4.1.8. Integral and Derivative Form

4.1.8.1. What are the conditions of applicability of the integral and differential forms of the conservation laws?

  • Integral form \(\longrightarrow\) same as limits of continuum model:

    • \(Kn \lt \lt 1\)

    • No mass-energy conversion

  • Differential form \(\longrightarrow\) same as limits of Gauss-Divergence theorem:

    • \(\phi\) must be continuous (no shock waves or free surfaces etc)

    • \(\partial \phi / \partial x\) must exist

    • \(\partial \phi / \partial x\) must be continuous

4.1.8.2. What is the Reynolds Transport Theorem?

\[{D \over {Dt}} \int_V \phi dV = {\partial \over {\partial t}} \int_V \phi dV + \int_S \phi (\vec{u} \cdot \vec{n}) dS\]

  • where: \(\phi\) is a scalar or vector and is a conserved quantity

4.1.8.3. What is the meaning of the Reynolds Transport Theorem?

\[\mathbf{Temporal} \ change \ of \ \phi \ following \ a \ \mathbf{moving} \ volume = \mathbf{Temporal} \ change \ of \ \phi \ in \ a \ \mathbf{fixed} \ volume + \mathbf{Spatial} \ change \ of \ \phi \ through \ \mathbf{fixed} \ surfaces\]

4.1.8.4. What are some examples of the Reynolds Transport Theorem?

  • Mass conservation \(\phi = \rho\) (mass per unit volume):

\[{D \over {Dt}} \int_V \rho dV = {\partial \over {\partial t}} \int_V \rho dV + \int_S \rho (\vec{u} \cdot \vec{n}) dS\]

  • Momentum conservation \(\phi = \rho \vec{u}\) (momentum per unit volume):

\[{D \over {Dt}} \int_V \rho \vec{u} dV = {\partial \over {\partial t}} \int_V \rho \vec{u} dV + \int_S \rho \vec{u} (\vec{u} \cdot \vec{n}) dS\]

  • Energy conservation \(\phi = \rho E\) (total specific energy per unit volume):

\[{D \over {Dt}} \int_V \rho E dV = {\partial \over {\partial t}} \int_V \rho E dV + \int_S \rho E (\vec{u} \cdot \vec{n}) dS\]

4.1.8.5. What is the total derivative?

  • From Reynolds Transport Theorem:

\[{D \over {Dt}} \int_V \phi dV = {\partial \over {\partial t}} \int_V \phi dV + \int_S \phi (\vec{u} \cdot \vec{n}) dS\]

  • Gauss Divergence:

\[{D \over {Dt}} \int_V \phi dV = {\partial \over {\partial t}} \int_V \phi dV + \int_V \nabla \cdot (\phi \vec{n}) dV\]

  • Equal Volumes:

\[{{D \phi} \over {Dt}} = {{\partial \phi} \over {\partial t}} + \nabla \cdot (\phi \vec{u})\]

  • By expansion:

\[{{D \phi} \over {Dt}} = {{\partial \phi} \over {\partial t}} + (\vec{u} \cdot \nabla) \phi + (\nabla \cdot \vec{u}) \phi\]

  • Incompressible \(\longrightarrow \nabla \cdot \vec{u} = 0\):

\[{{D \phi} \over {Dt}} = {{\partial \phi} \over {\partial t}} + (\vec{u} \cdot \nabla) \phi = {{\partial \phi} \over {\partial t}} + u_i {{\partial \phi} \over {\partial x_i}}\]

4.1.8.6. What is the meaning of the total derivative?

\[\mathbf{Temporal} \ change \ of \ \phi \ following \ a \ \mathbf{moving} \ point = \mathbf{Temporal} \ change \ of \ \phi \ at \ a \ \mathbf{fixed} \ point + \mathbf{Spatial} \ change \ of \ \phi \ at \ a \ \mathbf{fixed} \ point\]

Or

\[Total \ Acceleration = Unsteady \ Acceleration + Convective/Advective \ Acceleration\]