Dynamics of liquids with high viscosity contrast in unevenly rotating Hele-Shaw cell

The paper is devoted to the experimental study of the dynamics of an interface between two liquids with high viscosity contrast filling a vertical, circular narrow gap rotating about a horizontal axis at a speed modulated by librations. The equilibrium shape and stability of the interphase boundary are considered. In the absence of librations, under the action of the centrifugal force the boundary has an axisymmetric shape. At librations, at certain frequency ratios, the interface loses its axisymmetric position: in the cavity reference frame, it is displaced in the radial direction. It is theoretically shown that the discovered phenomenon is explained by the average action of gravity; the results of experiments and theory are consistent. The experiments reveal that with an increase in the modulation amplitude the circular interface loses stability in a threshold manner: an azimuthally periodic relief, quasi-stationary in the cavity frame, emerges. This is associated with the Kelvin-Helmholtz instability due to tangential oscillations of the less viscous liquid near the interface. A dimensionless parameter that determines the stability of the interface is obtained in the limit of high dimensionless libration frequency. The stability threshold increases with the decrease in the dimensionless frequency. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.