The Effect of Viscous Heat Dissipation on Convective Heat Transfer in Small-Scale Slipping Gaseous Flows

Abstract

The presence of slip in small-scale gaseous flows leads to shear work and dissipation at the boundary. This effect has been neglected in recent studies investigating the effect of viscous dissipation on convective heat transfer in small scale channels. In this paper we illustrate the effect of shear work at solid boundaries in small-scale gaseous flows through the solution of the constant-wall-heat-flux problem in the slip-flow regime. We show that dissipation at the boundary scales with the Brinkman number similarly to viscous heat dissipation inside the channel, and increases with increasing Knudsen number. As a result, it is incorrect to neglect this effect when viscous heat generation needs to be considered. An analytical expression for the fully developed slip-flow Nusselt number under constant-wall-heat-flux conditions in the presence of viscous heat dissipation is presented. This expression is verified by direct Monte Carlo solutions of the Boltzmann equation. An expression for the skin friction coefficient under fully developed flow conditions for arbitrary Knudsen numbers is presented. Simple approximate expressions for the skin friction coefficient in the ranges 0 ≤ Kn ⪝ 0 4 . and 0.4 ⪝ Kn ⪝ 3 are also presented. These expressions are in agreement with direct Monte Carlo solutions of the Boltzmann equation.