Viscosity and energy balance 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?
 A: 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.
A: 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.
