Drop Pinch-Off for Drop Pinch-Off for Discrete Flows from a Discrete Flows from a

CapillaryCapillaryFrank Bierbrauer

School of Computing, Mathematics and Digital Technology, Manchester Metropolitan University

Lancashire, M1 5GD, UK

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ContentsContents

1.1. Continuous FlowsContinuous Flows2.2. Discrete FlowsDiscrete Flows3.3. Critical Drop EjectionCritical Drop Ejection4.4. PhenomenologyPhenomenology

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1. Continuous Flows1. Continuous Flows

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Dripping from a CapillaryDripping 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 does not vary with time Typical dripping behaviour from a capillary

with constant internal pipe pressure gradient

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Continuous Pipe FlowContinuous Pipe Flowincompressible Newtonian fluidsincompressible 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

Δz

Δp

4μ

RU ,rR

R

Uru 0, u

2

max22

2max

zr

2

max2maxmax , ,

RUWe

gR

UFr

RURe

σ

μUCa,

σ

ρgRBo,

ρRσ

μOh max

2

Characteristic drop ejection parameters

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Continuous Dripping ModesContinuous Dripping ModesOperability DiagramOperability Diagram

• 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

Bo = 0.3

H.J. Subramani, H.-K. Yeoh, R. Suryo, Q. Xu, B. Ambravaneswaran, O. Basaran, Phys. Fluids, 18 (2006), 032106-1032106-13. 6

Static Drop ReleaseStatic Drop Release

• Harkins-Brown Analysis* – Very slow inflation of the pendant drop– Only the buoyancy FB = VFSg and surface tension force

FS = 2R 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– 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

3

4

6

2

g

RHr

g

RHV dd

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*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.

Free Surface Flow ModelFree 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– : surface curvature of the free surface– n : the unit normal vector to the surface

We

κ

Re

1p

0FrRe

1p

t

I

T

2

nuuIn

uj

uuuu

,

*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.

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Code Validation ICode Validation I

• s

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Code Validation IICode Validation II

• o

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2. Discrete Flows2. Discrete Flows

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Discrete Pipe Flow 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:

0tif1

0tif0tH

D

D22

20

D22

2max

z

Ttif0

TtifrRR

UtTHrR

R

Utr,u

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Comparison of FlowsComparison 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-off tp

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3. Critical Drop Ejection3. Critical Drop Ejection

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Stage 1: Building a Pendant DropStage 1: Building a Pendant Drop

• 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 Vpend for a known internal pipe velocity Umax

– That is: Tpend = 2Vpend/R2Umax

• Then allow the newly created pendant drop to relax to a quiescent state

t = Tpend

Umax

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Stage 2: Pendant Drop OscillationStage 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%

pendpend V3π8f

dpend τ1λ νf 2πVτ pend31

pendd

λt0eAtA

λ10lnT10%

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Stage 3: Ejecting a DropStage 3: Ejecting a DropIn 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

Dmax2R

0

T

0 D22

2max

R

0

T

0D

TUπR2

1drdttTHrR

R

U 2π

drdttr,u2πV

D

D

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Critical Drop EjectionCritical 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:• = 1000 kg/m3, = 0.024 Pas, = 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 = 2HR/g×mL• Choose Vpend = 13 L and see how this influences the time at which a

drop will be ejected as Umax is increased18

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

max20.1Ucrit 21.6eT

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Critical Injection and Pinch-Off VolumesCritical Injection and Pinch-Off Volumes• Corresponding to the critical injection time is the critical injection volume

and actual critical ejection volume: max20.1U

max2

injinjpendcrit eUR 21.6πVVVV ,

What we eventually expect to see is: Vinj = f(Vpend)R2Umax exp(-g(Vpend)Umax)20

Time to Pinch-Off 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

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Example Simulations Example Simulations

Umax = 0.1 m/s Umax = 0.6 m/s

Tinj = 1.306 ms, Vinj = 0.213 L Tinj = 0.0001 ms, Vinj = 0.0001 L

TPO = 194 ms TPO = 119 ms

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Characteristic Pinch-Off BehaviourCharacteristic Pinch-Off Behaviour

Factors•The size of the velocity jump U max

•The volume Vinj injected in time Tinj 23

4. Phenomenology4. Phenomenology

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Internal Drop Flow Internal Drop Flow Low and High Velocity CasesLow 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 = R2/• gravitational timescale: tgrav = (R/g)1/2

• capillary timescale: tcap = (R3/

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Capillary Wave Explanation ICapillary Wave Explanation I

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 = R2/ is much larger than the convective timescale e.g. For Umax = 0.1 m/s tconv = 10.2 ms, tvisc = 43.6 ms, for Umax = 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 = 0.1 m/s

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

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Capillary Wave Explanation II 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 = (R3/ ≈ 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

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Capillary Wave Explanation III 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

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Movement of the Centre-of-Mass (COM) of the Movement of the Centre-of-Mass (COM) of the Pendant DropPendant 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

msTmsT POPO 194120 1.06.0 LVLV injinj 2.00001.0 1.06.0

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Movement of the Limiting Length Movement of the Limiting Length LLdd

U = 0.1 m/s•the length increases almost linearly with an almost constant velocityU = 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 ms although 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

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Further WorkFurther 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 L close to the static pinch-off volume of 14 L

– 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 L?• 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?

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Umax = 0.4 m/s Umax = 0.3 m/s Tinj = 0.0204 ms, Vinj = 0.013 L Tinj = 0.068 ms, Vinj = 0.033 LTPO = 203 ms TPO = 193 ms

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U (m/s) T1 (non-dim) T1 (ms) V1 (mL) TPO (ms)

0.05 0.71610 14.608 1.1936 265

0.10 0.12800 1.3056 0.2133 194

0.15 0.06360 0.4324 0.1060 134

0.20 0.05403 0.2757 0.0901 192

0.25 0.04050 0.1652 0.0675 201

0.30 0.02000 0.0680 0.0333 193

0.35 0.02223 0.0647 0.0370 137

0.40 0.00800 0.0204 0.0133 203

0.45 0.00060 0.00136 0.0010 152

0.50 0.00050 0.00102 0.000833 118

0.55 0.00010 0.000185 0.000166 112

0.60 0.00005 0.000085 0.000085 119

Table 3.1: Table of Critical Ejection Times T1 , volumes V1 and associated

pinch off time TPO