Law of Conservation of Momentum and Energy

The momentum and energy conservation law states that a fluid property ($\phi$) following a fluid particle is a function of position (x, y, z) and time. And it experiences two rates of changes viz,

change in property concerning the time ($\frac{\partial \phi}{\partial t}$)

change is property concerning the location and movement ($\frac{\partial \phi}{\partial x} \frac{dx}{dt} + \frac{\partial \phi}{\partial y} \frac{dy}{dt} + \frac{\partial \phi}{\partial z} \frac{dz}{dt}$)

Considering $\phi$ is the property per unit mass, then the total derivative of $\phi$ (the rate of change of property per unit mass) can be expressed as

$\frac{D \phi}{Dt} = \frac{\partial \phi}{\partial t} + \frac{\partial \phi}{\partial x} \frac{dx}{dt} + \frac{\partial \phi}{\partial y} \frac{dy}{dt} + \frac{\partial \phi}{\partial z} \frac{dz}{dt}$

The change in x direction concerning time ($\frac{dx}{dt}$) is velocity u, y direction ($\frac{dy}{dt}$) is v, and z direction ($\frac{dz}{dt}$) is w.

$\frac{D \phi}{Dt} = \frac{\partial \phi}{\partial t} + u \frac{\partial \phi}{\partial x} + v \frac{\partial \phi}{\partial y} + w \frac{\partial \phi}{\partial z}$

Using the Gradient rule and dot product rule (refer divergence rule)

$\frac{D \phi}{Dt} = \frac{\partial \phi}{\partial t} + U . grad \phi$

When we multiply the rate of change of property per unit mass ($\frac{D \phi}{Dt}$) with density ($\rho$), we get the rate of change of property per unit volume.

$\rho \frac{D \phi}{Dt} = \rho (\frac{\partial \phi}{\partial t} + U . grad \phi)$

The equation for the conservation of mass for a fluid with a orbitrary conserved property can be written as below (refer Conservation of mass).

$\frac{\partial (\rho \phi)}{\partial t} + div(\rho \phi U)$

Using dot product rule

$= \rho \frac{\partial \phi}{\partial t} + \phi \frac{\partial \rho}{\partial t} + \phi div(\rho U) + \rho U . grad \phi$

$= \rho (\frac{\partial \phi}{\partial t} + U . grad \phi) + \phi (\frac{\partial \rho}{\partial t} + div(\rho U))$

The mass conservation law states that $\frac{\partial \rho}{\partial t} + div(\rho U)$ is equal to zero.

$= \rho (\frac{\partial \phi}{\partial t} + U . grad \phi)$

$= \rho \frac{D \phi}{Dt}$

so,

$\rho \frac{D \phi}{Dt} = \frac{\partial (\rho \phi)}{\partial t} + div(\rho \phi U)$

$\begin{Bmatrix} \text{The Rate Of Increase } \\ \text{Of } \phi \text{ for a}\\ \text{fluid particle} \end{Bmatrix} = \begin{Bmatrix} \text{The Rate Of Increase } \\ \text{Of } \phi \text{ of }\\ \text{fluid element} \end{Bmatrix} + \begin{Bmatrix} \text{Net rate of flow } \\ \text{Of } \phi \text{ out of }\\ \text{fluid element} \end{Bmatrix}$

Note the terms:

Fluid Particle = Lagrangian form => the fluid particle that is moving with the flow

Fluid Element = Eulerian form => the fluid element that is stationary in space

From the above equation we can construct the three momentum equation and energy equation as below.

Momentum	Notation	LHS	RHS
x-momentum	u	$\rho \frac{Du}{Dt}$	$\frac{\partial (\rho u)}{\partial t} + div(\rho u U)$
y-momentum	v	$\rho \frac{Dv}{Dt}$	$\frac{\partial (\rho v)}{\partial t} + div(\rho v U)$
z-momentum	w	$\rho \frac{Dw}{Dt}$	$\frac{\partial (\rho w)}{\partial t} + div(\rho w U)$
Energy	E	$\rho \frac{DE}{Dt}$	$\frac{\partial (\rho E)}{\partial t} + div(\rho E U)$