The application of boundary layer theory to the convective flow of a fluid with a depth dependent viscosity

Abstract

The boundary layer theory for high Rayleigh number convection has been generalized to study the convective flow structure of a fluid with a viscosity that increases exponentially with depth. This method is used to study convection in a square cell with free slip isothermal boundaries and with the top temperature less than the bottom temperature. The flow is driven by sidewall shear stresses approximating the rising and descending thermal plumes. Viscosity variations up to 10 4 across the cell are considered. Values from 10 5 to 10 9 for a Rayleigh number based on the surface viscosity are used in this work. The flow and thermal structure is found to differ from constant viscosity flows. The temperature drop across the bottom boundary layer is larger than across the top, and thus the interior temperature of the cell is less than the mean of the bottom and top temperatures as in the constant viscosity flows. The flow structure is no longer symmetric since the rising and descending plumes do not have equal strengths. The global heat transfer is less efficient than in a fluid with constant viscosity equal to the top viscosity in the present models. The fluid in the lower section of the cell moves more slowly relative to the upper section of the cell for a given viscosity contrast.