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3.5.1 Weak Formulation
The nodal unknowns in this formulation are and (or ), which may be solved using the momentum balance in eq.(2.11.1-1) and the kinematic constraint between and given in eq.(2.11.1-6). The virtual work integral for a Galerkin finite element formulation  is given by where is a virtual velocity and is a virtual energy density; is the fluid finite element domain and is a differential volume in . This virtual work statement may be directly related to the axiom of energy balance, specialized to conditions of isothermal flow of viscous compressible fluids (see Section 2.11.2↑). Using the divergence theorem, we may rewrite the weak form of this integral as the difference between external and internal virtual work, , where and Here, is the boundary of and is a differential area on , is the viscous component of the traction , and is the velocity normal to the boundary , with representing the outward normal on . From these expressions, it becomes evident that 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, .