Theory Manual Version 3.4
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Subsection 2.13.1: BFSI Governing Equations Up Section 2.13: Hybrid Biphasic Material Chapter 3: The Nonlinear FE Method

### 2.13.2 BFSI Continuous Variables

Jump conditions on the axioms of mass, momentum and energy balance are needed to determine which variables may be selected as nodal DOFs in the finite element implementation, and which tractions are naturally continuous across an interface. The full set of jump conditions for a hybrid biphasic material were derived in our recent study for the constitutive assumptions adopted in this formulation [46]. Here, we summarize the salient results, which apply to an interface defined on the porous solid matrix of the hybrid biphasic domain, which includes the shared faces of adjoining biphasic elements. Thus, the velocity of the interface is given by the velocity of the solid constituent of the hybrid biphasic material. We employ the notation to denote the jump in the function across the interface , with and denoting the values of on either side of . The unit normal on is , which points away from the side. A variable which is continuous across satisfies .
Based on the jump condition on the axiom of mass balance, the normal component of the mass flux of the fluid relative to the solid is continuous across , . Furthermore, a sufficient condition to satisfy the jump on the axiom of energy balance is to enforce continuity of the fluid specific free enthalpy (also known as the Gibbs function), , where is the fluid specific free energy. This jump condition applies only when there is fluid on both sides of the interface . In an isothermal framework the specific free enthalpy is a function of state that only depends on , therefore this energy jump condition implies that must be continuous across , thus also implying that . Given (2.13.1-1), it follows that and the mass balance jump condition reduces to , implying that the relative fluid flux component normal to must be continuous. For the tangential component of on we appeal to the analysis of Hou et al. [47], who showed that a valid pseudo-noslip condition requires this tangential component to be continuous. Combining these two jump conditions produces
The momentum jump condition requires that the mixture traction be continuous across , thus . Since based on the energy jump, this mixture momentum jump condition reduces to Finally, another relation which is sufficient to satisfy the jump condition on the energy balance is the continuity of the true fluid traction (force acting on fluid per fluid area), This jump condition (2.13.2-4), which also applies only if fluid is present on both sides of , is interesting because it implies that the viscous stress (and thus, the viscosity) of a fluid flowing in a porous solid matrix scales with the porosity of that medium, such that where would be the true fluid viscous stress. Thus, we can use FEBio's various constitutive relations for the viscous stress of Newtonian or non-Newtonian fluids and adapt those models to a biphasic mixture where is evaluated as . Accordingly, the contribution of would properly reduce to zero in the limit as fluid content reduces to zero (), in which case the mixture momentum jump in (2.13.2-3) would reduce to , since .
Letting and be nodal DOFs automatically enforces the jump conditions (2.13.2-1) and (2.13.2-2), acting as essential boundary conditions, along with the solid displacement . Finally, subtracting (2.13.2-4) from (2.13.2-3), we obtain the momentum jump condition for the solid constituent,
A BFSI domain may be reduced to a FSI domain by letting and ; the solid matrix would still be ascribed a material response but its stiffness would need to be negligible. The number of nodal DOFs would remain the same. The CFD domain is a special case of the FSI domain where the mesh displacement is uniformly .
However, when the biphasic domain interfaces with a non-porous solid domain across , the jump conditions (2.13.2-1) on and (2.13.2-5) on don't apply. In that case, the jump condition (2.13.2-2) on the relative fluid volumetric flux should reduce to for the BFSI domain on , although the user would have to enforce this no-slip condition explicitly by prescribing it as an essential boundary condition. The mixture momentum jump implies that the mixture traction on the BFSI side of an interface with a solid is equal to the solid traction on the solid side. This jump condition would also need to be enforced explicitly.
Subsection 2.13.1: BFSI Governing Equations Up Section 2.13: Hybrid Biphasic Material Chapter 3: The Nonlinear FE Method