Fluid Mechanics/Conservation Equations

Conservation Equations of Mass, Momentum and Energy

Equation of Continuity

• A differential mass balance relating density change to velocity.

$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot \left(\rho 𝐕\right)=0$

• For incompressible fluids the equation of continuity reduces to:

$\rho (\mathrm{\nabla }\cdot 𝐕)=0$

since

$\frac{\partial \rho }{\partial t}=0$

for all incompressible fluids

Euler's Equation

• applies conservation of momentum to inviscid, incompressible flow.

$\rho 𝐠-\mathrm{\nabla }p=\rho \frac{d𝐕}{dt}$

Stokes' Equation

• applies conservation of momentum in creeping flow limit (low Reynold's Number)

$\mathrm{\nabla }p=\mu {\mathrm{\nabla }}^2𝐕$