface thickness is sufficiently small relative to the ridge's characteristic dimensions. The dimensions of the wetting ridge are proportional (with a factor close to unity) to the elasto-capillary length scale composed of the ratio of surface tension to the shear modulus.…”

A unified numerical model for wetting of soft substrates

“…where the center x C of the spherical cap is placed such that its distance to the surface of the substrate is |R C cos |. 36 As in Reference 15, we choose the following surface tensions:…”