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.

It is well known that the constant injection rate flow in radial Hele-Shaw cells leads to the formation of highly branched patterns, where finger tip-splitting events are plentiful. Different kinds of patterns arise in the lifting Hele-Shaw flow problem, where the cell's gap width grows linearly with time. In this case, the morphology of the emerging structures is characterized by the strong competition among inward moving fingers. By employing a mode-coupling theory we find that both finger tip-splitting and finger competition can be restrained by properly adjusting the injection rate and the time-dependent gap width, respectively. Our theoretical model approaches the problem analytically and is capable of capturing these important controlling mechanisms already at weakly nonlinear stages of the dynamics.

Variable-gap Hele-Shaw flows consider viscous fluid displacements resulting from the lifting or squeezing of the upper cell plate, while the lower plate remains at rest. Conventionally, researchers focus on the situation in which the cell plates are perfectly parallel. We study a slightly different version of the problem, where the upper plate is gently inclined so that the plates are no longer parallel. Within this tapered Hele-Shaw cell context we examine how the presence of such a small gap gradient affects the stability properties of the fluid-fluid interface. Linear stability analysis indicates that the existence of the taper offers a simple geometric way to control the development of interfacial fingering instabilities under both lifting and squeeze flow circumstances.

The injection of a fluid into another of larger viscosity in a Hele-Shaw cell usually results in the formation of highly branched patterns. Despite the richness of these structures, in many practical situations such convoluted shapes are quite undesirable. In this Brief Report, we propose an efficient and easily reproducible way to restrain these instabilities based on a simple piecewise-constant pumping protocol. It results in a reduction in the size of the viscous fingers by one order of magnitude.