4.1.10. Cauchy Momentum Equation

4.1.10.1. What is the derivation of the Cauchy Equation?

  • Temporal change in momentum:
\[{\partial \over {\partial t}} \int_V \rho \vec{u} dV\]
  • Spatial change in momentum:
\[\int_S \rho \vec{u} (\vec{u} \cdot \vec{n}) dS\]
  • Surface forces:
\[\int_S (\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} \cdot \vec{n}) dS\]
  • Body forces:
\[\int_V \rho \vec{g} dV\]
  • Newton’s 2nd Law:
\[\text{Temporal change in momentum + Spatial change in momentum = Surface forces + Body forces}\]
\[{\partial \over {\partial t}} \int_V \rho \vec{u} dV + \int_S \rho \vec{u} (\vec{u} \cdot \vec{n}) dS = \int_S (\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} \cdot \vec{n}) dS + \int_V \rho \vec{g} dV\]

4.1.10.2. What is the derivation of the conservation of momentum in differential form?

  • Gauss divergence theorem:
\[\int_S \rho \vec{u} (\vec{u} \cdot \vec{n}) = \int_V \nabla \cdot (\rho \vec{u} \otimes \vec{u}) dV\]
\[\int_S (\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} \cdot \vec{n}) dS = \int_V \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} dV\]
  • From integral form:
\[{\partial \over {\partial t}} \int_V \rho \vec{u} dV + \int_V \nabla \cdot (\rho \vec{u} \otimes \vec{u}) dV = \int_V \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} dV + \int_V \rho \vec{g} dV\]
  • Volume is fixed and arbitary:
\[{\partial \over {\partial t}} (\rho \vec{u}) + \nabla \cdot (\rho \vec{u} \otimes \vec{u}) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
  • In Einstein notation:
\[{\partial \over {\partial t}} (\rho u_j) + {\partial \over {\partial x_i}} (\rho u_j u_i) = {\partial \over {\partial x_i}} {\sigma_{ij}} + \rho g_j\]

4.1.10.3. What is the non-conservative form of the momentum equation (Cauchy Equation) for incompressible flow?

  • Differential form:
\[{\partial \over {\partial t}} (\rho \vec{u}) + \nabla \cdot (\rho \vec{u} \otimes \vec{u}) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
  • Differentiation:
\[\vec{u} {{\partial \rho} \over {\partial t}} + \rho {{\partial \vec{u}} \over {\partial t}} + (\vec{u} \cdot \nabla) \rho \vec{u} + (\nabla \cdot \vec{u}) \rho \vec{u} = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
  • Incompressible \(\longrightarrow\) \(\partial \rho / \partial t = 0\), \(\rho = constant\) and \(\nabla \cdot \vec{u}=0\):
\[\rho {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \rho \vec{u} = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]

4.1.10.4. What is represented by the Cauchy Equation?

  • From non-conservative, differential form:
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
  • \(\rho {{\partial \vec{u}} \over {\partial t}}\) = Temporal change in momentum/unit volume at a fixed point
  • \(\rho (\vec{u} \cdot \nabla) \vec{u}\) = Spatial change in momentum/unit volume at a fixed point
  • \(\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma}\) = Total stress force/unit volume
  • \(\rho \vec{g}\) = Gravity force/unit volume