Law of Conservation of Momentum and Energy

by Dasa . 29 Aug 2023

The momentum and energy conservation law states that a fluid property (ϕ\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 (ϕt\frac{\partial \phi}{\partial t})

change is property concerning the location and movement (ϕxdxdt+ϕydydt+ϕzdzdt\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 ϕ\phi is the property per unit mass, then the total derivative of ϕ\phi (the rate of change of property per unit mass) can be expressed as

DϕDt=ϕt+ϕxdxdt+ϕydydt+ϕzdzdt\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 (dxdt\frac{dx}{dt}) is velocity u, y direction (dydt\frac{dy}{dt}) is v, and z direction (dzdt\frac{dz}{dt}) is w.

DϕDt=ϕt+uϕx+vϕy+wϕz\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)

DϕDt=ϕt+U.gradϕ\frac{D \phi}{Dt} = \frac{\partial \phi}{\partial t} + U . grad \phi

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

ρDϕDt=ρ(ϕt+U.gradϕ)\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).

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

Using dot product rule

=ρϕt+ϕρt+ϕdiv(ρU)+ρU.gradϕ= \rho \frac{\partial \phi}{\partial t} + \phi \frac{\partial \rho}{\partial t} + \phi div(\rho U) + \rho U . grad \phi

=ρ(ϕt+U.gradϕ)+ϕ(ρt+div(ρU))= \rho (\frac{\partial \phi}{\partial t} + U . grad \phi) + \phi (\frac{\partial \rho}{\partial t} + div(\rho U))

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

=ρ(ϕt+U.gradϕ)= \rho (\frac{\partial \phi}{\partial t} + U . grad \phi)

=ρDϕDt= \rho \frac{D \phi}{Dt}


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

{The Rate Of Increase Of ϕ for afluid particle}={The Rate Of Increase Of ϕ of fluid element}+{Net rate of flow Of ϕ out of fluid element}\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 ρDuDt\rho \frac{Du}{Dt} (ρu)t+div(ρuU)\frac{\partial (\rho u)}{\partial t} + div(\rho u U)
y-momentum v ρDvDt\rho \frac{Dv}{Dt} (ρv)t+div(ρvU)\frac{\partial (\rho v)}{\partial t} + div(\rho v U)
z-momentum w ρDwDt\rho \frac{Dw}{Dt} (ρw)t+div(ρwU)\frac{\partial (\rho w)}{\partial t} + div(\rho w U)
Energy E ρDEDt\rho \frac{DE}{Dt} (ρE)t+div(ρEU)\frac{\partial (\rho E)}{\partial t} + div(\rho E U)

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