HydrostaticPressure

The basic equation for the hydrostatic pressure is: \begin{align} \frac{\left( \nabla ^{T}\mathbf{v} \right)_{hyd}^{n+1}}{\Delta t}+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{hyd}^{n+1} \right) \right) = \Pi ^{n+1}+\nabla ^{T}\mathbf{g} + \Theta^{n+1}_{hyd}\end{align} This pressure might be different from zero even if there is no motion of the fluid. It might be due to gravity (depth pressure), internal forces (elasticity), etc. We represent the compression part \( \nabla ^{T}\mathbf{v}\) by the expressions found in DeriveDivergenceOfVelocity : \begin{align} \begin{array}{*{35}{l}} \frac{\left( \nabla ^{T}\mathbf{v} \right)_{hyd}^{n+1}}{\Delta t}+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{hyd}^{n+1} \right) \right) & = \Pi ^{n+1}+\nabla ^{T}\mathbf{g} + \Theta^{n+1}_{hyd} \\ -\frac{1}{\Delta t}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}\left( p_{hyd}^{n+1}-p_{hyd}^{n} \right) \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla \left( p_{hyd}^{n+1} \right) \right) & = \Pi ^{n+1}+\nabla ^{T}\mathbf{g} + \Theta^{n+1}_{hyd} \\ -\frac{1}{\Delta t}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}p_{hyd}^{n+1} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla p_{hyd}^{n+1} \right) & =-\frac{1}{\Delta t}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}p_{hyd}^{n} \right)+\Pi ^{n+1}+\nabla ^{T}\mathbf{g} + \Theta^{n+1}_{hyd} \\ \end{array}\end{align} the final equation is \begin{align} -\frac{1}{\Delta t}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}p_{hyd}^{n+1} \right)+\nabla ^{T}\left( \frac{1}{\rho }\nabla p_{hyd}^{n+1} \right) = -\frac{1}{\Delta t}\left( \frac{1}{\rho }\frac{\partial \rho }{\partial p}p_{hyd}^{n} \right)+\Pi ^{n+1}+\nabla ^{T}\mathbf{g} + \Theta^{n+1}_{hyd}\end{align} and its numerical discretization is found in HydrostaticPressureAlgorithm .