If the viscous dissipation term is taken into accountDisplay Formula

$\u2223Q\u0307C\u2223=\u2223Q\u0307H\u2223+\u222bV(volumetricviscousdissipationrate)dV$

(3)

violating the energy conservation principle by the reasons detailed above. Thus, if the viscous dissipation is taken into account an

*additional* term needs to be taken into account. The complete thermal energy conservation equation (for a clear fluid) can be obtained from Ref.

3; this additional term is the work of pressure forces, and as

$\u2223Q\u0307H\u2223=\u2223Q\u0307C\u2223$ it is

Display Formula$\u222bV(volumetricviscousdissipationrate)dV+\u222bV(volumetricrateofworkofpressureforces)dV=0$

(4)

Locally, the volumetric viscous dissipation rate can be different from the volumetric rate of work of pressure forces, and Eq.

4 applies to the overall enclosure. Viscous dissipation is always positive, and the work of pressure forces can be positive or negative depending if the fluid is contracting or expanding, respectively (

3). Viscous dissipation results from the fluid motion, in natural convection problems fluid motion results from the expansion∕contraction experienced by the fluid, and both the viscous dissipation and the work of pressure forces need to be taken into account in order to have the unique consistent energy conservation formulation.