Law of Conservation of Momentum and Energy
. 29 Aug 2023
The momentum and energy conservation law states that a fluid property () 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 ()
change is property concerning the location and movement ()
Considering is the property per unit mass, then the total derivative of (the rate of change of property per unit mass) can be expressed as
The change in x direction concerning time () is velocity u, y direction () is v, and z direction () is w.
Using the Gradient rule and dot product rule (refer divergence rule)
When we multiply the rate of change of property per unit mass () with density (), we get the rate of change of property per unit volume.
The equation for the conservation of mass for a fluid with a orbitrary conserved property can be written as below (refer Conservation of mass).
Using dot product rule
The mass conservation law states that is equal to zero.
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.