$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The nodal unknowns in this formulation are $\mathbf{v}$ and $J$ (or $e$ ), which may be solved using the momentum balance in eq.(2.11.1-1) and the kinematic constraint between $J$ and $\mathbf{v}$ given in eq.(2.11.1-6). The virtual work integral for a Galerkin finite element formulation [23] is given by $$\begin{equation} \begin{aligned}\delta W & =\int_{\Omega}\delta\mathbf{v}\cdot\left(\divg\boldsymbol{\sigma}+\rho\left(\mathbf{b}-\mathbf{a}\right)\right)dv\\ & +\int_{\Omega}\delta J\left(\frac{\dot{J}}{J}-\divg\mathbf{v}\right)dv\,, \end{aligned} \label{eq:virtual work} \end{equation}$$ where $\delta\mathbf{v}$ is a virtual velocity and $\delta J$ is a virtual energy density; $\Omega$ is the fluid finite element domain and $dv$ is a differential volume in $\Omega$ . 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, $\delta W=\delta W_{ext}-\delta W_{int}$ , where $$\begin{equation} \begin{aligned}\delta W_{int} & =\int_{\Omega}\boldsymbol{\tau}:\grad\delta\mathbf{v}\,dv+\int_{\Omega}\delta\mathbf{v}\cdot\left(\grad p+\rho\mathbf{a}\right)\,dv\\ & -\int_{\Omega}\left(\delta J\frac{\dot{J}}{J}+\grad\delta J\cdot\mathbf{v}\right)\,dv\,, \end{aligned} \label{eq:virtual-work-internal-1} \end{equation}$$ and $$\begin{equation} \delta W_{ext}=\int_{\partial\Omega}\delta\mathbf{v}\cdot\mathbf{t}^{\tau}\,da+\int_{\Omega}\delta\mathbf{v}\cdot\rho\mathbf{b}\,dv-\int_{\partial\Omega}\delta J\,v_{n}\,da\,.\label{eq:virtual-work-external-1} \end{equation}$$ Here, $\partial\Omega$ is the boundary of $\Omega$ and $da$ is a differential area on $\partial\Omega$ , $\mathbf{t}^{\tau}=\boldsymbol{\tau}\cdot\mathbf{n}$ is the viscous component of the traction $\mathbf{t}$ , and $v_{n}=\mathbf{v}\cdot\mathbf{n}$ is the velocity normal to the boundary $\partial\Omega$ , with $\mathbf{n}$ representing the outward normal on $\partial\Omega$ . From these expressions, it becomes evident that essential (Dirichlet) boundary conditions may be prescribed on $\mathbf{v}$ and $J$ , while natural (Neumann) boundary conditions may be prescribed on $\mathbf{t}^{\tau}$ and $v_{n}$ . 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, $\mathbf{v}=v_{n}\mathbf{n}+\mathbf{v}_{t}$ , where $\mathbf{v}_{t}=\left(\mathbf{I}-\mathbf{n}\otimes\mathbf{n}\right)\cdot\mathbf{v}$ . In particular, for inviscid flow, the viscous stress $\boldsymbol{\tau}$ and its corresponding traction $\mathbf{t}^{\tau}$ are both zero, leaving $v_{n}$ as the sole natural boundary condition; similarly, $J$ becomes the only essential boundary condition in such flows, since $\mathbf{v}_{t}$ is unknown a priori on a frictionless boundary and must be obtained from the solution of the analysis.