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Subsubsection 4.13.2.4: Prescribed Fiber Stress Up Chapter 4: Materials Subsection 4.14.1: General Specification of Fluid Materials

## 4.14 Viscous Fluids

Fluid mechanics analyses may be used to examine fluid flow over fixed domains. Fluid-structure interactions (FSI) may be used to examine fluid flow over deforming domains. FEBio's fluid and fluid-FSI modules treat the fluid as isothermal (constant and uniform temperature) and compressible; incompressible flow is simulated by prescribing a physically realistic bulk modulus to model the fluid's elastic compressibility (for example, for water at room temperature). The Cauchy stress in a fluid is given by where is the fluid pressure arising from the elastic response and is the viscous stress resulting from the fluid viscosity and its rate of deformation. The fluid pressure is evaluated from whre is the fluid volume ratio and is the dilatation (the relative change in volume). The fluid density varies with according to where is the fluid density in the reference configuration (when the pressure is zero). The dependence of on the fluid rate of deformation tensor , where fluid velocity, is given by a constitutive model which may be chosen from the list provided in Section 4.14.2↓. Though Newtonian and non-Newtonian fluids may be analyzed in this framework, all fluids are purely viscous (no viscoelasticity is included in this formulation). Setting the viscosity to zero is allowable, for the purpose of analyzing inviscid flow. The user is referred to the FEBio Theory Manual for a general description of this isothermal compressible viscous flow framework.
For fluid analyses, which are performed over a fixed mesh, FEBio solves for the components of the fluid velocity and fluid dilatation at each node. In contrast, fluid-FSI analyses are performed over deforming meshes and FEBio treats fluid-FSI domains as a mixture of a viscous fluid and a massless, frictionless porous solid . Thus, the fluid encounters zero resistance as it flows through the deforming mesh (porous solid). The solid constituent of a fluid-FSI domain regularizes the mesh deformation; it is a hyperelastic solid whose constitutive relation is selected by the user; the recommended choice is the unconstrained neo-Hookean solid defined in Section 4.1.3.16↑, with and set to a very small (but non-zero) value. For fluid-FSI analyses, FEBio solves for the solid displacement , the fluid velocity relative to the mesh, and the fluid dilatation . The fluid velocity is then calculated as where is the mesh velocity (the material time derivative of the solid displacement ). Since the mesh is fixed in a standard fluid analysis, and in that case. Therefore, we use to refer to the fluid velocity degrees of freedom for fluid and fluid-FSI analyses. For both types of analyses, the no-slip condition for viscous fluids flowing along a boundary is satisfied by setting on that boundary.
On any fluid boundary, the outward surface normal may be denoted by and the traction vector on the fluid is given by , where is the viscous traction. Essential (Dirichlet) boundary conditions may be prescribed on and , while natural (Neumann) boundary conditions may be prescribed on and . The appearance of velocity in both essential and natural boundary conditions may seem surprising at first. To better understand the nature of these boundary conditions, it is convenient to separate the velocity into its normal and tangential components, , where . In particular, for inviscid flow, the viscous stress and its corresponding traction are both zero, leaving as the sole natural boundary condition; similarly, becomes the only essential boundary condition in such flows, since is unknown a priori on a frictionless boundary and must be obtained from the solution of the analysis.
In general, prescribing is equivalent to prescribing the elastic fluid pressure, since is only a function of . On a boundary where no conditions are prescribed explicitly, we conclude that and , which represents a frictionless wall. Conversely, it is possible to prescribe and on a boundary to produce a desired inflow or outflow while simultaneously stabilizing the flow conditions by prescribing a suitable viscous traction. Prescribing essential boundary conditions and determines the tangential velocity on a boundary as well as the elastic fluid pressure , leaving the option to also prescribe the normal component of the viscous traction, , to completely determine the normal traction (or else naturally equals zero). Mixed boundary conditions represent common physical features: Prescribing and completely determines the velocity on a boundary; prescribing and completely determines the traction on a boundary. Note that and are mutually exclusive boundary conditions, and the same holds for and the tangential component of the viscous traction, .
For viscous fluids, the no-slip boundary condition at an impermeable wall is enforced by setting , whereas the wall impermeability condition implies . Therefore, these conditions may be combined by prescribing wx, wy and wz components of to zero. FEBio offers a range of options for conveniently prescribing natural and mixed conditions on boundary surfaces in fluid analyses (Section 3.12.2↑).

Subsubsection 4.13.2.4: Prescribed Fiber Stress Up Chapter 4: Materials Subsection 4.14.1: General Specification of Fluid Materials