Drop Pinch-Off for Discrete Flows from a Capillary

ESAIM: PROCEEDINGS, Juillet 2013, Vol. 40, p. 16-33 C. Bourdarias, S. Gerbi, Editors DROP PINCH-OFF FOR DISCRETE FLOWS FROM A CAPILLARY ∗, ∗∗ F. Bierbrauer 1 , N. Kapur 2 and M.C.T. Wilson 2 Abstract. The problem of drop formation and pinch-off from a capillary tube under the influence of gravity has been extensively studied when the internal capillary pressure gradient is constant. This ensures a continuous time independent flow field inside the capillary tube typically of the Poiseuille flow type. Characteristic drop ejection behaviour includes: periodic drop ejection, drop ejection with associated satellite production, complex dripping, chaotic behaviour and jetting. It is well known that this characteristic behaviour is governed by the Weber (We) and Ohnesorge (Oh) numbers (for a given Bond number) and may be delineated in a We verses Oh operability diagram. An in-depth physical understanding of drop ejection is also of great importance to industry where the tight control of drop size and ejection velocity are of critical importance in industrial processes such as sealants used in electronics assembly and inkjet printing. However, the use of such a continuous flow approach for drop ejection in industry is often impractical since such flows cannot be operator controlled. For this reason it is important to investigate so-called discrete pipe flows where the flow can be turned on and off at will. This means the flow inside the pipe is now time-dependent being controlled in a step-wise fashion. As a first stage in the investigation of drop pinch-off behaviour in discrete pipe flows this paper will study the critical pinch-off time required for drop ejection starting from a pendant drop. This is the discrete amount of time the pipe flow is turned on for in order for a drop to be ejected from the capillary. A Newtonian incompressible free-surface CFD flow code developed at the University of Leeds is used to investigate the critical pinch-off time for a range of internal pipe velocities (the central flow maximum in Poiseuille flow). It is found that the time required for drop ejection to occur decreases exponentially with internal pipe velocity. These characteristic times are also far smaller than typical static drop release times expected from Harkins and Brown analyses. The phenomenology of the process is due to the creation of a capillary wave at the pipe exit upon the sudden turning on of the flow inside the pipe. The capillary wave acts to transport fluid from the upper part of the forming pendant drop at the end of the capillary to the lower part of the drop both lowering the pendant drop centre-of-mass and thinning the neck region connecting the drop to the pipe. This allows the drop to be pinched off at an earlier than expected time as compared to static drop release times. 1. Introduction The ejection of drops from a capillary plays an important role in engineering and industry. Typical examples include spray painting of car door panels, water-spray cooling in the steel industry, sealants used in electronics ∗ The authors wish to thank the Technology Strategy Board for funding under Grant no. TS/H001166/1 ∗∗ The authors appreciate the use of the finite element free surface code developed by Oliver Harlen of the School of Mathematics, University of Leeds 1 School of Computing, Mathematics and Digital Technology, Manchester Metropolitan University, Manchester, M1 5GD, UK 2 School of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, UK c EDP Sciences, SMAI 2013 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201340002 ESAIM: PROCEEDINGS 17 assembly and more recently in ink-jet printing. Such systems are constituted by a capillary or spray nozzle attached to a pipe which provides a given liquid under pressure. The fluid is forced through the pipe so that, at the open end, a pendant drop starts to form (the drop formation stage); this grows until the forces at work detach the drop whether through drop ejection due to inertia or drop release due to gravitational forces (the drop ejection stage). The drop formation stage involves the growth of a pendant drop at the end of the capillary which forms with a moving interface or free surface. For the simplified case of free-surface flows where the external environment outside of the drop liquid is assumed to be a passive gas at constant pressure (or of density significantly less than the drop fluid) we have the situation shown in Figure 1. In this paper we study the process of ejecting a drop of incompressible Newtonian Figure 1. Drop ejection parameters and fluid dynamical scales. liquid, of density ρ, viscosity µ and surface tension σ, from a capillary tube under the influence of gravity. As seen in Figure 1 the capillary is oriented vertically down in the same direction as the force of gravity Fg . The capillary tube itself acts to define a natural set of flow parameters including the tube radius R, the central flow velocity U within the pipe as well as the flow rate Q. The developed flow itself is assumed axisymmetric and the typical flow field within the capillary may be expressed in cylindrical polar coordinates (r, z). The flow is generally axisymmetric so that the flow velocity components (ur , uz ), radial and axial components, within the pipe are given by   U R2 ∆p ur = 0, uz (r) = 2 (R2 − r2 ), U =− (1) R 4µ ∆z This is the well known Poiseuille flow in a pipe with the velocity scale U defined in terms of the pressure gradient within the pipe. This pipe flow velocity scale is also the maximum velocity along the symmetry line of the flow. The pressure gradient acts to “push” the fluid out of the pipe. Fluid within the pipe is acted on by various forces including inertia, the pressure gradient, gravity (with acceleration g = 9.81 m/s2 ), internal frictional forces or viscosity and, once a free surface has formed at the end of the pipe, the surface tension force which acts on the fluid boundary to lower surface energy and restore drop equilibrium. The physical scales of the flow are then the length scale R, velocity scale U , advective time scale Tadv = R/U as well as a natural pressure scale ρU 2 . These scales typically characterise the non-dimensional fluid parameters such as the Weber number W e = ρRU 2 /σ (the ratio of inertial to surface tension forces), the Reynolds number Re = ρRU/µ (the ratio of inertial to viscous forces) and the Froude number F r = U 2 /gR (the ratio of inertial to gravitational √ √ forces). A related set of parameters can be defined through the Ohnesorge number Oh = W e/Re = µ/ ρRσ 18 ESAIM: PROCEEDINGS which measures the ratio of viscous to inertial and surface forces, the Bond number Bo = ρgR2 /σ (the ratio of gravitational to surface tension forces) and the Capillary number Ca = µU/σ(the ratio of viscous to surface tension forces). 1.1. Drop Formation from a Capillary The best known case o (...truncated)