Formation and evolution of bubbly screens in confined oscillating bubbly liquids

We consider the dynamics of dilute monodisperse bubbly liquid confined by two plane solid walls and subject to small-amplitude high-frequency oscillations normal to the walls. The initial state corresponds to the uniform distribution of bubbles and motionless liquid. The period of external driving is assumed much smaller than typical relaxation times for a single bubble but larger than the period of volume eigenoscillations. The time-averaged description accounting for the two-way coupling between the liquid and the bubbles is applied. We show that the model predicts accumulation of bubbles in thin sheets parallel to the walls. These singular structures, which are formally characterized by infinitely thin width and infinitely high concentration, are referred to as bubbly screens. The formation of a bubbly screen is described analytically in terms of a self-similar solution, which is in agreement with numerical simulations. We study the evolution of bubbly screens and detect a one-dimensional stationary state, which is shown to be unconditionally unstable.

Figure 1

(Color online) A sketch of the problem geometry. Bubbly liquid filling the space between solid parallel walls subject to normal oscillations with respect to the walls.

Figure 2

(Color online) The comparison of numerical calculations (lines) and the asymptotic solution given by Eqs. (5, 13, 14) (circles) presented for ${\alpha }^2=12$ $\left(t_0=0.401\right)$. The concentration of bubbles (a) and velocity potential (b) at $t=0.350$ (solid lines and closed circles) and at $t=0.360$ (dashed lines and open circles).

Figure 3

(Color online) The fields of concentration $\Phi$ $\left(t<t_0=0.401\right)$ or its regular part $\varphi$ $\left(t>t_0\right)$ (a) and velocity potential $\psi$ (b) plotted for ${\alpha }^2=12$ at different times before and after the emergence of a bubbly screen. The dotted, solid, and dashed lines correspond to $t=0.35$, $t=0.425$, and $t=0.615$, respectively.

Figure 4

(Color online) Evolution of the concentration plotted at ${\alpha }^2=12$ $\left(t_0=0.401\right)$. Variation in maximal concentration, ${\Phi }_m$ (solid line) and mean value of concentration, $⟨\varphi ⟩$ (dashed-dotted line) with $t$ (a). The vertical dashed line shows the moment of formation of the bubbly screen, $t=t_0$. At $t>t_0$, only the regular part of the concentration, $\varphi$, contributes into $⟨\varphi ⟩$. Characteristics of the bubbly screen (b): the power $A$ (solid line) and position $z_c$ (dashed-dotted line).

Figure 5

(Color online) Growth rates $\lambda$ as a function of $k$ plotted for ${\Phi }_a=0$, $Q=1$, $A_0{\alpha }^2=5$. The solid and dashed-dotted lines correspond to the odd and even modes, respectively. Vertical asymptotes are determined by Eqs. (39). Asymptotical formulas (37, 38), are shown by circles.