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 depthhand lengthl, respectively, of the channel), one velocity scale (the mean velocityVof the fluid along the channel), the thermal conductivityk, thermal diffusivitykand viscosityVof the fluid, and the temperature coefficientbof the viscosity. From these, two important dimensionless groups arise, the Graetz number (Gz = Vh²/kl) and the Nahme–Griffith number (G= μV²b/k). In the case of steady flow withG⁻¹[Lt ]Gz⁻¹[Lt ] 1 a thin thermal boundary layer of thickness proportional toGz^−½arises at each wall with an even thinner shear layer, detached from the wall and embedded in the thermal boundary layer, of thickness proportional toGz^−½(lnG)⁻¹, coinciding with the region of maximum temperature (lnG)/b. The similarity variable is (Pe^½y/x^½) wherePeis the Péclet number (Vh/k) andyandxare measured away from and along (either) boundary wall. The analogous unsteady uniform flow solution is also given.