where u is the velocity, P is the pressure, ρf is the fluid density, Tf is the fluid temperature, τ is the viscous stress tensor, f is a force vector, q˙f is a volumetric heat source in the fluid, Cp,f is the fluid isobaric specific heat capacity, and kf is the fluid thermal conductivity.
Passive Scalars
A passive scalar is any quantity s which satisfies an equation of the form
a(∂t∂s+ui∂xi∂s)=∂xi∂(b∂xi∂s)+q˙
where a and b are coefficients. For instance, temperature is a passive scalar if a=ρCp and b=kf. Other quantities which obey passive scalar equations include many aspects of mass transport.