# 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 $$\rho = constant$$:

$\nabla \cdot \vec{u} = 0$