# Heat equation from Navier Stokes eqns?

• A
Can you derive the heat conduction equation from the navier stokes equations (particularly the energy eqn)?

Gold Member
The energy equation contains a heat conducting term already (the one containing ##\kappa \nabla T##), so then if you zeroed out all the other terms, then yes, you could get the heat equation. Since conduction is actually one of several parts of the energy equation, though, I don't know why you would want to do this. There are many other factors in energy transport.

The energy equation contains a heat conducting term already (the one containing ##\kappa \nabla T##), so then if you zeroed out all the other terms, then yes, you could get the heat equation. Since conduction is actually one of several parts of the energy equation, though, I don't know why you would want to do this. There are many other factors in energy transport.
Thanks. Question.

Are equations like potential flow, Euler's, heat conduction equation formulated before the Navier Stokes equations? I see that Euler existed before Navier&Stokes.
If so, then were the Navier Stokes equations formulated on the basis of these simplified equations?

Andy Resnick
Thanks. Question.

Are equations like potential flow, Euler's, heat conduction equation formulated before the Navier Stokes equations? I see that Euler existed before Navier&Stokes.
If so, then were the Navier Stokes equations formulated on the basis of these simplified equations?

A minor comment- the Navier Stokes equation(s) is/are concerned with momentum transport. Just as mass transport is handled by a different equation-the continuity equation- energy transport is handled by it's own equation. All three equations are, to some degree, independent:

http://www.ldeo.columbia.edu/~mspieg/mmm/Conserveq.pdf

A minor comment- the Navier Stokes equation(s) is/are concerned with momentum transport. Just as mass transport is handled by a different equation-the continuity equation- energy transport is handled by it's own equation. All three equations are, to some degree, independent:

http://www.ldeo.columbia.edu/~mspieg/mmm/Conserveq.pdf
Wait I thought the Navier Stokes equations refers to the 3 conversation equations for mass, momentum, and energy? And hence the pluralness of "equationS?" If its just the momentum equation then why is it not called the Navier Stokes Equation?

Gold Member
Momentum conservation results in one equation per spatial dimension, hence there are three momentum equations.

Really it's just a semantics debate, though.

Momentum conservation results in one equation per spatial dimension, hence there are three momentum equations.

Really it's just a semantics debate, though.
Oh yeah I forget it is a vector equation. So the NS equations technically include the continuity and energy eqn?

Chestermiller
Mentor
Oh yeah I forget it is a vector equation. So the NS equations technically include the continuity and energy eqn?
No. It only addresses momentum. The continuity equation can be combined with the NS equation to obtain a different form of the equation, but its primary focus is still momentum. It is also possible to obtain the "mechanical energy balance equation" by dotting the NS equation with the velocity vector. This can be subtracted from the "overall energy balance equation " to yield the so-called "thermal energy balance equation".