Variable-viscosity flows in channels with high heat generation

Abstract

This paper presents a similarity solution for plane channel flow of a very viscous fluid, whose viscosity is exponentially dependent upon temperature, when heat generation is very large. A dimensionless formulation of the problem involves two length scales (the depth h and length l, respectively, of the channel), one velocity scale (the mean velocity V of the fluid along the channel), the thermal conductivity k, thermal diffusivity k and viscosity V of the fluid, and the temperature coefficient b of the viscosity. From these, two important dimensionless groups arise, the Graetz number (Gz = Vh²/kl) and the Nahme–Griffith number (G = μ V²b/k). In the case of steady flow with G⁻¹ [Lt ] Gz⁻¹ [Lt ] 1 a thin thermal boundary layer of thickness proportional to Gz^−½ arises at each wall with an even thinner shear layer, detached from the wall and embedded in the thermal boundary layer, of thickness proportional to Gz^−½(ln G)⁻¹, coinciding with the region of maximum temperature (ln G)/b. The similarity variable is (Pe^½y/x^½) where Pe is the Péclet number (Vh/k) and y and x are measured away from and along (either) boundary wall. The analogous unsteady uniform flow solution is also given.