Hydrodynamic assist and the dynamic contact angle in the coalescence of liquid drops

^** Email: decentsp{at}for.mat.bham.ac.uk The coalescence of two viscous liquid drops in an inviscid gasor in a vacuum is studied using the interface formation model.In the very early stages of coalescence during the formationof the ‘liquid-bridge’ connecting the two drops,this model predicts a moving contact line and a dynamic contactangle. This paper examines the dynamic evolution of this contactangle, and for small Reynolds number and small Capillary number,relevant particularly in micro-fluidics, a non-linear differentialequation is derived for the contact angle and solved computationally.It is found that the contact angle evolution can only be evaluatedby determining information about the flow away from the contactline. This is a manifestation of so-called hydrodynamic assist,studied experimentally in the context of curtain coating byBlake et al. (1999 Experimental evidence of non-local hydrodynamicinfluence on the dynamic contact angle. Phys. Fluids, 11, 1995–2007).For small Capillary number and small Reynolds number, the free-surfaceevolution is determined for the coalescence of two cylindersof equal radius. Finally, some comments are made on experimentsin coalescence, as well as on issues arising in a computationalsolution of the full model described here.     

The formation of a liquid bridge during the coalescence of drops

This paper examines a mathematical model for the coalescence of two viscous liquid volumes in an inviscid gas or in a vacuum which removes the pressure singularity at the instant of impact inherent in the classical formulation of the continuum model. The very early stages of coalescence are examined in order to study the formation of the liquid bridge in two cases: (i) for two infinitely long, coalescing liquid cylinders; and (ii) for two coalescing spheres. Numerical solutions are computed for the velocity and pressure fields in the flow in both cases, and they confirm the removal of the pressure singularity. Also, the free-surface position at small times is determined.     

On the instability of microjets

The instability of an axisymmetric viscous liquid jet in a gas or in a vacuum is examined using the interface formation theory. This model allows for variable surface tension at constant temperature, generalising the classical continuum formulation by using irreversible thermodynamics. Steady-state solutions are determined and found to be unstable to a travelling wave that propagates down the liquid jet, causing the jet to break-up into drops. The linear instability results are compared to those of the classical formulation. These are especially found to differ when the jets are on the micron scale. This will give rise to significantly revised predictions in some parameter ranges for the break-up length and droplet sizes produced by microjets. Comparisons with molecular dynamics simulations are also presented, with encouraging results. Finally, the dependence of the results on the initial conditions is discussed. PACS 68.03.Cd     

The trajectory and stability of a spiralling liquid jet: Viscous theory

We examine a spiralling slender viscous jet emerging from a rapidly rotating orifice, extending Wallwork et al. [I.M. Wallwork, S.P. Decent, A.C. King, R.M.S.M. Schulkes, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory, J. Fluid Mech. 459 (2002) 43–65] by incorporating viscosity. The effects of viscosity on the trajectory of the jet and its linear instability are determined using a mixture of computational and asymptotic methods, and verified using experiments. A non-monotonic relationship between break-up length and rotation rate is demonstrated with the trend varying with viscosity. The sizes of the droplets produced by this instability are determined by considering the most unstable wave mode. It is also found that there is a non-monotonic relationship between droplet size and viscosity. Satellite droplet formation is also considered by analysing very short wavelength modes. The effects of long wavelength modes are examined, and a wave which propagates down the trajectory of the jet is identified for the highly viscous case. A comparison between theoretical and experimental results is made, with favourable agreement. In particular, a quantitative comparison is made between droplet sizes predicted from the theory with experimental observations, with encouraging agreement obtained. Four different types of break-up are identified in our experiments. The experimentally observed break-up mechanisms are discussed in light of our theory.     

Surface-tension-driven flow in a slender cone

After a droplet has broken away from a slender thread or jetof liquid, the tip of the thread or jet recoils rapidly. Atthe moment of break-off, the tip of the thread/jet is observedto have the shape of a cone close to the bifurcation point.In this paper, we study the evolution of an ideal fluid whichis initially conical, where the only force acting on the fluidis due to surface tension. We find an asymptotic solution tothe problem in terms of the aspect ratio of the cone which isassumed to be small. Using a similarity transformation, whichis valid for small times after the bifurcation, we identifya rapidly oscillating non-linear wave which propagates awayfrom the tip, as observed in experiments.     

Influence of surfactant upon air entrainment hysteresis in curtain coating

The onset of air entrainment for curtain coating onto a pre-wetted substrate was studied experimentally in similar parameter regimes to commercial coating (Re = ρQ/μ = O(1), We = ρQ u _c /σ = O(10), Ca = μU/σ = O(1)). Impingement speed and viscosity were previously shown to be critical parameters in correlating air entrainment data with three qualitatively different regimes of hydrodynamic assist identified (Marston et al. in Exp Fluids 42(3):483–488, 2007a). The interaction of the impinging curtain with the pre-existing film also led to a significant hysteretic effect throughout the flow rate-substrate speed parameter space. For the first time, results considering the influence of surfactants are presented in attempt to elucidate the relative importance of surface tension in this inertia-dominated system. The results show quantitative and qualitative differences to previous results with much more complex hysteretic behaviour which has only been reported previously for rough surfaces.     

Influence of viscosity and impingement speed on intense hydrodynamic assist in curtain coating

The onset of air entrainment for curtain coating onto a pre-wetted substrate was studied experimentally in similar parameter regimes to commercial coating [Re = ρ Q/μ = O(1), We = ρ Q u _c /σ = O(10), Ca = μ U/σ = O(1)]. As with coating onto dry substrates, a large flow-rate dependence was observed; termed hydrodynamic assist. The influence of the impingement speed and viscosity are shown to be critical and three qualitatively different regimes of assist are identified. The interaction of the impinging curtain with the pre-existing film also led to a significant hysteretic effect throughout the flow rate—substrate speed parameter space.     

The effect of surfactants on the instability of a rotating liquid jet

It has long been known that the presence of surfactants on the free surface of a liquid jet can create surface tension gradients along the interface. The resulting formation of tangential stresses along the surface lead to Marangoni type flows and greatly affect the resulting dynamics of rupture. In this way surfactants can be used to manipulate the breakup of a liquid jet and control the size of droplets produced. In this paper we investigate the effects of insoluble surfactants on the breakup of rotating liquid jets with applications to industrial prilling. Using a long wavelength approximation we reduce the governing equations into a set of one-dimensional equations. We use an asymptotic theory to find steady solutions and then carry out a linear instability analysis on these solutions. We show that steady state centreline solutions are independent of viscosity to leading order and that the most unstable wavenumber and growth rate of disturbances decrease as the effectiveness of surfactants is increased. We also numerically solve these equations using a finite difference scheme to investigate the effects of changing the initial surfactant concentration and other fluid parameters. Our results show that differences in breakup lengths between rotating surfactant-laden jets and surfactant-free jets increase with the rate of rotation. Moreover, we find that satellite droplet sizes decrease as the rate of rotation is decreased with the effect of surfactants amplifying the reduction in sizes. Furthermore, the presence of surfactants at fixed rotation rates is shown to produce larger main droplets at low disturbance wavenumbers whilst satellite droplets are smaller for moderate disturbance wavenumbers κ≈0.7.