Pattern formation on the surface of the granular medium in a horizontal rotating cylinder filled with fluid

The stability of the interface between a low-viscosity fluid and a granular medium in a horizontal rotating cylinder is experimentally studied. We consider a moderate rotation when heavy granules are able to form an annular layer near a cylindrical wall under the action of centrifugal force. The gravitational force acting on the particles of the granular material fluidizes the granular bed and induces the rotation of the particles at the interface with a velocity different from that of the rotating fluid. The effect of gravity can be characterized by the ratio of the gravitational force to the inertial centrifugal force $\mathrm{\Gamma }\equiv \frac{g}{{\left(2\pi f\right)}^{2}a}$, and it increases with the decrease of the cylinder rotation rate. The observations revealed previously unobserved regular ripples at the interface in a narrow range of rotation rates. At a sufficiently low rotation rate, a large number of granules become suspended, and regular ripples disappear under the action of gravitational force. In the present study, the geometric characteristics and spatiotemporal dynamics of regular ripples are studied in a wide range of granular bed thickness, and a possible mechanism of the ripple formation is discussed.

Figure 1

Photo of the experimental setup

Figure 2

Photograph of the granular medium in the cylinder rotating at different rates: $f$ = 3.50 (a) and 1.15 (b) rps. The thickness of the granular bed $h$ = 2 mm. The cylinder rotates clockwise.

Figure 3

Ripple formation obtained in the experiments with various amount of granular material: $h=3$ mm and $f=1.40$ rps (a), $h=5$ mm and $f=1.55$ rps (b), $h=15$ mm and $f=1.80$ rps (c), $h=33$ mm and $f=2.05$ rps (d). Reflection and absorption of light in the granular bed affects the visible color of the granules. The cylinder rotates clockwise.

Figure 4

Regular ripples containing large amount of suspended particles: $f$ = 1.47 rps (a) and 1.50 rps (b); $h$ = 17 mm. The cylinder rotates clockwise.

Figure 5

Dependence of the ripple length $\lambda$ on the azimuthal coordinate $\varphi$: $h$ = 2 mm, $f$ = 1.2 rps.

Figure 6

Dependence of azimuthal coordinate $\mathrm{\Phi }$ of a ripple relative to the rotating cylinder on dimensionless time $tf$ for various rotation rates ($h$ =33 mm).

Figure 7

Dependence of azimuthal length $\lambda$ of granular ripples on rotation rate $f$ of the cylinder for various thicknesses of granular bed.

Figure 8

Photo showing the thicknesses used to measure the porosity of the granular bed in the fluidized state: $h$ = 5 mm, $f$ = 4 rps (left) and 1.55 rps (right).

Figure 9

Dependence of dimensionless length $\mathrm{\Lambda }$ of granular ripples on dimensionless parameter $\frac{\mathrm{\Delta }{F}^2}{(\rho -\frac{1}{\rho })}$. The symbols correspond to those in Fig. 7. The solid line indicates the predictions of the theory of the oscillatory Kelvin-Helmholtz instability.

Figure 10

Dependence of the ratio of the wavelength to the thickness of the fluidized layer of granular medium on the dimensionless acceleration. The empty symbols correspond to those in Fig. 7. The filled circles indicate the data obtained from Fig. 4.