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When solving Navier Stokes equations for viscous fluid over rigid surface, the viscous term in the momentum equation accounts for the momentum transfer between the fluid and surface in the near wall region, i.e. part of the fluid momentum is extracted by the act of viscosity. This is manifested as friction, wall shear stress, drag or flow resistance (you name it).

The dilemma that faces me know is the energy balance in this process. Since the kinetic energy and momentum are related by:

$$K=\frac{1}{2}\frac{P^2}{m} $$

Then momentum transfer must implies energy transfer. Since the surface will remain stationary, the kinetic energy should be transformed into other kind of enrgy. I expect that it would transformed into heat.

But since for incompressible flows we usually don't solve the energy equation I'm wondering how can we account for this energy transfer and transformation? and does neglecting that may affect the solution of momentum equation?

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    $\begingroup$ Check out the Brinkman number, it will give an indication if viscous dissipation is negligible or not for a viscous flow. If it is then temperature increases due to viscous dissipation are also negligible. $\endgroup$
    – nluigi
    Dec 15, 2017 at 10:56

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The Navier Stokes system model is incomplete with regards to including other paths in which the energy can flow. You need to include thermodynamics equations as part of your model. Yes, the energy losses due to friction causes heat, not only on the walls but within the fluid itself. And that changes temperature at both locations.

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    $\begingroup$ Which, for incompressible (as in constant-density) flows, the additional heat is negligible with respect to the background flow. If that is not the case, then the incompressible equations cannot be used and either low-Mach (density function of temperature) or fully-compressible equations are needed. $\endgroup$
    – tpg2114
    Dec 14, 2017 at 19:23
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    $\begingroup$ @tpg2114 This is not correct. You appear to be biased by your experience only with low viscosity compressible flows. In high viscosity incompressible fluids like corn syrup, for example, viscous heating can be very significant, and can result in increases on the order of 10's of degrees C at the walls of the duct. Such viscous heating is routinely observed in the polymer processing industries. $\endgroup$
    – Chet Miller
    Jan 2, 2018 at 15:53
  • $\begingroup$ @ChesterMiller Sure -- but, if the change in temperature doesn't affect density or transport properties, then it doesn't affect the momentum equation and so neglecting it isn't a problem. If it does affect density, then low-Mach equations are needed at the least. I'm not familiar with flows of things like corn syrup, but looking at the density variation with temperature, it is negligibly small. So the temperature changes won't affect the flow either. $\endgroup$
    – tpg2114
    Jan 2, 2018 at 18:32
  • $\begingroup$ Which is not to say it isn't important for other things, like heat transfer to the walls or something else. But that wasn't what I thought OP was asking about when I left my comment. $\endgroup$
    – tpg2114
    Jan 2, 2018 at 18:33
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    $\begingroup$ @tpg2114 My point regarding incompleteness. Energy doesn't restrict itself to one set of equations like humans often do. It always finds a way to balance the books. It knows all equations. $\endgroup$
    – docscience
    Jan 2, 2018 at 18:54
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The viscous dissipation of mechanical energy to internal energy is occurring not only at the walls of the duct, but throughout the duct. The local rate at which this is occurring is proportional to the viscosity times the second invariant of the rate of deformation tensor (typically, the square of the shear rate). This is usually referred to as "viscous heat generation." This viscous heat generation certainly does occur in viscous fluids, and, in high viscosity fluids, can cause substantial temperature increase (particularly locally, near the duct wall, where the shear rate is highest). For a complete discussion of the details on this, see Transport Phenomena by Bird, Stewart, and Lightfoot.

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