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Drop Pinch-Off for Discrete Flows from a Capillary

Drop Pinch-Off for Discrete Flows from a Capillary. Frank Bierbrauer School of Computing, Mathematics and Digital Technology, Manchester Metropolitan University Lancashire, M1 5GD, UK. Contents. Continuous Flows Discrete Flows Critical Drop Ejection Phenomenology. 1. Continuous Flows.

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Drop Pinch-Off for Discrete Flows from a Capillary

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  1. Drop Pinch-Off for Discrete Flows from a Capillary Frank Bierbrauer School of Computing, Mathematics and Digital Technology, Manchester Metropolitan University Lancashire, M1 5GD, UK

  2. Contents • Continuous Flows • Discrete Flows • Critical Drop Ejection • Phenomenology

  3. 1. Continuous Flows

  4. Dripping from a Capillary • This means that the driving pressure gradient is constant • This means the velocity at inflow is typically Poiseuille pipe flow • The inflow doesnot vary with time Typical dripping behaviour from a capillary with constant internal pipe pressure gradient

  5. Continuous Pipe Flowincompressible Newtonian fluids • Nozzle inflow (Axisymmetric Poiseuille flow) • Fluid parameters • Reynolds no. : inertial to viscous forces • Froude no. : inertial to gravitational forces • Weber no. : inertial to surface forces • Alternatively • Ohnesorge no. : viscous to inertial and surface forces • Bond no. : gravitational to surface forces • Capillary no. : viscous to surface forces Characteristic drop ejection parameters

  6. Continuous Dripping ModesOperability Diagram Bo = 0.3 • P1 • characterised by periodic dripping whereby a drop is expelled at equal times and equal pinch-off lengths • P1S • as for P1 except for the expulsion of a satellite droplet as part of the primary drop pinch-off • CD • complex dripping characterised by drop expulsion at varying periods, e.g. P2 dripping, period doubling occurs so that two successive drops are expelled with two separate pinch-off lengths and periods H.J. Subramani, H.-K. Yeoh, R. Suryo, Q. Xu, B. Ambravaneswaran, O. Basaran, Phys. Fluids, 18 (2006), 032106-1032106-13.

  7. Static Drop Release • Harkins-Brown Analysis* • Very slow inflation of the pendant drop • Only the buoyancy FB = VFSgDrand surface tension force FS= 2psR are assumed to be acting • VFS : is the total liquid volume attached to nozzle tip at the instant at which the net force on the pendant drop equals zero • Vd : is the volume of the drop formed and rd its radius • g: is the acceleration due to gravity • Dr :the density difference between the drop liquid and the ambient gas • H : the Harkins-Brown factor which corrects for the fraction of liquid volume which remains attached to the nozzle after drop break off *Scheele, G.F., Meister, B.J., Drop Formation at Low Velocities in Liquid-Liquid Systems: Part 1 – prediction of drop volume, AIChE J. (1968) 14, 9-15.

  8. Free Surface Flow Model • Solve the scaled, time dependent, viscous incompressible Navier-Stokes equations • A free-surface finite element flow code developed in the School of Mathematics at the University of Leeds, UK by Oliver Harlen* and Neil Morrison, modified by the author • I : represents the free surface or interface • k : surface curvature of the free surface • n : the unit normal vector to the surface *Harlen, O.G., Rallison, J.M., Szabo, P., A Split Lagrangian-Eulerian Method for Simulating Transient Viscoelastic Flows, J. non-Newtonian Fluid. Mech., 60 (1995), 81-104.

  9. Code Validation I • s

  10. Code Validation II • o

  11. 2. Discrete Flows

  12. Discrete Pipe Flow • Discrete Pipe Flow • Time dependent pressure gradient within the pipe, so that: uz(r,t) = uz(r)ut(t) • Where ur is Poiseuille flow and ut the time dependent aspect • This is a new area with little experimental or computational data • There are no operability diagrams for this case • Pulsed, Step-wise Flow • which suffers a rapid, time dependent, change in pipe flow velocity • Discontinuous flows • Defined through a Heaviside function: • Approximately discontinuous flows with very rapid velocity changes • The velocity uz(r,t) is now given in terms of a discrete (D) flow pulse:

  13. Comparison of Flows • Continuous pipe flow • Continuous Pinch-off time tp • The time after the flow is turned on at which a continuous flow pinches off • Discrete pipe flow • Discrete flow timescale TD • The total amount of time that fluid is being injected into the capillary • TD > tp:discrete flow timescale ends before natural pinch-off can occur • TD < tp:discrete flow timescale exceeds the natural pinch-off timescale necking pinch-offtp

  14. 3. Critical Drop Ejection

  15. Stage 1: Building a Pendant Drop t = Tpend • Build a pendant drop at the end of the pipe exit nozzle • For a given zero pendant drop initial condition (see Figure) • Turn on the flow for a time Tpend to build a drop of desired volume Vpendfor a known internal pipe velocity Umax • That is: Tpend = 2Vpend/pR2Umax • Then allow the newly created pendant drop to relax to a quiescent state Umax

  16. Stage 2: Pendant Drop Oscillation • It is known that a pendant drop oscillates with a frequency equal to • This oscillation dies out with a decay rate given by • The amplitude decay given in the form means that the amplitude will decay to 10% of its value in the time • This tells us for how long we must allow the drop to oscillate in order to reach a state where its oscillation has been reduced by 90% • The total time needed is Tpend + T10%

  17. Stage 3: Ejecting a Drop In practice • There is an existent pendant drop of volume Vpend already at the pipe exit nozzle (stages 1 &2) • To this is added a finite volume VD determined by the time TD the discrete flow is turned on for • Volume conservation implies: Vpend + VD = Vdrop + Vres + (Vsat) • This discrete volume VD may be calculated from standard pipe flow turned on for a time TD

  18. Critical Drop Ejection • Critical Injection Time • One way to study this process is to determine the time Tinj that fluid has to be injected into the capillary in order for a drop to be ejected for a given internal pipe velocity Umax • Or equivalently, the injection volume Vinj injected into the pipe during time Tinj to eject a drop for a given internal pipe flow velocity Umax • We compare this with the Harkins-Brown approach • Choosing Vpend • We start with a known value of Vpend which is already close to the static pinch-off volume • We use a fluid with the following physical properties: • r = 1000 kg/m3, m = 0.024 Pas, s = 0.044 N/m, R = 1.02 mm • For this set of fluid parameters and a Harkins-Brown factor H = 0.5 we get Vstatic = 2HpsR/gDr = 1.44 × 10-8m3 = 14.4 mL • Choose Vpend = 13 mL and see how this influences the time at which a drop will be ejected as Umax is increased

  19. Critical Injection Time We Find • an exponential decrease in the critical injection time required as the internal pipe velocity increases, i.e.

  20. Critical Injection and Pinch-Off Volumes • Corresponding to the critical injection time is the critical injection volume and actual critical ejection volume: What we eventually expect to see is: Vinj = f(r,s,m,Vpend)pR2Umax exp(-g(r,s,m,Vpend)Umax)

  21. Time to Pinch-Off Time to Pinch-Off (PO) • This is the total time needed for the drop to be ejected after the end of Tcrit • This critical pinch-off time can be a good deal longer than the critical drop injection time since after the fluid has been injected it acts to create an instability which eventually ejects a drop • We expect that as Umax decreases the drop takes longer to be ejected • Note the increase in TPO for very small Umax

  22. Example Simulations Umax = 0.1 m/sUmax = 0.6 m/s Tinj = 1.306 ms, Vinj = 0.213 mL Tinj= 0.0001 ms, Vinj = 0.0001 mL TPO = 194 ms TPO = 119 ms

  23. Characteristic Pinch-Off Behaviour Factors • The size of the velocity jump U max • The volume Vinj injected in time Tinj

  24. 4. Phenomenology

  25. Internal Drop Flow Low and High Velocity Cases • The diagram shows how the internal drop flow for two different internal pipe velocities: U = 0.1 and 0.6 m/s, effect the drop formation process • U = 0.1 m/s: • the flow is initially free to move into the lower part of the pendant drop • The flow is mostly vertically down • U = 0.6 m/s: • the flow can no longer move freely into the lower part of the pendant drop. There is flow resistance • The flow is forced sideways thereby creating a bulge of fluid on either side of the pipe exit • In order to conserve mass this bulge draws fluid upwards towards the nozzle exit • This creates an unstable system and generates a capillary wave which attempts to lower the surface energy of the drop • Timescales • convective timescale: tconv = R/U • viscous timescale: tvisc = rR2/m • gravitational timescale: tgrav = (R/g)1/2 • capillary timescale: tcap = (rR3/s)1/2

  26. Capillary Wave Explanation I Umax = 0.1 m/s Explanation • The fluid attempts to leave the pipe exit nozzle and enter the pendant drop, the time during which fluid flows into the drop is the convective timescale tconv = R/U • If the drop velocity is high the inertia of the drop resists the imposed flow, this happens when the viscous timescale tvisc = rR2/mis much larger than the convective timescale e.g. For Umax= 0.1 m/s tconv = 10.2 ms, tvisc = 43.6 ms, forUmax= 0.6 m/s: tconv = 1.7 ms, • We see that for Umax= 0.1 m/s the convective and viscous timescales are of the same order of magnitude whereas for Umax= 0.6 m/s the convective timescale is an order of magnitude smaller than the viscous timescale Umax = 0.6 m/s t = 0 0.85 1.70 2.55 3.40 4.25 5.10 5.95 6.80 7.65 8.50 9.35 10.20 11.05 11.90 ms

  27. Capillary Wave Explanation II Explanation Continued (Umax= 0.6 m/s) • A capillary wave is only generated if the flow is forced sideways pushing fluid into bulges on either side of the top part of the pendant drop and pinned to the exit nozzle, t = 0.85 ms • Typically, the capillary wave travels on the boundary of the drop and dissipates on a timescale of tcap = (rR3/s)1/2 ≈ 5 ms • High velocity regions form on either side of the drop where the capillary wave is initiated • Fluid is transported by the capillary wave to the lower part of the pendant drop, a low velocity region (blue) persists at the lower part of the drop until t = 6.8 ms • The fluid transported by the capillary wave arrives at the bottom of the drop and replaces the low velocity region with a higher velocity region (yellow), t = 8.5 ms

  28. Capillary Wave Explanation III Explanation continued • This higher velocity region maintains the drop in its current shape instead of oscillating back to a more stable shape until about t = 10.2 ms • By this stage gravitational effects start to pull the drop down with a timescale of tgrav = (R/g)1/2= 10 ms • NB it is clear that for the largest values of Umax that only a very small injection time and volume is required. This means that the capillary wave generates enough of an instability to eject a drop by itself • The capillary wave explains how fluid is transported to the bottom of the drop as well as showing why high velocity flows pinch-off before low velocity ones even though the amount of fluid Vinj entering the drop is much smaller for the high velocity flow than the low velocity one

  29. Movement of the Centre-of-Mass (COM) of the Pendant Drop Centre-of-Mass • The velocity discontinuity clearly indicates the time at which drop ejection occurs so that . It is also clear that . • e.g. the position of the COM at t = 0.1 s for U = 0.6 m/s is 2.6 mm whereas for U = 0.1 m/s it is 2.3 mm

  30. Movement of the Limiting Length Ld U = 0.1 m/s • the length increases almost linearly with an almost constant velocity U = 0.6 m/s • the length increases slowly up to t = 3.6 ms, the bulge near the exit nozzle draws fluid upwards and decreases Ld between t:4-6 ms, after this period Ld increases between 6-12 msalthough the speed of increase slows down after t = 9 ms. • The velocity of the endpoint is much higher for the U = 0.6 m/s case than for the U = 0.1 m/s case reaching a peak at t = 8.8 ms • A high velocity region is created at this endpoint but dies out after t = 8.8 ms

  31. Further Work • Remark • This is preliminary research and requires more research, many questions still to be answered • The work so far has used a Vpend = 13 mL close to the static pinch-off volume of 14 mL • A capillary wave forms only because of the sudden change in pipe injection velocity from a low to a high velocity • Questions • How does the critical pinch-off phenomenology change when Vpend is not close to the static pinch-off volume, e.g. Vpend = 5 mL? • does a capillary wave still form and act to transport fluid? • In reality it takes time for a flow to change from zero to a given velocity in incompressible flow this has an associated impulsive timescale • Does this mean that a capillary wave will not form? • A capillary wave will still form provided the impulsive timescale is less than the viscous timescale • Are these capillary waves experimentally observable? • Aim to obtain experimental evidence over the next couple of months • What happens when the flow is injected for longer than the critical pinch-off time? • If so does the flow behave as the continuous case?

  32. Umax = 0.4 m/sUmax = 0.3 m/s Tinj = 0.0204 ms, Vinj = 0.013 mL Tinj = 0.068 ms, Vinj = 0.033 mL TPO = 203 ms TPO = 193 ms

  33. Table 3.1: Table of Critical Ejection Times T1 , volumes V1 and associated pinch off time TPO

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