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.

The dynamics of liquid bridges are relevant to a wide variety of applications including high-speed printing, extensional rheometry, and floating-zone crystallization. Although many studies assume that the contact lines of a bridge are pinned, this is not the case for printing processes such as gravure, lithography, and microcontacting. To address this issue, we use the Galerkin/finite element method to study the stretching of a finite volume of Newtonian liquid confined between two flat plates, one of which is stationary and the other moving. The steady Stokes equations are solved, with time dependence entering the problem through the kinematic boundary condition. The contact lines are allowed to slip, and we evaluate the effect of the capillary number and contact angle on the amount of liquid transferred to the moving plate. At fixed capillary number, liquid transfer to the moving plate is found to increase as the contact angle on the stationary plate increases relative to that on the moving plate. When the contact angle is fixed and the capillary number is increased, the liquid transfer improves if the stationary plate is wetting, but worsens if it is nonwetting. The presence of a cavity on the stationary plate significantly affects the contact line motion, often causing pinning along the cavity wall. In these cases, liquid transfer is controlled primarily by the cavity shape, suggesting that the effects of surface topography dominate over those of surface wettability. At low capillary numbers, bridge breakup can be understood in terms of the Rayleigh–Plateau stability limit, regardless of the combination of contact angles or the plate geometry. At higher capillary numbers, the bridge is able to stretch beyond this limit although the deviation from this limit appears to depend on contact line pinning, and not directly on the combination of contact angles or the plate geometry.