Rotating Hele-Shaw cell with a time-dependent angular velocity

Despite the large number of existing studies of viscous flows in rotating Hele-Shaw cells, most investigations analyze rotational motion with a constant angular velocity, under vanishing Reynolds number conditions in which inertial effects can be neglected. In this work, we examine the linear and weakly nonlinear dynamics of the interface between two immiscible fluids in a rotating Hele-Shaw cell, considering the action of a time-dependent angular velocity, and taking into account the contribution of inertia. By using a generalized Darcy's law, we derive a second-order mode-coupling equation which describes the time evolution of the interfacial perturbation amplitudes. For arbitrary values of viscosity and density ratios, and for a range of values of a rotational Reynolds number, we investigate how the time-dependent angular velocity and inertia affect the important finger competition events that traditionally arise in rotating Hele-Shaw flows.

Figure 1

Illustrative sketch of a rotating Hele-Shaw cell.

Figure 2

Linear growth rate $\lambda (n,t)$ as a function of time $t$, for $\alpha ={10}^{-3},\beta =0$, and $B=3.0\times {10}^{-3}$. The action of both a constant $\left(\gamma \to \infty \right)$, and a time-dependent $\left(\gamma =2\right)$ angular velocity is examined. (a) $\mathrm{Re}=0.03,n=2,n=4$, and $n=8$. (b) $n=8$, and two Reynolds numbers $(\mathrm{Re}=0.03$, and $\mathrm{Re}=0.2)$ are considered.

Figure 3

Time overlaid plots (top panels) illustrating the typical evolution of the two-fluid interfaces obtained in rotating Hele-Shaw flows, for a time-dependent angular velocity given by Eq. (28) with $\gamma =2$ and three increasing values of the Reynolds number: (a) $\mathrm{Re}=0.03$, (b) $\mathrm{Re}=0.16$, and (c) $\mathrm{Re}=0.2$. The resulting fingering patterns (shaded areas) obtained at the final time $t=t_f$ are shown in the middle panels. The snapshots are taken for times (a) $t_f=1.9107$, (b) $t_f=2.072$, and (c) $t_f=2.1599$, where $t_f$ is defined as the time at which the amplitude of the fundamental mode has reached the value $a_n(t=t_f)=0.15$, for each value of $\mathrm{Re}$. The small arrows point in the direction of inward or outward motion of the competing fingers. The dashed circles are added to facilitate visualization of the competition among the fingers. The corresponding time evolution of the perturbation amplitudes $a_n\left(t\right),a_{n/2}\left(t\right)$, and $b_{n/2}\left(t\right)$, for modes $n=10$ and $n/2=5$ is depicted in the bottom panels. We set $\alpha ={10}^{-3},\beta =0$, and $B=3.0\times {10}^{-3}$.

Figure 4

Behavior of the perturbation amplitudes $a_{n/2}$ and $b_{n/2}$ as the Reynolds number $\mathrm{Re}$ is changed under constant angular velocity $(\gamma \to \infty$, represented by the dashed curves), and under a time-dependent angular velocity condition $(\gamma =2$, symbolized by the solid curves). All physical parameters and initial conditions are the same as the ones used in Fig. 3.

Figure 5

Variation of the perturbation amplitudes $a_{n/2}$ and $b_{n/2}$ as the Reynolds number $\mathrm{Re}$ is varied, under constant angular velocity $(\gamma \to \infty$, expressed by the dashed curves), and under a time-dependent angular velocity condition $(\gamma =2$, pictured by the solid curves). Three representative values for the viscosity ratio are considered: (a) $\beta =0$, (b) $\beta =1$, and (c) $\beta =100$. Here we take $\alpha =0.5,n=8$, and $n/2=4$. All other physical parameters and initial conditions are exactly the same as the ones employed in Fig. 3.