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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.

Dynamic wetting is crucial to processes where liquid displaces another fluid along a solid surface, such as the deposition of a coating liquid onto a moving substrate. Numerous studies report the failure of dynamic wetting past some critical process speed. However, the hydrodynamic factors that influence the transition to wetting failure remain poorly understood from an empirical and theoretical perspective. The objective of this investigation is to determine the effect of meniscus confinement on the onset of dynamic wetting failure. A novel experimental system is designed to simultaneously view confined and unconfined wetting systems as they approach wetting failure. The experimental apparatus consists of a scraped steel roll that rotates into a bath of glycerol. Confinement is imposed via a gap formed between a coating die and the roll surface. Flow visualization is used to record the critical roll speed at which wetting failure occurs. Comparison of the confined and unconfined data shows a clear increase in the relative critical speed as the meniscus becomes more confined. A hydrodynamic model for wetting failure is developed and analysed with (i) lubrication theory and (ii) a two-dimensional finite-element method (FEM). Both approaches do a remarkable job of matching the observed confinement trend, but only the two-dimensional model yields accurate estimates of the absolute values of the critical speeds due to the highly two-dimensional nature of the stress field in the displacing liquid. The overall success of the hydrodynamic model suggests a wetting failure mechanism primarily related to viscous bending of the meniscus.

This work investigates the onset of wetting failure for displacement of Newtonian fluids in parallel channels. A hydrodynamic model is developed for planar geometries where an advancing fluid displaces a receding fluid along a moving substrate. The model is evaluated with three distinct approaches: (i) the low-speed asymptotic theory of Cox [J. Fluid Mech. 168, 169–194 (1986)], (ii) a one-dimensional (1D) lubrication approach, and (iii) a two-dimensional (2D) flow model solved with the Galerkin finite element method (FEM). Approaches (ii) and (iii) predict the onset of wetting failure at a critical capillary number Cacrit, which coincides with a turning point in the steady-state solution family for a given set of system parameters. The 1D model fails to accurately describe interface shapes near the three-phase contact line when air is the receding fluid, producing large errors in estimates of Cacrit for these systems. Analysis of the 2D flow solution reveals that strong pressure gradients are needed to pump the receding fluid away from the contact line. A mechanism is proposed in which wetting failure results when capillary forces can no longer support the pressure gradients necessary to steadily displace the receding fluid. The effects of viscosity ratio, substrate wettability, and fluid inertia are then investigated through comparisons of Cacrit values and characteristics of the interface shape. Surprisingly, the low-speed asymptotic theory (i) matches trends computed from (iii) throughout the entire investigated parameter space. Furthermore, predictions of Cacrit from the 2D flow model compare favorably to values measured in experimental air-entrainment studies, supporting the proposed wetting-failure mechanism.

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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.

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SUMMARYWe discuss the use of polygonal finite elements for analysis of incompressible flow problems. It is well‐known that the stability of mixed finite element discretizations is governed by the so‐called inf‐sup condition, which, in this case, depends on the choice of the discrete velocity and pressure spaces. We present a low‐order choice of these spaces defined over convex polygonal partitions of the domain that satisfies the inf‐sup condition and, as such, does not admit spurious pressure modes or exhibit locking. Within each element, the pressure field is constant while the velocity is represented by the usual isoparametric transformation of a linearly‐complete basis. Thus, from a practical point of view, the implementation of the method is classical and does not require any special treatment. We present numerical results for both incompressible Stokes and stationary Navier–Stokes problems to verify the theoretical results regarding stability and convergence of the method. Copyright © 2013 John Wiley & Sons, Ltd.