Energy Equation in Microgeometries

The relaxation of PoiseuiUe flow to obtain an expression for the rate of viscous dissipation of energy, — is presented in the previous section. In this section, let us analyze the steady-state PoiseuiUe flow. Here, the pressure P dSi) to the left-hand side on dS is higher than the pressure P dS2) to the right-hand side on dS2 given as 

For such a flow, the velocity field is constant and consequently the kinetic energy of the fluid is constant. The rate j Wyjsc of heat generation by viscous friction is balanced by the mechanical power Wmech put into the fluid by the pressure force at steady-state condition, which can be expressed as 

For incompressible flow and constant density, the N-S equation is given by 

Note that the left-hand side of this equation disappears due to steady-state translation invariance behavior of the PoiseuiUe flow. The first term in the right-hand side of equation (2.100) contributes to the mechanical energy and the second term contributes to the viscous dissipation. 

We can determine the rate of change in Wmech by multiplying the pressure term in equation (2.100) by V and integrating over volume. 

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