Drop formation dynamics of constant low-viscosity, elastic fluids

https://doi.org/10.1016/S0377-0257(02)00084-8Get rights and content

Abstract

The dynamics of drop formation under gravity has been investigated as a function of elasticity using a set of low-viscosity, ideal elastic fluids and an equivalent Newtonian glycerol–water solution. All solutions had the same shear viscosity, equilibrium surface tension, and density, but differed greatly in elasticity. The minimum drop radius in the early stages of drop formation (necking) was found to scale as expected from potential flow theory, independent of the elasticity of the solutions. Thus, during this stage of drop formation when viscous force is still weak, the dynamics are controlled by a balance between inertial and capillary forces, and there is no contribution of elastic stresses of the polymer. However, upon formation of the pinch regions, there is a large variation in the drop development to break-off observed between the various solutions. The elastic solutions formed secondary fluid threads either side of a secondary drop from the necked region of fluid between the upper and lower pinches, which were sustained for increasing amounts of time. The break-off lengths and times increase with increasing elasticity of the solutions. Evolution of the filament length is, however, identical in shape and form for all of the polymer solutions tested, regardless of differing elasticity. This de-coupling between filament growth rate and break-up time (or equivalently, final filament length at break-up) is rationalised. A modified force balance to that of Jones and Rees [48] is capable of correctly predicting the filament growth of these low-viscosity, elastic fluids in the absence of any elastic contributions due to polymer extension within the elongating filament. The elongation of the necked region of fluid (which becomes the filament) is dominated by the inertia of the drop, and is independent of the elasticity of the solution. However, elasticity does strongly influence the resistance of the pinch regions to break-off, with rapid necking resulting in extremely high rates of surface contraction on approach to the pinch point, initiating extension of the polymer chains within the pinch regions. This de-coupling phenomenon is peculiar to low-viscosity, elastic fluids as extension does not occur prior to the formation of the pinch points (i.e. just prior to break-up), as opposed to the high viscosity counterparts in which extension of polymers in solution may occur even during necking. Once steady-state extension of the polymers is achieved within the pinch at high extension rates, the thread undergoes elasto-capillary break-up as the capillarity again overcomes the viscoelastic forces. The final length at detachment and time-to-break-off (relative to the equivalent Newtonian fluid) is shown to be linearly proportional to the longest relaxation time of the fluid.

Introduction

The dynamics and associated mechanisms of drop formation and fluid rupture from a nozzle under the influence of gravity alone has seen much interest since the mid 1800s [1], [2], [3]. These early works established that as a fluid column begins to exit a nozzle, the speed of the growth and shape of the emerging drop are primarily controlled by the opposing forces of surface tension and gravity. While gravity causes the fluid to exit the nozzle, surface tension strives to attain a minimum surface area, which initially slows the fluid from exiting the nozzle. As gravity overcomes surface tension, the fluid emerges and surface tension forces cause the formation of a neck, from which a primary filament and drop evolve. Inertia obviously plays a major role in the development, elongation and breaking the drop. Even for highly viscous fluids, the effects of inertia cannot be ignored as the filament becomes thin and the drop approaches break-up [4], [5]. Break-up of the drop from a suspended column of fluid attached to the primary thread is a result of the propagation of surface oscillations, caused by previous drop detachment or other external sources, along the length of this thread (the classic Rayleigh instability phenomenon [3]). As the drop detaches and falls, surface tension creates a sphere to enclose the maximum volume of fluid in the minimum surface area.

In the early 1970s, with the development of ink-jet printing technology, researchers began investigating the dynamics of high-speed drop formation in more detail. Early mathematical models of continuous jet streams of differing fluid properties (viscosity, surface tension and density) were presented (e.g. Bruce [6]), and these ideas were also applied to drop-on-demand ink-jet technology. Focuses were on the drop-on-demand production of drops of consistent volume and velocity, coping with the problem of small satellite droplets which often follow the primary drop, and understanding the effects of variables such as fluid viscosity and density, nozzle diameter and the thickness of the nozzle wall (e.g. Young [7]).

Peregrine et al. [8] re-sparked significant interest in this field with their qualitative discussion and pictures of drop formation with low-viscosity Newtonian fluids up to and past the ‘pinch’ point. The ‘pinch’ point is reached after a period of rapid necking of the drop downstream of the nozzle, and marked by the formation of a secondary thread or microthread, far thinner than the primary thread (formed from the necked fluid), just prior to the final break-up event. Shi et al. [9] provided a computational analysis of what Peregrine et al. [8] had discussed, and compared it with further experimental work. It was found that after the initial pinch point or secondary thread formation, this secondary liquid thread could lead to a series of smaller threads with still thinner diameters [9].

Shi et al. [9], Zhang and Basaran [10] and Wilkes et al. [11] studied the effect of viscosity on the drop formation and break-up of Newtonian fluids. Zhang and Basaran [10] observed the thread diameter and elongation length of the forming drop, and the volumes of primary and satellite droplets formed. The effects of nozzle geometry, liquid flowrate, viscosity and surfactants were systematically investigated. In comparing water with an 85% glycerol solution, Zhang and Basaran [10] found that increasing viscosity increased the maximum length of the primary thread prior to break-up. This agrees with much earlier experimental reports by Narasinga Rao et al. [12] and Edgerton et al. [13], and with the finite element computations of Wilkes et al. [11]. It is now well understood that viscosity gives rise to a viscous pressure in the thinning filament, which opposes the capillary pressure, dampening surface oscillations and, hence, increasing the lifetime (and length) of the primary thread [10], [14]. Additionally, Brenner et al. [15] found that viscosity has a significant effect on the shape of the fluid interface at the point of attachment between the drop and the filament, immediately before rupture.

As mentioned earlier, surface tension plays a considerable role in the drop formation process. Zhang and Basaran [10] found that increasing the surface tension caused a drop to appear more spherical during formation, and increased the volume of the drop and the length of the drop at detachment. However, Badie and de Lange [16] report that surface tension has little effect on the volume of the primary drop from a drop-on-demand ink-jet. The liquid flowrate and the nozzle geometry are also known to affect drop formation, with larger drops being formed from larger nozzles and at higher flowrates.

Henderson et al. [17] found that there are two distinct pinch methods. Between the nozzle fluid and the primary thread, the thread gradually necks to form an upper pinch. However, between the primary thread and the drop, there is not gradual necking, but rather the distinct rapid formation of the lower pinch via a secondary or microthread, as was seen by Peregrine et al. [8] and Shi et al. [9]. Both threads are subject to Rayleigh instabilities and break in multiple locations, forming small secondary satellite droplets in most cases. The pinch point between the primary thread and drop usually breaks first, followed swiftly by break-up at the upper pinch point, which is enhanced by the capillary waves propagating up the primary thread to the nozzle fluid; although the reverse has also been observed [9].

Numerous studies have provided simplified Navier–Stokes equations for incompressible flow with a free surface in an effort to describe drop formation [18], [19], [20], [21]. The effects of fluid flow within [21] and surrounding [22] the forming drop have also been considered. While more recently, the solutions of the complete Navier–Stokes equations to describe drop formation have been given by Wilkes et al. [11] and Gueyffier et al. [23]. The finite element calculations of Wilkes et al. [11] were able to capture both the gross features of the drop formation, including the limiting length of a drop at break-up, and its fine features such as the secondary thread that forms from the main thread at certain fluid conditions.

For drop formation from a nozzle under gravity of viscoelastic fluids, the recent experimental results of Amarouchene et al. [24] are very relevant. Although their work looked at low-viscosity, elastic fluids similar to those used in this study, they only considered the behaviour of very high molecular weight polymers (4 and 8×106 g/mol) in solution. Unfortunately, they also did not give any indication of the magnitude of the relaxation time scale of their solutions. In addition, although not quantified by these workers, such high molecular weights would generally result in the solutions being shear-thinning to different extents. In contrast to the solutions of Amarouchene et al. [24], the fluids used here are a set of well characterised and matched constant low-viscosity, elastic fluids [25], of equivalent density, surface tension and constant shear viscosity, with the only difference being the addition of different molecular weight polymers at dilute concentrations. The result of such well designed solutions is a set of fluids which vary significantly only in their elastic properties and, hence, relaxation times. These elastic properties and relaxation times are also quantified independently and related to the associated drop formation process. Thus, the current paper provides a detailed insight into the effects of varying degrees of elasticity on the dynamics of drop development, elongation and break-up in low-viscosity solutions. The variation in drop length and minimum drop radius is investigated, and the similarities and differences amongst the various solutions compared. It will be shown that the length at detachment and time-to-break-off of the fluid thread can be correlated with the relaxation time. Finally, the dynamics of the drop evolution up to break-up of the solutions with differing elasticity is discussed and the mechanism that helps sustain the drop from break-up is proposed.

Section snippets

Newtonian fluids

Experimental observations by previous investigators utilising Newtonian fluids, e.g. Shi et al. [9], Kowalewski [26], have shown that free-surface shapes of a given fluid are very similar when approaching the pinch point (i.e. within the pinch region), totally independent of the initial conditions, such as the nozzle radius [5]. The dynamics of break-up are characteristic of the non-linear properties of the equations of motion, and as the flow near the break-up point accelerates, only the fluid

Image capture and measurement of drop formation dynamics

Images of drop formation were taken using a high-speed drum camera (Fig. 1), as used by Crooks and Boger [51] and Crooks et al. [52] to study drop impact on dry surfaces. Fluid was fed from a reservoir via a capillary tube to a nozzle of 4.0 mm o.d., 2.0 mm i.d. Prior to image capture, several drops of fluid were allowed to form to ensure a constant liquid flowrate.

The drum camera consists of a NIKON F66 camera, fitted with an AF Micro Nikkor 105 mm lens. Film, which was attached to the perimeter

Results and discussion

Fig. 5a–d show the time evolution of the drop formation experiments for the 50% glycerol–water solution, the 300 000, 600 000 and the 1 000 000 molecular weight PEO solutions. The time scale shown is the “relative time” or time-to-break-off (tdt), which is the time difference between the point at which the extending drop detaches or undergoes break-off from the nozzle fluid or fluid thread, the detachment time, td, and the actual time t of interest. Hence, time-to-break-off (tdt) is large at the

Conclusion

The effect of fluid elasticity on drop dynamics during drop formation at a nozzle under gravity has been investigated using a set of constant low-viscosity, elastic fluids and compared directly with an equivalent Newtonian glycerol–water solution. All solutions have the same shear viscosity, equilibrium surface tension, and density, but differ greatly in elasticity. It was found that the early stages of drop formation, i.e. necking, were the same for all solutions, regardless of the elasticity,

Acknowledgements

The non-Newtonian fluid mechanics program at the University of Melbourne has been funded by a Special Investigator Grant and in part by the Particulate Fluids Processing Centre. We would like to thank the reviewers for their insightful and detailed comments.

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