Flow and heat transfer between co/counter-rotating cone-plate apparatus: full solutions

Abstract

This study revisits the cone-disk apparatus, considering the novel scenario where both the cone and the disk can rotate simultaneously, either in the same or opposite directions, about the axis of rotation. We demonstrate that an ideal peripheral velocity and temperature field develops for the incompressible Newtonian fluid within the gap region between the cone and the disk. We derive exact formulas for the peripheral velocity and temperature distribution as functions of latitudinal and radial coordinates. These formulas simplify to existing data when one device remains stationary. However, with simultaneous rotation, the momentum and thermal behaviors deviate from the traditional case. By analyzing the generated velocity field, we extract the progression of wall shears on both surfaces and the torque required to maintain steady rotation. Interestingly, these quantities exhibit a linear relationship with the rotation ratio parameter. Our viscous heating analysis reveals that the temperature within the gap grows proportionally to the square of the rotation ratio parameter. Consequently, the rate of heat transfer from both the cone and disk surfaces is formulated as the square of this parameter. The presented explicit expressions also allow for the straightforward identification of threshold cone/disk angles at which distinct phenomena emerge in the velocity and temperature fields. From a physical perspective, our findings indicate that flow reversal occurs at a critical gap angle when counter-rotation is present. Additionally, narrower cone-disk configurations experience higher temperatures, and enhanced heat transfer rates occur from the cone, further amplified by the rotation ratios.