A theoretical and experimental investigation of two-dimensional (2-D) liquid curtains (gravitationally thinning liquid sheets) is provided under conditions where the curtain issues from a thin slot whose centreline is inclined with respect to the vertical. This analysis is motivated in part by recent works where it has been proposed that oblique liquid curtains (those exiting a non-vertical slot) may bend upwards against gravity when the relevant Weber number at the slot is less than unity ($We <1$). By contrast, Weinstein et al. (J. Fluid Mech., vol. 876, 2019, R3) have proposed that such $We<1$ curtains must be vertical and downward falling regardless of the inclination of the slot. Under low-Reynolds-number ($Re$) conditions typical of liquid film coating operations, our experiments show that the curtain shape follows the classic ballistic (parabolic) trajectory in the supercritical regime ($We>1$). In subcritical conditions ($We<1$), experiments show that the downward-falling curtain is vertical except in a relatively small region near the slot, where the combined effects of viscosity and surface tension induce the so-called teapot effect. These experimental results are confirmed by 2-D numerical simulations, which predict the curtain behaviour ranging from highly viscous ($Re = O(1)$) to nearly inviscid conditions. The one-dimensional (1-D) inviscid model of Weinstein et al. is recast in a different form to facilitate comparisons with the 2-D model, and 1-D and 2-D results agree favourably for supercritical and subcritical conditions. Despite the large parameter range explored, we have found no evidence that upward-bending curtains exist in an oblique configuration.

We consider the time-dependent response of a gravitationally thinning inviscid liquid sheet (a coating curtain) leaving a vertical slot to sinusoidal ambient pressure disturbances. The theoretical investigation employs the hyperbolic partial differential equation developed by Weinstein et al. (Phys. Fluids, vol. 9, issue 12, 1997, pp. 3625–3636). The response of the curtain is characterized by the slot Weber number, $W_{e_0} = \rho q V/2\sigma$, where $V$ is the speed of the curtain at the slot, $q$ is the volumetric flow rate per unit width, $\sigma$ is the surface tension and $\rho$ is the fluid density. Flow disturbances travel along characteristics with speeds relative to the curtain of $\pm \sqrt {uV/W_{e_0}}$, where $u = \sqrt {V^{2} + 2gx}$ is the curtain speed at a distance $x$ downstream from the slot. Here g is the acceleration of gravity. When the flow is subcritical ($W_{e_0} < 1$), upstream travelling disturbances near the slot affect the curtain centreline, and the slope of the curtain centreline at the slot oscillates with an amplitude that is a function of $W_{e_0}$. In contrast, all disturbances travel downstream in supercritical curtains ($W_{e_0} > 1$) and the slope of the curtain at the slot is vertical. Here, we specifically examine the curtain response under supercritical and subcritical flow conditions near $W_{e_0} = 1$ to deduce whether there is a substantial change in the overall shape and magnitude of the curtain responses. Despite the local differences in the curtain solution near the slot, we find that subcritical and supercritical curtains have similar responses for all imposed sinusoidal frequencies.

In a recent paper (J. Fluid Mech., vol. 861, 2019, pp. 328–348), Benilov derived equations governing a laminar liquid sheet (a curtain) that emanates from a slot whose centreline is inclined to the vertical. The equations are valid for slender sheets whose characteristic length scale in the direction of flow is much larger than its cross-sectional thickness. For a liquid that leaves a slot with average speed, $u_{0}$, volumetric flow rate per unit width, $q$, surface tension, $\unicode[STIX]{x1D70E}$, and density, $\unicode[STIX]{x1D70C}$, Benilov obtains parametric equations that predict steady-state curtain shapes that bend upwards against gravity provided $\unicode[STIX]{x1D70C}qu_{0}/2\unicode[STIX]{x1D70E}<1$. Benilov’s parametric equations are shown to be identical to those derived by Finnicum, Weinstein, and Ruschak (J. Fluid Mech., vol. 255, 1993, pp. 647–665). In the latter form, it is straightforward to deduce an alternative solution of Benilov’s equations where a curtain falls vertically regardless of the slot’s orientation. This solution is consistent with prior experimental and theoretical results that show that a liquid curtain can emerge from a slot at an angle different from that of the slot centreline.

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

The shape of a two-dimensional liquid curtain issuing from a slot and falling under the influence of gravity is predicted theoretically and verified experimentally for cases where a pressure is applied to the curtain. A set of approximate equations is derived which governs the location of the curtain for a liquid having surface tension σ, density ρ, volumetric flow per unit width Q, and local free-fall velocity V. These equations possess a singularity at the point where the local Weber number, We = ρQV/2σ, is equal to 1. Despite the fact that previous work on the stability of two-dimensional curtains shows that curtains having locations where We < 1 are unstable to small disturbances, our experiments show that these curtains can exist over a wide range of flow conditions. Thus, it is necessary to consider how the singularity is resolved when a pressure is applied.

It is found that the singularity can be eliminated from the governing equations if the curtain assumes a definite direction as it leaves the slot. By contrast, if the curtain leaves the slot such that We > 1, there is no such restriction, and experimentally it is found that the curtain leaves parallel to the slot walls. The theoretical predictions of the curtain shapes are in agreement with those measured experimentally for all Weber numbers investigated.

The problem of viscous fingering in a Hele-Shaw cell with moving contact lines is considered. In contrast to the usual situation where the displaced fluid coats the solid surface in the form of thin films, here, both the displacing and the displaced fluids make direct contact with the solid. The principal differences between these two situations are in the ranges of attainable values of the gapwise component of the interfacial curvature (the component due to the bending of the fluid interface across the small gap of the Hele-Shaw cell), and in the introduction of two additional parameters for the case with moving contact lines. These parameters are the receding contact angle, and the sensivity of the dynamic angle to the speed of the contact line. Our objective is the prediction of the shape and widths of the fingers in the limit of small capillary number, Uμ/σ. Here, U denotes the finger speed, μ denotes the dynamic viscosity of the more viscous displaced fluid, and σ denotes the surface tension of the fluid interface. As might be expected, there are similarities and differences between the two problems. Despite the fact that different equations arise, we find that they can be analysed using the techniques introduced by McLean & Saffman and Vanden-Broeck for the thin-film case. The nature of the multiplicity of solutions also appears to be similar for the two problems. Our results indicate that when contact lines are present, the finger shapes are sensitive to the value of the contact angle only in the vicinity of its nose, reminiscent of experiments where bubbles or wires are placed at the nose of viscous fingers when thin films are present. On the other hand, in the present problem at least two distinct velocity scales emerge with well-defined asymptotic limits, each of these two cases being distinguished by the relative importance played by the two components of the curvature of the fluid interface. It is found that the widths of fingers can be significantly smaller than half the width of the cell.