Generalized elastica patterns in a curved rotating Hele-Shaw cell

We study a family of generalized elasticalike equilibrium shapes that arise at the interface separating two fluids in a curved rotating Hele-Shaw cell. This family of stationary interface solutions consists of shapes that balance the competing capillary and centrifugal forces in such a curved flow environment. We investigate how the emerging interfacial patterns are impacted by changes in the geometric properties of the curved Hele-Shaw cell. A vortex-sheet formalism is used to calculate the two-fluid interface curvature, and a gallery of possible shapes is provided to highlight a number of peculiar morphological features. A linear perturbation theory is employed to show that the most prominent aspects of these complex stationary patterns can be fairly well reproduced by the interplay of just two interfacial modes. The connection of these dominant modes to the geometry of the curved cell, as well as to the fluid dynamic properties of the flow, is discussed.

Figure 1

Typical stationary fluid-fluid interface shapes that emerge during the conventional (i.e., flat), rotating Hele-Shaw problem [29, 31]: (a) threefold closed elastica shape having rounded fingers with wide necks; (b) fivefold structure with racket-shaped fingers possessing thin necks. The same class of planar elasticalike patterns appear in various other contexts [7, 8, 9, 10, 11].

Figure 2

Illustrative examples of warped-cone surfaces with (a) mild surface undulations and (b) more intense surface distortions.

Figure 3

Schematic illustration of the fluid flow on a curved rotating Hele-Shaw cell. An inner fluid of density ${\rho }_1$ and viscosity ${\eta }_1$ is surrounded by an outer fluid of density ${\rho }_2$ and viscosity ${\eta }_2$. The fluids are located in a warped-cone Hele-Shaw cell that rotates with constant angular velocity $\mathrm{\Omega }$ around a rotation axis (dashed line). We consider that ${\rho }_1>{\rho }_2$, in such a way that the centrifugal force acts against capillary effects, inducing interfacial deformations at the fluid-fluid interface.

Figure 4

Representative family of generalized elastica patterns that arise in a rotating warped-cone Hele-Shaw cell when the geometric parameter $C$ is increased. The values we use for $C$ are: (a) 0; (b) 0.1677; (c) 0.5556; (d) 0.6495; (e) 0.7025; (f) 0.7462; (g) 0.7813; (h) 0.8046; (i) 0.8234; (j) 0.8865; (k) 0.9030; and (l) 0.9289. The remaining parameters are $a=-12$, $N_{\mathrm{\Omega }}=22.5$, $r(s=0)=1$, $r_s(s=0)=0$, $\phi (s=0)=0$, and $m=4$.

Figure 5

Comparison between some of the full spectrum generalized elastica shapes presented in Fig. 4, and the corresponding two modes solutions, for which only the largest amplitude modes ($n$ and $n^{\prime }$) are taken into account. (a) The full spectrum shape is given by the pattern shown in Fig. 4, and the two modes solution considers the sole action of modes $n=8$ and $n^{\prime }=13$; (b) The full spectrum pattern is the interface plotted in Fig. 4, while the two modes solution takes into consideration just the modes $n=8$ and $n^{\prime }=16$.

Figure 6

(a) Absolute value of the cosine Fourier amplitudes $|a_n|$ as a function of the azimuthal mode number $n$. (b) Comparison between numerical and linear solutions for the stationary interface shape. Here $C=0.58$ and $m=2$. Note that $n=2m=4$ and $n^{\prime }=8$.

Figure 7

(a) Absolute value of the cosine Fourier amplitudes $|a_n|$ as a function of the azimuthal mode number $n$. (b) Comparison between numerical and linear solutions for the stationary interface shape. Here $C=0.4965$ and $m=3$. Note that $n=2m=6$ and $n^{\prime }=9$.