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2.11.1 Mass and Momentum Balance
In a spatial (Eulerian) frame, the momentum balance equation for a continuum is where is the density, is the Cauchy stress, is the body force per mass, and is the acceleration, given by the material time derivative of the velocity in the spatial frame, where is the spatial velocity gradient. The mass balance equation is where the material time derivative of the density in the spatial frame is Let denote the deformation gradient (the gradient of the motion with respect to the material coordinate). The material time derivative of is related to via Let denote the Jacobian of the motion (the volume ratio, or ratio of current to referential volume, ); then, the dilatation (relative change in volume between current and reference configurations) is given by . Using the chain rule, 's material time derivative is which, when combined with eq.(2.11.1-5), produces a kinematic constraint between and , Substituting this relation into the mass balance, eq.(2.11.1-3), produces , which may be integrated directly to yield where is the density in the reference configuration (when ). Since is obtained by integrating the above material time derivative of , it is an intrinsic material property that must be invariant in time and space.
The Cauchy stress is given by where is the identity tensor, is the viscous stress, is the pressure arising from the elastic response, and is the free energy density of the fluid (free energy per volume of the continuum in the reference configuration). The axiom of entropy inequality dictates that cannot be a function of the rate of deformation . In contrast, the viscous stress is generally a function of and .
Boundary conditions may be derived by satisfying mass and momentum balance across a moving interface . Let divide the material domain into subdomains and and let the outward normal to on be denoted by . The jump condition across derived from the axiom of mass balance requires that where on and is the velocity of the interface . Thus, represents the velocity of the fluid relative to . The double bracket notation denotes , where and represent the value of on in and , respectively. This jump condition implies that the mass flux normal to must be continuous. In particular, if is a fluid domain and is a solid domain, and denotes the solid boundary (e.g., a wall), we use , for the fluid, and for the solid, such that eq.(2.11.1-10) reduces to . The jump condition derived from the axiom of linear momentum balance similarly requires that This condition implies that the jump in the traction across must be balanced by the jump in momentum flux normal to . In addition to jump conditions dictated by axioms of conservation, viscous fluids require the satisfaction of the no-slip condition, which implies that the velocity component tangential to is continuous across that interface.
In our finite element treatment we use and as nodal variables, implying that our formulation automatically enforces continuity of these variables across element boundaries, thus and . Based on eqs.(2.11.1-7) and (2.11.1-9), it follows that the density and elastic pressure are continuous across element boundaries in this formulation, and . Thus, the mass jump in eq.(2.11.1-10) is automatically satisfied, and the momentum jump in eq.(2.11.1-11) reduces to , requiring continuity of the traction, or more specifically according to (2.11.1-8), the continuity of the viscous traction , since is automatically continuous.