The thermal boundary layer due to viscous dissipation in impulsively started Poiseuille flow

Abstract

Viscous dissipation occurs in the boundary layers on the walls of a channel in which a flow is accelerated from rest by the sudden imposition of a pressure gradient. We analyse the thermal boundary layer due to this dissipative heating, obtaining numerical solutions and also asymptotic solutions for the cases of both large and small Prandtl number, with both isothermal and adiabatic wall conditions. With large $\mathrm{Pr}$ the temperature rise is controlled by the viscous layer, so is independent of $\mathrm{Pr}$ and of the wall condition. With small $\mathrm{Pr}$ heat is conducted away from the viscous layer more rapidly, so the temperature rise is reduced as $\mathrm{Pr}$ decreases.