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I would like to know more about the heat distribution over time in a flowing liquid. To this end, I consider the Navier-Stokes equation (where the coefficients may be temperature dependent) and the heat equation $$ \rho c_p \left(\frac{dT}{dt} + u \cdot \nabla T\right) = \sigma + S(u) $$ where $T$ is the temperature and $u$ the flow velocity. The term $\sigma$ may be some artificial heat source.

The term $S(u)$ is now of interest to me: It corresponds to the heat generated by the friction of particles (I think), and I believe it must be proportional to the dynamic viscosity of the medium. I don't recall the specifics though. Can anyone help out on what $S(u)$ should be?

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Viscous Heating? – Alan Rominger Apr 5 '13 at 21:36
up vote 2 down vote accepted

See for example Eq. 13.74 of here:

Basically the velocity shear tensor squared and multiplied by the viscosity.

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It is typically called the viscous work term. – tpg2114 Apr 6 '13 at 1:11

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