Boundary conditions at a gel-fluid interface

Hydrogels consist of a polymer skeleton hydrated by an aqueous solvent, and their hydrodynamics is often described by a coarse-grained poroelasticity model where the boundary conditions between the hydrogel and a surrounding solvent require careful consideration. Young et al. [Phys. Rev. Fluids 4, 063601 (2019)] used the energy dissipation principle to derive a set of boundary conditions regarding the velocity jumps at the interface. However, when applied to an external shear flow over a gel layer, these conditions predict no entrained flow inside the gel, in contrast to the prediction of a previous model by Minale [Phys. Fluids 26, 123102 (2014)]. We adapt the procedure of Young et al. to derive an alternative set of boundary conditions that does allow an external shear flow to induce shear inside the gel and compare the velocity profile to that of Minale. We also derive the limiting form of the boundary conditions in a Darcy medium.

Figure 1

A schematic of a drop of poroelastic gel in a Newtonian viscous fluid. $\mathrm{\Gamma }$ is the fluid-gel interface.

Figure 2

A schematic showing the shear-flow geometry considered by Minale [19]. A layer of gel lies atop a rigid substrate, and a viscous fluid undergoes a simple shear above the gel parallel to the gel-fluid interface.

Figure 3

For a gel with porosity $\varphi =0.8$ and a large pore size ($\gamma =25$), the model predicts an interfacial velocity $V_i$ on the pure-fluid side that increases with the slip ratio $\alpha$. Interestingly, the interfacial velocity $v_i$ on the gel side exhibits a maximum at $\alpha \approx 0.04$. The interfacial slip $V_i-v_i$ also increases with $\alpha$.

Figure 4

Velocity profile in the 1D shear-flow solution for a gel layer with large pores ($\gamma =25$), large slip ($\alpha =0.01$) and moderate porosity $\varphi =0.8$. The $y$ axis is the dimensionless coordinate $y\sqrt{\varphi /K}$, with the interface at $y\sqrt{\varphi /K}=\gamma =25$ (dotted horizontal line). On the pure-fluid side, only a small portion of the linear profile is plotted. The solid curves are the velocity profile for BC2 in this work, and the dash-dotted curves are prediction of Minale's model.