Influence of centrifugal force on convective flow in a spherical gap under a central force field.

Abstract

The study of large-scale convective flows within a spherical gap has been the focus of numerous theoretical and numerical investigations because of its relevance to geophysical applications. This is particularly true in scenarios where the inner surface is warmer than the outer surface, and the fluid is influenced by a radial force field. Various researchers have extensively analyzed the flow patterns that emerge in this simplified model, examining both nonrotating and rotating cases. In this study we present the results of numerical investigations of thermal convection within a rotating spherical gap. The inner surface maintains a higher temperature than the outer surface, T_{in}>T_{out}. The dielectrophoretic effect induces a radial force field. The convective flow can be controlled by two parameters, ΔT and V_{rms}, where ΔT=T_{in}-T_{out}, and V_{rms} is the mean square voltage between surfaces. In the nonrotating case, the Rayleigh number Ra_{E}∼V_{rms}^{2}ΔT, serves as the sole control parameter. The situation becomes much more complex if the system rotates. The centrifugal force can be interpreted as a buoyancy term. The key question in this work concerns whether ΔT is constant and V_{rms} varies or vice versa. In the first case, the centrifugal force can be expressed as f_{c}∼Ta, where Ta=(2Ωd^{2}/ν)^{2} is the Taylor number, d=R_{out}-R_{in} is the width of the gap, Ω is the rotation rate, and ν is the kinematic viscosity. In the second case, the centrifugal force follows the relationship f_{c}∼Ra_{E}Ta. A numerical investigation is performed to analyze both cases in detail. After examining the behavior of the steady two-dimensional basic flow, a linear instability analysis is carried out to calculate the critical Rayleigh number and critical frequencies of the most dominant perturbations as functions of the Taylor number, denoted Ra_{Ec}(Ta) and ω_{c}(Ta), respectively. According to instability analysis, the critical Rayleigh number increases with the Taylor number. Moreover, the basic flow becomes unstable concerning nonaxisymmetric perturbations. In all the cases considered, the instability sets in as a Hopf bifurcation. Furthermore, according to the numerical analysis of the three-dimensional flow, this bifurcation is supercritical. Heat transfer is expressed in terms of the Nusselt number, which increases significantly when the Rayleigh number surpasses a critical value.