S.P. Decent

The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory

We examine a spiralling slender inviscid liquid jet which emerges from a rapidly rotating orifice. The trajectory of this jet is determined using asymptotic methods, and the stability using a multiple scales approach. It is found that the trajectory of the jet becomes more tightly coiled as the Weber number is decreased. Unstable travelling wave modes are found to grow along the jet. The breakup length of the jet is calculated, showing good agreement with experiments.

Free jets spun from a prilling tower

A mathematical model of the dynamics of an inviscid liquid jet, subjected to both gravity and surface tension, which emerges from rotating drum is derived and analysed using asymptotic and computational methods. The trajectory and linear stability of this jet is determined. By use of the stability results, the break up length of the jet is calculated. Such jets arise in the manufacture of pellets (for example, of fertilizer or magnesium) using the prilling process. Here the drum would contain many thousands of holes, and the molten liquid would be pumped into the rotating drum. After the jet has broken up into droplets, these droplets solidify to form pellets. The jets in this prilling process are curved in space by both gravity and surface tension.

The Trajectory and Stability of a Spiralling Liquid Jet

We examine a spiralling slender inviscid liquid jet with a curved centre line which emerges from a rotating orifice. The trajectory of this jet is determined using asymptotic methods, and the stability using a multiple scales approach. Some novel results will be presented and discussed.

The Instability of Shear Thinning and Shear Thickening Spiralling Liquid Jets: Linear Theory

The linear instability of a power law liquid emerging as a jet from an orifice on the surface of a rotating container is investigated, with applications to industrial prilling. Asymptotic methods are used to examine the growth rate and wavenumber of the most unstable traveling wave mode for different flow index numbers. Comparison with Newtonian liquids show that for small rotation rates shear thinning liquids are most stable to disturbances. In contrast for higher rotation rates we find shear thickening liquids are more stable than shear thinning liquids. The influence of viscosity, surface tension, and rotation rate on the growth rates and most unstable wavenumbers associated with both types of liquids are also examined.

Influence of the flow field in curtain coating onto a prewet substrate

The onset of air entrainment for curtain coating onto a surface prewetted with the coating fluid was studied. The substrate used was a polished, scraped steel wheel and coating was performed over ranges of dimensionless parameters observed in commercial coating processes (Reynolds number, 0.14<Re=ρQ∕μ<33.02; Capillary number, 0.19<Ca=μU∕σ<25.07). The substrate velocity for the onset of air entrainment was obtained as a function of the curtain flow rate per unit width of curtain (1<Q<9cm2s−1), fluid dynamic viscosity (0.0326<μ<0.878Pas), curtain height (0.035<h<0.095m), and thickness of the prewet film (1×10−7<c<3×10−5m). A remarkable and strong dependence of the onset of air entrainment on curtain flow rate was observed (hydrodynamic assist) and the general features of the hydrodynamics were very similar to those observed for previous works onto dry substrates. However, the presence of the prewet film led to higher maximum substrate velocities at the onset of air entrainment than observed for dry substrat...

The nonlinear damping of parametrically excited two-dimensional gravity waves

Parametrically excited waves are usually modelled with a nonlinear amplitude equation. It has recently been demonstrated that the behaviour of these waves depends critically upon the coefficient of the cubic damping term in the nonlinear amplitude equation, and especially upon the sign of this coefficient (see Decent and Craik [J. Fluid Mech. 293 (1995) 237]. However, very little work has been carried out on theoretically determining the value of this coefficient. This paper derives the coefficient of cubic damping for the single-mode nonlinear amplitude equation which models two-dimensional gravity waves in a narrow rectangular container. Energy dissipation in the main body of the fluid and in boundary layers at the sidewalls and at the surface is considered. Theoretical results agree fairly well with an experiment carried out by Decent and Craik (1995).

Hysteresis and non-uniqueness in the speed of the onset of instability in curtain coating

The maximum speed of stable coating is determined experimentally by the onset of air entrainment in curtain coating onto a pre-wetted surface over a broad range of dimensionless parameters (Reynolds number: 0.14 < Re < 33.02; capillary number: 0.19<Ca<25.07). We show not only that the substrate speed at the onset of instability is flow-rate dependent for a pre-wetted surface, but that the speed at which instability occurs is not always unique for a given flow rate.

Breakup of inviscid compound liquid jets falling under gravity

In this paper, we examine the instability of an axisymmetric inviscid compound liquid jet (subject to axisymmetric disturbances) emerging from a concentric tube and falling vertically under the effects of gravity. We formulate the governing equations and then use a slender jet approximation to consider a one-dimensional model of jet breakup. We determine the steady-state solutions of a falling compound jet. We then numerically simulate breakup and droplet formation using an explicit two-step finite difference scheme. The effects of changing the surface tension and density ratio between the inner and outer fluids is investigated, as is the effect of altering the Froude number.

Sideband instability and modulations of Faraday waves

Abstract Three-mode interaction equations are derived for Faraday waves in a long rectangular container. Two water depths are studied, revealing very different behaviours. Our equations include conservative and non-conservative cubic nonlinear terms, and quintic conservative nonlinear terms. Instability of a single standing wave to neighbouring modes is examined. Resultant three-mode interactions display rich structure, with fast and slow timescales. For the smaller depth, but not the greater, recurrent nearly calm intervals are separated by strong wave activity. Our results agree quite well with experimental findings of Craik and Armitage [Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank, Fluid Dyn. Res. 15 (1995) 129–143] and with unpublished observations by Armitage, Craik and Sterratt, here briefly described.

Temporal instability analysis of inviscid compound jets falling under gravity

Compound liquid jets can be used in a variety of industrial applications ranging from capsule production in pharmaceutics to enhance printing methods in ink-jet printing. An appreciation of how instability along compound jets can lead to breakup and droplet formation is thus critical in many fields in science and engineering. In this paper, we perform a theoretical analysis to examine the instability of an axisymmetric inviscid compound liquid jet which falls vertically under the influence of gravity. We use a long-wavelength, slender-jet asymptotic expansion to reduce the governing equations of the problem into a set of one-dimensional partial differential equations, which describe the evolution of the leading-order axial velocity of the jet as well as the radii of both the inner and the outer interfaces. We first determine the steady-state solutions of the one-dimensional model equations and then we perform a linear temporal instability analysis to obtain a dispersion relation, which gives us useful inf...