4.1.9. Conservation of Mass

4.1.9.1. What is the difference between compressible and incompressible flows?

  • Incompressible flow:
    • \(p=f(u)\)
    • \(\partial p\), \(\partial \rho\) and \(\partial T\) (small changes)
  • Compressible flow:
    • \(p=f(\rho, T)\)
    • \(\Delta p\), \(\Delta \rho\) and \(\Delta T\) (large changes)

4.1.9.2. What is the derivation of the conservation of mass in integral form?

  • Temporal change of mass:
\[{\partial \over {\partial t}} \int_V \rho dV\]
  • Spatial change of mass:
\[\int_S \rho(\vec{u} \cdot \vec{n})dS\]
  • Conservation of mass:
\[\text{Temporal change of mass} + \text{Spatial change of mass} = 0\]
\[{\partial \over {\partial t}} \int_V \rho dV + \int_S \rho(\vec{u} \cdot \vec{n})dS = 0\]
  • If incompressible \(\partial \rho / \partial t = 0\) and \(\rho=constant\):
\[\int_S (\vec{u} \cdot \vec{n})dS = 0\]

4.1.9.3. What is the derivation of the conservation of mass in differential form?

  • Gauss divergence theorem:
\[\int_S \rho (\vec{u} \cdot \vec{n})dS = \int_V \nabla \cdot (\rho \vec{u}) dV\]
  • From Integral Form:
\[{\partial \over {\partial t}} \int_V \rho dV + \int_V \nabla \cdot (\rho \vec{u})dS = 0\]
  • Volume fixed:
\[\int_V \left( {{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec{u}) \right) dV = 0\]
  • Volume is arbitary (this is for compressible flow):
\[{{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec{u}) = 0\]
  • By expansion:
\[{{\partial \rho} \over {\partial t}} + (\vec{u} \cdot \nabla) \rho + (\nabla \cdot \vec{u}) \rho = 0\]
  • If incompressible \({{\partial \rho} \over {\partial t}} = 0\) and \(\nabla \rho = 0\) and \(\rho = constant\):
\[\nabla \cdot \vec{u} = 0\]