Effects of circular rigid boundaries and Coriolis forces on the interfacial instability in a rotating annular Hele-Shaw cell

We report analytical results for the development of instability of an interface between two immiscible, Newtonian fluid layers confined in a rotating annular Hele-Shaw cell. We perform a linear stability analysis and focus our study on the influence of both Coriolis force and curvature parameters on the interface instability growth rate. The results show that the Coriolis force does not alter the stability of a disturbance with a particular wave number but reduces the maximum growth rate. The results related to the role played by the confinement of the liquid layers are also shown to provide a modification of the fastest-growing mode and its corresponding linear growth rate.

Figure 1

Sketch of a Hele-Shaw cell with two confined liquid layers at the equilibrium. The aspect ratio $ϵ=e∕R_0,ϵ⪡1$.

Figure 2

The effect of boundaries on the instability. (a) ${\delta }_1=0$, ${\delta }_2\to \infty$ [4], (b) ${\delta }_1=0$, ${\delta }_2=1.1$, (c) ${\delta }_1=0.95$, ${\delta }_2\to \infty$, and (d) ${\delta }_1=0.95$, ${\delta }_2=1.1$. All data are for $B=200$, $A=0$, and $ϵ=0.05$.

Figure 3

The effect of contrast viscosity with the presence of the inner boundary on the instability. All data are for $B=200$, $ϵ=0.05$, ${\delta }_1=0.95$, and ${\delta }_2\to \infty$.

Figure 4

Comparison of growth rate for different values of contrast viscosity and for ${\rho }_1∕{\rho }_2=1000$, ${\delta }_1=0$, and $\delta =0$. Note that the curve $A=-1$ should produce the same results of Waters and Cummings [5].

Figure 5

Comparison of growth rate for different values of contrast viscosity and for ${\rho }_1∕{\rho }_2=1.5$, ${\delta }_1=0$, and $\delta =0$. These results can be obtained from the recent paper of Waters and Cummings [12].

Figure 6

Water-air interface ($\sigma =0.072\mathrm{N}∕\mathrm{m}$, $R_0=10\mathrm{cm}$, $e=2\mathrm{mm}$): variation of the dimensional growth rate versus $n$ (a) without Coriolis forces [3, 4] and (b) with Coriolis forces.

Figure 7

Water-air interface ($\sigma =0.072\mathrm{N}∕\mathrm{m}$, $R_0=1\mathrm{cm}$, $e=1.5\mathrm{mm}$, $B=200$, ${\delta }_1=0$, ${\delta }_2\to \infty$): variation of the dimensional growth rate versus $n$ in presence of Coriolis effects. (a) Dashed line: without viscous stresses obtained either by Eq. (3.26) [12] or by Eq. (13) of the present study with $\delta =0$. (b) Solid line: with Coriolis forces and viscous stresses obtained by Eq. (13) of the present study with $\delta =1$.