The effects of a strongly temperature-dependent viscosity on slow flow past a hot sphere

Abstract

This work determines analytically the drag on, and heat flux out of, a hot sphere that translates steadily in a fluid of strongly temperature-dependent viscosity. There is no dissipative heating. The essentials are illustrated by an exact solution for the flow induced by slowly squeezing two parallel planes together. The lower plane is hot and stationary; the upper is cold and advances in a direction normal to itself at uniform speed U . The gap is completely filled by a fluid of strongly temperature-dependent viscosity. We find the temperature and velocity profiles, and determine the Nusselt number N and Péklet number P as functions of the normal force D on the lower plane. The large viscosity variation tries to concentrate the flow into a relatively thin softened layer in which the viscosity is of order its value μ 0 at the hot plane. In the limit of infinite viscosity ratio (fixed P ), it succeeds (lubrication limit): if P [Lt ] 1, the width of the softened layer is determined by conduction and D ∝ μ 0 U ; but D ∝ μ 0 U 4 when forced convection is important. If P → ∞ (fixed viscosity ratio), the softened layer is so thin that it chokes, and all the deformation occurs outside the thermal layer in the fluid of uniform viscosity μ ∞ (Stokes limit); then D ∝ μ ∞ U . These mechanisms appear as three distinct legs in our plot of log P against log D . There are similar transitions in the plot of log N against log D . The solution gives an estimate of the drag on a sphere. We test this estimate against an analytical solution for the sphere in the lubrication limit. Then we extend the solution to cover power-law fluids, and apply it to a model (by Marsh) of magma transport beneath island-arc volcanoes. The results suggest that the magma covers the first 50 km of its ascent by an isoviscous mechanism, with the lubrication mechanism operating in the remaining 50 km. To open a fresh pathway from the source to the surface takes about 10 6 years and uses about 10 27 erg.