Transient Interface Shape of a Two-Layer Liquid in an Abruptly Rotating Cylinder

Abstract

A description is made of the transient shape of interface of a two-layer liquid in an abruptly rotating circular cylinder. The density of the lower layer is higher than that of the upper layer, but the viscosities may assume arbitrary values. The overall Ekman number is much smaller than unity, and the cylinder aspect ratio is 0(1). The classical Wedemeyer model, which deals with the spin-up from rest of a homogeneous fluid, is extended to tackle the two-layer liquid system. If the upper-layer fluid is of higher viscosity, the interface, at small and intermediate times, rises (sinks) in the center (periphery). After reaching a maximum height at the center, the interface tends to the parabolic shape characteristic of the final-state rigid-body rotation. If the lower-layer fluid is of higher viscosity, the interface, at small and intermediate times, sinks (rises) in the center (periphery). The deformation at the center reaches a minimum height, after which the interface approaches the final-state parabola. The gross adjustment process is accomplished over the spin-up time scale, En−1/2Ω−1, where En and Ω denote the lower value of the Ekman numbers of the two layers and the angular velocity of the cylindrical container, respectively. These depictions are consistent with the physical explanations offered earlier. A turntable experiment is performed to portray the transient interface shape. The model predictions of the interface form are in satisfactory agreement with the laboratory measurements.