4.1.11. Euler Momentum Equation

4.1.11.1. What is the derivation of the Euler Momentum Equation from the Cauchy Equation?

  • Cauchy Equation:
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
  • Inviscid flow \(\longrightarrow \overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau}=0\):
\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = \begin{bmatrix} \sigma_{11} & 0 & 0 \\ 0 & \sigma_{22} & 0 \\ 0 & 0 & \sigma_{33} \end{bmatrix}\end{split}\]
  • Normal stresses = applied pressure:
\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = -p \delta_{ij} = -p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} = \begin{bmatrix} -p & 0 & 0 \\ 0 & -p & 0 \\ 0 & 0 & -p \end{bmatrix}\end{split}\]
  • Expansion of divergence of shear stress:
\[\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = \nabla \cdot (-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I}) = -(\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{I})p -(\overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \cdot \nabla)p = -\nabla p\]
  • Hence:
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) = -\nabla p + \rho \vec{g}\]

4.1.11.2. What is the meaning of the terms in the Euler momentum equation?

\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) = -\nabla p + \rho \vec{g}\]
  • \(\rho {{\partial \vec{u}} \over {\partial t}}\) = Temporal change of momentum/unit volume at a fixed point
  • \(\rho(\vec{u} \cdot \nabla) \vec{u}\) = Spatial change of momentum/unit volume at a fixed point
  • \(-\nabla p\) = Pressure force/unit volume at a fixed point
  • \(\rho \vec{g}\) = Gravity force/unit volume at a fixed point

4.1.11.3. How are all the terms in the Euler Equations expanded in 3D?

  • x-direction:
\[\rho \left( {{\partial {u}} \over {\partial t}}+ {u} {{\partial u} \over {\partial x}} + {v} {{\partial u} \over {\partial y}} + {w} {{\partial u} \over {\partial z}} \right)= -{{\partial {p}} \over {\partial x}} + \rho g_x\]
  • y-direction:
\[\rho \left( {{\partial {v}} \over {\partial t}}+ {u} {{\partial v} \over {\partial x}} + {v} {{\partial v} \over {\partial y}} + {w} {{\partial v} \over {\partial z}} \right)= -{{\partial {p}} \over {\partial y}} + \rho g_y\]
  • z-direction:
\[\rho \left( {{\partial {w}} \over {\partial t}}+ {u} {{\partial w} \over {\partial x}} + {v} {{\partial w} \over {\partial y}} + {w} {{\partial w} \over {\partial z}} \right)= -{{\partial {p}} \over {\partial z}} + \rho g_z\]
  • Can also add source terms to the RHS \(S_x\), \(S_y\) and \(S_z\)