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:
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?