Fingering instability transition in radially tapered Hele-Shaw cells: Insights at the onset of nonlinear effects

We investigate the effect of the capillary number $\mathrm{Ca}$ on the interfacial viscous fingering instability in radially tapered Hele-Shaw cells. By employing a perturbative weakly nonlinear approach, we manage to identify a fingering instability transition in the system at the onset of nonlinearities. We find that for low $\mathrm{Ca}$ the interface in tapered situations is stabilized (destabilized) in converging (diverging) cells, with respect to the equivalent behavior occurring in a parallel-plate (uniform) Hele-Shaw cell. However, for large $\mathrm{Ca}$, we observe that the relative stability behavior changes, so that converging cells destabilize the interface in comparison to uniform cells, while diverging cells lead to relatively more stable interfaces. Moreover, we verify that finger tip-splitting is favored for large $\mathrm{Ca}$, and restrained in the low-$\mathrm{Ca}$ regime. Our weakly nonlinear results are qualitatively consistent with recent intensive numerical simulations in the literature in which such an instability transition was examined at fully nonlinear stages of the flow.

Figure 1

Illustrative sketch of the side view of the injection-driven fluid displacements in three kinds of radial Hele-Shaw cells: (a) converging ($\alpha <0$), (b) uniform ($\alpha =0$), and (c) diverging ($\alpha >0$), where $\alpha =db\left(r\right)/dr$ denotes the small gap gradient, $b\left(r\right)$ is the variable gap thickness, $r$ is the radial coordinate, and $b_0=b(r=0)$. The time-dependent unperturbed interface radius is represented by $R\left(t\right)$. Moreover, $Q$ denotes the volumetric injection flow rate, $\eta$ is the outer (displaced) fluid viscosity, while the inner (displacing) fluid (in gray) has negligible viscosity.

Figure 2

Characteristic lengths defining the finger length function $\mathrm{\Delta }\mathcal{R}$ as defined in Eq. (43): ${\mathcal{R}}_{\max }$ is the maximum value of the radial coordinate of the deformed interface (located at a finger tip), and ${\mathcal{R}}_{\min }$ is the corresponding minimum radius (located at a finger valley).

Figure 3

Snapshot of the weakly nonlinear interfaces at time $t=t_f=13$ (top panels), illustrating typical fingering patterns for injection-driven radial flow in diverging ($\alpha ={10}^{-3}$), uniform ($\alpha =0$), and converging ($\alpha =-{10}^{-3}$) Hele-Shaw cells. The following values of the capillary number are analyzed: (a) $\mathrm{Ca}=1200$, (b) $\mathrm{Ca}=1637$, (c) $\mathrm{Ca}=1905$, and (d) $\mathrm{Ca}=3000$. The interfaces are obtained by considering the nonlinear coupling between modes $n$ and $2n$, where $n=5$. The corresponding time evolution of the finger length $\mathrm{\Delta }\mathcal{R}\left(t\right)$ for $0\le t\le t_f$ is shown in the bottom panels.

Figure 4

Snapshots of the evolving interface for the most unstable situation examined in Fig. 3 [i.e., the case for $\mathrm{Ca}=3000$, and $\alpha =-{10}^{-3}$ illustrated in Fig. 3], plotted at equal time intervals ($\mathrm{\Delta }t=1$) when (a) $0\le t\le t_f$, and (b) $0\le t\le t_f+5$, where $t_f=13$. Interface self-intersections are evident in panel (b). Since these self-intersections are not observed in radial Hele-Shaw cell systems in which $\beta \to \infty$, we use $t=t_f$ as the upper bound time for the validity of our weakly nonlinear perturbative approach.

Figure 5

Determination of the critical values of the capillary number at which the relative instability transitions. This is done by plotting the variation of finger length evaluated at final time $t=t_f$, $\mathrm{\Delta }\mathcal{R}\left(t_f\right)$ as $\mathrm{Ca}$ is increased for $\alpha =-{10}^{-3}$, $\alpha =0$, and $\alpha ={10}^{-3}$. The critical capillary number ${\mathrm{Ca}}_{\mathrm{crit}}^{\alpha >0}$ (${\mathrm{Ca}}_{\mathrm{crit}}^{\alpha <0}$) is determined by the point at which the curves for $\alpha =0$ and $\alpha >0$ ($\alpha <0$) cross one another, as indicated by the vertical dashed lines. Here all physical parameters and initial conditions are identical to those utilized in (a) Fig. 3 $[n=5$, $a_n\left(0\right)={10}^{-2}$, and $t_f=13]$, and (b) Fig. 6 $[n=4$, $a_n\left(0\right)=2.5\times {10}^{-2}$, and $t_f=12]$.

Figure 6

Snapshot of the weakly nonlinear interfaces at time $t=t_f=12$ (top panels), displaying characteristic fingering patterns for flow in diverging ($\alpha ={10}^{-3}$), uniform ($\alpha =0$), and converging ($\alpha =-{10}^{-3}$) Hele-Shaw cells. The following values of the capillary number are examined: (a) $\mathrm{Ca}=1200$, (b) $\mathrm{Ca}=1687$, (c) $\mathrm{Ca}=1925$, and (d) $\mathrm{Ca}=4000$. The interfaces are produced by taking into consideration the nonlinear coupling between modes $n$ and $2n$, where $n=4$. The corresponding time evolution of the finger length $\mathrm{\Delta }\mathcal{R}\left(t\right)$ for $0\le t\le t_f$ is depicted in the bottom panels.