The problem of the dynamics and stability of interphase boundaries is relevant in scientific and applied aspects [1,2]. The stability of rimming flows in rotating cavities is actual for fundamental science and is an important technological problem, the study of which has been the subject of many works [3–7]. The instability of the interface can be excited by gravity in various rotating hydrodynamic systems, two low-viscosity immiscible liquids of different densities [8], and the surface of a granular medium in a liquid [9,10]. Of great interest in this regard is the effect of rotational vibration on the shape of the boundary discovered in [11]. It was found in [11] that when modulating the rotation speed of a cylindrical cavity containing two liquids, when the denser liquid covering the lateral boundary has a high viscosity, the interphase axisymmetric boundary loses stability. With an increase in the amplitude of the rotation speed modulation, a quasi-stationary relief in the form of a “frozen” wave appears on the interface in a threshold manner. The relief has the form of an azimuthally periodic system of two-dimensional crests and troughs extended along the axis of rotation, which are rotating together with the container and performing slight azimuthal oscillations relative to the cavity with the libration frequency. It was shown that the appearance of the frozen two-dimensional wave on the surface of a viscous liquid film is associated with the development of Kelvin–Helmholtz oscillatory instability. The physical model proposed in [11], which describes the development of the frozen wave on the surface of the layer (film) of the viscous liquid on the inner wall of the rotating horizontal cylinder, is based on the results of systematic theoretical [12–16] (and later [17]) and experimental [18–20] studies of the Kelvin–Helmholtz oscillatory instability without rotation. A feature of the cited works is that the instability of the plane interphase boundary is studied during tangential oscillations of liquids relative to the interface caused by translational vibrations. The experimental results [11] are in agreement with the model constructed, which reflects the behavior of the interphase boundary only to certain approximations. The latter include: relatively slow rotation, when the Coriolis force does not play a role; high viscosity of the liquid covering the cavity wall, when the entire layer of viscous liquid is completely entrained by the wall during its azimuthal oscillations; high-frequency modulation of the rotation speed, at which the low-viscosity liquid filling the remaining volume of the cavity rotates at a constant speed due to the absence of viscous interaction with the denser liquid. The latter means that the thickness of the Stokes layer in the low-viscosity liquid near the interface is negligibly thin due to the high contrast of liquid viscosities. Finally, the radius of the interface in the rotating cylinder is assumed large compared to the capillary length, while the length of the frozen wave is small compared to the azimuthal length of the interface. The above makes it relevant to study the influence, on the effect of excitation of the quasi-stationary relief at the boundary of the viscous-liquid layer during the non-monotonous rotation of the cavity, of such factors as: the thickness of the viscous liquid layer, the contrast of viscosities and the rotation speed of the cavity.

The relevance of the study of interface dynamics is determined by the importance of the fundamental and technological problem of controlling mass transfer across the interface [21,22]. At the same time, vibrational methods for controlling the interface dynamics are of particular interest due to the convenience and simplicity of the vibration effect on hydrodynamic systems [23,24]. Note that the effect of rotational vibrations (rotation rate modulation) on the shape of the interphase boundary, discovered experimentally in [11], is original and qualitatively new; the excitation of the oscillating motion of liquids near the interphase boundary in this case is determined by the different interactions of liquids with the boundaries of the cavity and with each other, due to high contrast of viscosities. At the same time, the experimental results and the physical model [11] refer to the case of relatively slow rotation, when rotation (Coriolis force) does not play an important role.

The present study is an advancement over [11]; the objective of this work is the experimental investigation of the dynamics of the boundary of a viscous-liquid layer with variations in the thickness of the viscous liquid layer and the rotation rate. The studies are performed with liquids of lower contrast of viscosities in a wide range of amplitudes and frequencies of rotation rate modulation with varying thicknesses of the viscous liquid layer. The novel and important result: it is found that the rotation has a stabilizing effect, which can be characterized by a dimensionless rotation rate, and the threshold of axisymmetric interface stability is determined by the product of vibrational parameter and the dimensionless rotation rate to some power.

The appearance of a frozen wave at the interface is associated with the development of oscillatory Kelvin–Helmholtz (K–H) instability, excited by an oscillating tangential velocity discontinuity at the boundary. This phenomenon is known and studied experimentally and theoretically in the case of tangential oscillations of liquids relative to a flat interface. Such oscillations of liquids are excited, for example, under the action of horizontal translational harmonic oscillations of a cavity filled with liquids. Liquids of different densities oscillate along the interface in an oscillating inertial field. A theoretical model of this phenomenon was developed in [12] for non-viscous liquids. Subsequently, the stability of the interface between viscous liquids was examined theoretically and experimentally. In [14,16,19], it is shown that an increase in viscosity contrast leads to a significant decrease in the stability threshold.

In [11], the excitation of oscillatory instability was discovered in a non-uniformly rotating cylinder with two liquids with a high viscosity contrast. The rotation was carried out at a fairly high speed, and the system was in a centrifuged state with an axisymmetric interface. In this case, the more viscous liquid had a higher density and was located near the outer boundary of the cavity. The nature of the occurrence of an oscillating tangential velocity discontinuity at the interface between liquids was that the viscous liquid near the boundary of the cavity, with uneven rotation, was completely entrained by the cavity wall and performed rotational oscillations along with it, that is, the thickness of the liquid layer was less than the thickness of the Stokes layer. At the same time, the low-viscosity liquid in the middle part of the cavity rotated uniformly at a constant speed, since the Stokes boundary layers in it were of negligible thickness. The theoretical model built in [11], based on the results of studies in the case of translational oscillations [13,14,19], included the approximation Ωrot << Ωlib ; ω2≡Ωlibh02/ν2 << 1 , ω1≡Ωlibh02/ν1 >> 1 , λcap << R . Thus, in the description of the phenomenon, the curvature of the surface is neglected (it is assumed that the wavelength is much less than the circumference), and the rotation speed is assumed to be small, that is, the effect of the Coriolis force is not taken into account. Note that in the described phenomenon, a high viscosity contrast (high viscosity of one of the liquids) plays a fundamental role, since it is the viscous liquid that performs tangential oscillations along the interface as a result of viscous drag by the oscillating wall.

The vibrational parameter B, which determines the threshold for the development of Kelvin–Helmholtz oscillatory instability [12,13], is transformed under the conditions of the present problem. It characterizes the amplitude of the oscillating tangential velocity discontinuity at the interface between two fluids when one fluid oscillates relative to the other in an averaged centrifugal force field and, in accordance with [11], is determined by the expression

(1) B=πρRinε22(1−ρ2)(1+0.5ε2)λcap where, ρ≡ρ1ρ2 , λcap=2πσ(ρ2−ρ1)Ωrot2Rin(1+0.5ε2) .

In this case, the role of the static force field in a rotating system is played by the centrifugal force averaged over the period, and the effect of the gravity field on the fluid in the cavity, which rotates quite rapidly relative to the horizontal axis, is neglected. Note that the results of experiments [11], carried out on liquids with a high viscosity contrast and on a relatively thin layer of viscous liquid, at high libration frequency were in satisfactory agreement with theoretical and experimental results for translational vibrations of the cavity in the absence of rotation [14,19].

The following questions remains: a) what is the role of the relative thickness of the viscous fluid layer, b) what role does the contrast of liquids viscosities play, c) how does the Coriolis force affect oscillatory K–H instability in rotating cylinder. The dimensionless frequency ω2≡Ωlibh02/ν2 is responsible for the first point, which characterizes the ratio of the thickness of the viscous fluid layer to the thickness of the Stokes layer, and therefore the degree of entrainment of the viscous fluid layer by the oscillating wall. It is proposed to characterize the influence of the Coriolis force by the dimensionless rotation speed ωrot≡Ωroth02/ν1 , this parameter characterizes the ratio of the Coriolis force to viscous forces (in this form–in a layer of low-viscosity fluid).

By analogy with [11], we will plot the dependence of the threshold value of the vibration parameter B on the relative frequency of librations Ωlib/Ωrot (Fig. 8). The figure shows that the stability threshold (critical value of parameter B) at a given rotation speed decreases with the relative frequency of librations Ωlib/Ωrot in all cases. At the same time, in a separate series of experiments performed on a layer of viscous liquid of a fixed thickness, the threshold curves are stratified by rotation speed; with increasing f_rot, the threshold curves rise, maintaining their shape.

Comparison with the results of [11], obtained on a pair of liquids “glycerol–PMS-0.65”, indicates that reducing the viscosity contrast increases stability (Fig. 8). An increase in the thickness of viscous liquid layer is also accompanied by an increase in stability; with an increase in the layer thickness h₀ = 3.3 mm (Fig. 8a) to h₀ = 4.6 mm (Fig. 8b), the family of curves corresponding to different rotation speeds shifts upward. With relatively slow rotation Ωlib/Ωrot > 2 , the stability threshold ceases to change with the relative frequency of librations. Note that in Fig. 8, the points in case of libration and rotation frequencies coincidence, at Ωlib/Ωrot=1 , fall out of the monotonic dependences and locate significantly lower.

The dimensionless wavelength λ/λ_cap, as can be seen from Fig. 9, at close values of h₀ depends on the contrast of liquid viscosities: λ/λ_cap goes up with the viscosity contrast. The dependences obtained for different pairs of liquids have a similar shape. It should be noted that a significant scatter of experimental points is determined by the condition of an integer number of waves falling on the perimeter of the interface.

From Fig. 9, it follows that for each pair of liquids, the points obtained at similar thicknesses fit within one dependence within the error. In the case of K = 12 the curve lies higher than with K = 29. It is worth noting that at low values of rotation speed, the dimensionless wavelength ceases to change, and the value of λ/λ_cap approaches a value close to unity. This value agrees satisfactorily with the theoretical one in the case of translational oscillations of liquids (thick layers) near a flat boundary [12,14].

Let us dwell on the stratification of the threshold curves (Fig. 8) by rotation speed (dependence of the stability threshold on the rotation speed)–with increasing rotation speed, the threshold curves shift to the region of higher values of the vibrational parameter B. As noted above, the physical model of the development of oscillatory Kelvin–Helmholtz instability does not take into account the action of the Coriolis force. Let us use the dimensionless rotation speed ωrot=Ωroth02/ν1 , which characterizes the ratio of the thickness of the layer of a viscous liquid to the thickness of the Ekman layer in a low-viscosity liquid (more precisely, the square of this ratio) in a rotating cavity. From the analysis of the results obtained, it follows that the stabilizing effect of rotation on the stability of the interphase boundary in the experiment has a power-law dependence, and the stability threshold can be described by the complex B/ωrot0.35 (Fig. 10).

All points obtained at different rotation speeds and stratified in Fig. 8 are in satisfactory agreement on the plane of these parameters (Fig. 10) (symbols in Figs. 8 and 10 coincide). This indicates that rotation has a stabilizing effect on the stability of the interphase boundary, and the stabilizing effect could be characterized by dimensionless rotation speed ωrot .

The threshold points are located much lower at Ωlib/Ωrot=1 . This is explained by the fact that at this the frozen waves excite only on a part of the interphase boundary, where the layer thickness is much smaller than h₀, which means that the dimensionless rotation speed, being calculated through the local thickness of the viscous liquid layer, is also smaller. This could explain the decrease in the threshold value of B/ωrot0.35 at Ωlib/Ωrot=1 .

The behavior of a viscous liquid film on the wall of rapidly rotating cylinder subjected to angular vibrations is experimentally studied. The cavity is filled with an immiscible low-viscosity liquid of lower density. It is found that with an increase in the amplitude of rotational vibrations, the axisymmetric interphase boundary loses stability–а periodic 2D frozen wave appears on the liquid-film interface in a threshold manner. The frozen wave excitation is associated with the oscillatory Kelvin–Helmholtz instability, and the stability threshold is determined by vibrational parameter. The experiments carried out allow us to draw two important conclusions. The first concerns the role of the viscosity contrast coefficient K, which characterizes the ratio of the thicknesses of Stokes layers in liquids on opposite sides of the interface. It is shown that as the viscosity contrast decreases, the threshold for excitation of instability increases. This conclusion is qualitatively consistent with the results of theoretical and experimental studies of the oscillatory instability of the K–H in plane geometry [14] in a different formulation: the layers of the liquid are considered thick, there is no rotation, and tangential oscillations of liquids are caused by uniform oscillating external force fields. The second important result concerns the role of cavity rotation. It was discovered that rotation plays a stabilizing role and leads to an increase in the threshold value of the vibrational parameter B∗ (at a given relative frequency of librations) according to a power law B∗∼ωrotn . Under the conditions of the experiments performed (at sufficiently high values of the viscosity contrast K = 12 and K = 29) n≈0.35 . The use of a dimensionless complex B/ωrot0.35 makes it possible to coordinate experimental points obtained at different rotation speeds at a certain value of the contrast in liquid viscosities. Note that the found dependence well approximates the results of the present experiments. At the same time, one can expect a transformation of the dependence (in particular, the exponent n) with a change in K.

The results obtained shed light on the features of a new phenomenon discovered in [11]–instability of the interface between liquids with a high contrast of viscosities in a cylindrical cavity undergoing modulated rotation. The discovered stabilizing effect of dimensionless rotation rate on the oscillatory instability of the interface is qualitatively new and promising in terms of applied development of vibrational methods for controlling the interfaces; it requires further systematic experimental and theoretical research. To develop this direction, the following is planned: a systematic experimental study of the dynamics of the interface depending on the relative volume of liquids and their properties with the viscosity contrast and dimensionless rotation speed variation; theoretical modeling of the found phenomenon.