Aero-breakup of elastic liquids as a competition of aerodynamics and elasticity

Constitutive dimensionless numbers:

Elasticity number E=gv2/G

Breakup Criterion: f(E, De)=?

Deborah number De=(g/l)v/d

Formation of high strain rates in the liquids

• Water gun accelerates liquid up to velocity of few hundreds meters per second

• Liquid jet impacts a small disk- or cone-like target, forming liquid sheet

Formation of high strain rates in the liquids

• In the liquid sheet the liquid element is subjected to extension =r/ro~100 during few milliseconds

• Elongational strain rate is of order of v0/r0~104 s-1

2r(t)

1 cm____

Jet: v0=18 m/sro=1 mm

Creation of high strain rates in the liquids

• In the liquid sheet the liquid element is subjected to extension =r/ro~100 during few milliseconds

• Elongational strain rate is of order of v0/r0~104 s-1

r(t)

1 cm____

Jet: v0=18 m/sro=1 mm

Dynamics of the liquid sheet for elastic liquids

• Equation of motion: d2r/dt2=-/r

• Elastic liquid: =G(r/r0)2

• Rim trajectory: d2r/dt2=-Gr/r02

• Initial conditions: t=0, r=0, dr/dt=v0:

• Solution: r=v0(G/r02)-1/2sin((G/r0

2)1/2t)

2r(t)

Best fitting for PEO M=4m, c=10k ppm

0 1 2 3 4 50

20

40

60

80

100

120

140

v0=119.82543+-6.29914

G=150.73159+-276.84532

4m10k d0=1mm

d35.01mm

d, m

m

t, ms

0 1 2 3 4 50

20

40

60

80

100

120

140

v0=26.43246+-0.28981 m/s

G=55.67196+-9.03244 Pa

V0=27.85047+-0.29021 m/s

G=270.21323+-25.76959 Pa

V0=11.90995+-0.18646 m/s

G=139.11487+-5.05467 Pa

4m10k d0=2mm datm03mm d32mm d30.02mm

d, m

m

t, ms

Best fitting for PEO M=4m, c=40k ppm

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

10

20

30

40

50

60

70

80

90

v0=27.63564+-0.79777 m/s

G=336.58413+-22.35137 Pa

v0=27.30765+-0.58619 m/s

G=168.55215+-6.69943 Pa

v0=21.28694+-0.86056 m/s

G=2126.10619+-121.05046 Pa

4m40k d0=1mm

datm01mm datm02mm d35.01mm

d, m

m

t, ms

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

10

20

30

40

50

60

70

80

90

v0=18.46431+-0.22804 m/s

G=938.60894+-23.73136 Pa

v0=22.94689+-0.58632 m/s

G=2129.46892+-105.86929 Pa

v0=12.0342+-0.41168 m/s

G=955.65238+-67.67377 Pa

v0=17.94101+-0.12605 m/s

G=588.1234+-10.47192 Pa

v0=22.70478+-0.38704 m/s

G=292.7273+-10.8891 Pa

4m40k d0=2mm datm01mm datm02mm d35.03mm d35.04mm d35.05mm

d, m

m

t, ms

Best fitting for PEO M=4m, c=100k ppm

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

5

10

15

20

25

30

35

40

45

50

v0=31.17302+-1.76901 m/s

G=506.12227+-64.6102 Pa

v0=24.47811+-0.48776 m/s

G=1892.82301+-46.03937 Pa

4m100k d0=1 mm datm01mm d35.01mm

d, m

m

t, ms

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

5

10

15

20

25

30

35

40

45

50

V0=12.3286+-0.48698 m/s

G=7157.55237+-437.30952 Pa

V0=25.15419+-0.71695 m/s

G=8076.59352+-328.71657 Pa

4m100k d0=2mm datm00mm d35mm

d, m

m

t, ms

Elastic modulus as function of concentration at high strain rate

1000 10000 100000 10000001

10

100

1000

10000

Entov et al (1988)

Polyethylene oxide, M=4 m

G=10̂ (-3.08)*c^1.293

G,

Pa

c, ppm

The bell shapes in the vacuum shall be used for estimation of the liquid relaxation time

3.8 %PSBMA-in-TBP d0=2 mm

The bell shapes in the vacuum shall be used for estimation of the liquid relaxation time

PEO M=4m c=10k ppm d0=2 mm

Used to matchPSBMA

The effect of air resistance

PEO M=4m c=10k ppm d0=2 mm

Conclusions

• Modeling of the aero-breakup of the viscoelastic liquids requires knowledge of the liquid rheological parameters

• Liquid rheological parameters strongly depend on strain rate of the liquid

• Liquid sheets and liquid bells can be used as sources of high strain rate

• Rheological parameters can be estimated by means of best fitting analysis of data of high speed video-monitoring of sheet and bell liquid flows