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

$Total \ Acceleration = Unsteady \ Acceleration + Convective/Advective \ Acceleration$