Energy Conservation

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\frac{\partial}{\partial t} (\frac{ρ v^{2}}{2}) = - \nabla \cdot [\vec{v} (\frac{ρ v^{2}}{2} + P) - ([σ^{'}] \cdot \vec{v})] + ρ \vec{v} \cdot \vec{f} - {σ^{'}}_{i j} \frac{\partial v_{i}}{\partial x_{j}}

\frac{d}{d t} ∭_{V} \frac{ρ v^{2}}{2} d V = - \iint_{S} \frac{ρ v^{2}}{2} (\vec{v} \cdot \hat{n}) d S - \iint_{S} P (\vec{v} \cdot \hat{n}) d S + \iint_{S} ([σ^{'}] \cdot \hat{n}) \cdot \vec{v} d S + ∭_{V} ρ \vec{f} \cdot \vec{v} d V - ∭_{V} {σ^{'}}_{i j} \frac{\partial v_{i}}{\partial x_{j}} d V

Fluid Fluency