# 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