Comparison of four boundary conditions for the fluid-hydrogel interface

Abstract

In adopting a poroelastic model for a hydrogel, one views its constituent fluid and solid phases as interpenetrating continua, thereby erasing the "pore-scale" geometry. This gives rise to the need for additional boundary conditions (BCs) at the interface between a hydrogel and a clear fluid to supplement the momentum equations for the fluid and solid phases in the hydrogel. Using a thermodynamic argument on energy dissipation, we propose three sets of boundary conditions for the gel-fluid interface that link the normal and tangential velocity jumps across the interface to the normal and tangential stresses on either side of the interface. Using several flow problems-one-dimensional compression, two-layer Couette and Poiseuille shear flows, and deformation of a gel particle by a planar extension flow-as tests, we compare the predictions of these three BCs with that of a previously proposed BC. Some differences are stark and reveal flaws in certain BCs. Others are subtler and will require quantitative experimental data for validation. Based on these results, we recommend one set of BCs over the other three for computing the flow and deformation of hydrogels in contact with a clear fluid. In addition, we suggest benchmark experiments to validate the BCs and our recommendation.

A theory of hydrogel mechanics that couples swelling and external flow

“…In those limits, we should reproduce respectively the solutions for quiescent swelling 10,19 and for flow through hydrogels. 42,43 This is one way to demonstrate how our model unifies the two aspects that have been previously studied separately. Toward the diffusion-only limit, we need only reduce V 0 toward 0.…”

“…This behavior closely resembles our previous simulations of 1D compression without the swelling effect. 42,43 However, a major distinction is that in the current unified model, the chemical potential gradient drives a diffusion flux, which is absent from the earlier studies. Moreover, the chemical potential modifies the pressure jump across the boundary via eqn (38).…”

“…37–40 Using irreversible thermodynamics, we have proposed such BCs and tested them in relatively simple flows involving a solvent–gel interface. 41–43 These results clearly suggest the importance of convection as a distinct mode of solvent transport, which may give rise to novel phenomena. For example, as the flow compresses the gel in a capillary, the solvent flux may vary non-monotonically with the pressure drop.…”

“…In the convective limit ( j f = 0), Young et al 40,41,43 used the principle of positive entropy production to derive the following BCs: μ ( V − v s )· n = η ( n · Σ · n − P + p ), μ ( V − v s )· t = β n · Σ · t ,where t is the tangential vector along the interface, and the positive coefficients η and β are the interfacial penetration length and slip length, respectively. These conditions correspond to BC2 of ref.…”

“…These conditions correspond to BC2 of ref. 41 and 43, with some notational simplifications as explained in Appendix A. For the current purpose, we follow a similar procedure to extend eqn (21) to account for diffusive flux across the gel–solvent interface (see Appendix A for details).…”