Minimization of Viscous Fluid Fingering: A Variational Scheme for Optimal Flow Rates

Conventional viscous fingering flow in radial Hele-Shaw cells employs a constant injection rate, resulting in the emergence of branched interfacial shapes. The search for mechanisms to prevent the development of these bifurcated morphologies is relevant to a number of areas in science and technology. A challenging problem is how best to choose the pumping rate in order to restrain the growth of interfacial amplitudes. We use an analytical variational scheme to look for the precise functional form of such an optimal flow rate. We find it increases linearly with time in a specific manner so that interface disturbances are minimized. Experiments and nonlinear numerical simulations support the effectiveness of this particularly simple, but nontrivial, pattern controlling process.

Figure 1

Sketch of the injection rate as a function of time for the optimal injection $Q(t)$ (solid line). The equivalent constant injection rate $Q_0$ is represented by the horizontal dashed line. The total volume of injected fluid in the interval [0, $t_f$] should be the same for both pumping rates.

Figure 2

The solid curve represents the integral $I$ as a function of mode $n$, calculated for the optimal injection rate (7). The dashed curve shows the corresponding quantity $I_0$ for the constant injection process (8).

Figure 3

Amplitude ratio ${\zeta }_0/\zeta$ as a function of $T_f$ for $\mathrm{Ca}=100$, 130. Here ${\zeta }_0$ ($\zeta$) denotes the maximum amplitude for constant (optimal) injection at $t=t_f$.

Figure 4

Typical experimental patterns at time $t_f=20\mathrm{s}$ for (a) $Q_0=2.08{\mathrm{cm}}^2/\mathrm{s}$, and (b) its corresponding ideal injection rate. For (a) and (b) $T_f=65$ and $\mathrm{Ca}=112$. The experimental shapes shown in (c) and (d) are taken at $t_f=6.7\mathrm{s}$ for $Q_0=3.75{\mathrm{cm}}^2/\mathrm{s}$, and for the equivalent optimal pumping rate, respectively. In (c) and (d) $T_f=40$ and $\mathrm{Ca}=200$.

Figure 5

Nonlinear numerical simulations depicting the time evolution of the interfacial patterns for constant injection $Q_0=1.25{\mathrm{cm}}^2/\mathrm{s}$ (left), and equivalent optimal injection (right). Here $t_f=240\mathrm{s}$, $R_f=9.8\mathrm{cm}$, and $R_0=1.0\mathrm{cm}$. The interfaces are plotted in intervals of $t_f/5$. Here $T_f=95$ and $\mathrm{Ca}=95$.