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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
1
Stability of liquid jets
immersed in another liquid
Univ.

Prof. Dr. Günter Brenn
Ass.

Prof. Dr. Helfried Steiner
Part of the CONEX project
„Emulsions with Nanoparticles for New Materials“
Conex mid

term meeting, Oct. 28 to 30
2004, Warsaw
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
2
Contents
•
Introduction
–
break

up of submerged jets in
emulsification
•
Description of jet dynamics
•
Linear stability analysis by Tomotika
•
Dispersion relation
•
Limitations to the applicability of the relation
•
Further work in the project
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
3
Introduction
–
Jet instability and break

up in another viscous liquid
Modes of drop formation
Dripping
Jetting
Transition dripping
–
jetting Transition
nomogram
jet drip jet drip
v
cont
= 0.39 m/s 0.36 m/s 0.49 m/s 0.46 m/s
v
disp
= 0.18 m/s 0.03 m/s
C. Cramer, P. Fischer, E.J. Windhab:
Drop formation in a co

flowing ambient fluid.
Chem. Eng. Sci. 59 (2004), 3045

3058.
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
4
Description of jet dynamics
Basic equations of motion
(u
–
r

velocity, w
–
z

velocity)
Continuity
r

momentum
z

momentum
Definition of stream function
For solution introduce the disturbance stream function to satisfy continuity
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
5
Elimination of pressure and linearization
with the differential operator
Eliminating the pressure from the
momentum equations yields
Linearization: neglect products of velocities and products
of velocities and their derivatives
Final equation for the stream function reads
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
6
Solutions of the differential equation
This differential equation is satisfied by functions
1
and
2
which are solutions of the two following equations
We make the ansatz for wavelike solutions of the form
and obtain the amplitude functions
where l
2
=k
2
+i
/
General solution of the linearised equation
where i = 1, 2
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
7
Inner and outer solutions and boundary conditions
where l
´
2
= k
2
+i
/
´
,
where l
2
= k
2
+i
/
,
Inner and outer solutions are specified from the general solution
by excluding Bessel functions diverging for r
→0 and for r→
, respectively
Boundary conditions
Continuity of tangential stress
Jump of radial stress by surface tension
where
Velocities at the interface equal in the two sub

systems
u
´

r=a
= u
r=a
w
´

r=a
= w
r=a
inner
outer
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
8
Determinantal dispersion relation from boundary conditions
The boundary conditions lead to the following dispersion relation
with the functions F
1
through F
4
reading
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
9
Specialisation for low inertial effects
Dispersion relation for neglected densities
and
´
with the functions G
1
, G
2
, and G
4
reading
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
10
Graph of special dispersion relation for low inertia
Dispersion relation for low inertia and
´
/
=0.91 (Taylor)
Consequences
•
Wavelength for maximum wave growth
is
= 5.53
2a, since ka
opt
= 0.568.
•
Drop size is D
d
=2.024
2a.
•
Cut

off wavelength unchanged against
the Rayleigh case of jet with
´
=0 in a
vacuum.
The dispersion relation is
where
S. Tomotika: On the instability of a cylindrical
thread of a viscous liquid surrounded by
another viscous fluid.
Proc. R. Soc. London A 150 (1935), 322

337.
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
11
Comparison with Taylor’s experiment
Flow situation: jet of lubricating oil in syrup
Dynamic viscosity ratio
´
/
=0.91
Calculation of the function (1

x
2
)
(x) yields the
maximum at ka = ka
opt
= 0.568
Measurements on photographs by Taylor yield
a = 0.272 mm,
= 3.452 mm
→ ka = 0.495
Deviation of

13%
→ Tomotika claims satisfactory agreement
Oil
Syrup
Syrup
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
12
Problems with applications of the Tomotika results
•
Undisturbed relative motion of the two fluids not accounted for
•
Most results derived from Tomotika in the literature without inertia
Dispersion relation with relative motion of jet in an inviscid host medium
C. Weber: Zum Zerfall eines
Flüssigkeitsstrahles.
ZAMM 11 (1931), 136

154.
•
Inviscid host medium allows for top

hat velocity profiles
•
Analytical derivation of dispersion relation is therefore possible
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
13
Remedy
–
dispersion relation from generalized approach
Continuity
r

momentum
z

momentum
Introduce into conservation the equations
the correct disturbance approaches u = U + u
´
and w = W + w
´
with the quantities U and W of the undisturbed coaxial flow of a jet in
its host medium,
cancel terms of the undisturbed flow and neglect
small quantities of
higher order.
→
This leads again to a linearization of the momentum equations
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
14
Conservation equations for the disturbances
Continuity
r

momentum
z

momentum
Procedure for the calculation
–
further work in the project
•
Calculate U (r,z) and W(r,z) for the undisturbed flow in both fluids (possibly
using a similarity approach ?)
•
Eliminate the pressure disturbance from the above momentum equations
•
Introduce stream function of the disturbance in a wavelike form
The disturbance approach with non

parallel flow (U
≠
0) yields
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Institute of Fluid Mechanics and Heat Transfer
Conex Mid

Term Meeting, Warsaw, October 2004
15
Summary, conclusions and further work
•
Instability of jets in another liquid is described by a
determinantal dispersion relation
•
Maximum wave growth rate at ka
≈ 0.57 for viscosity
ratio close to one (Taylor’s experiment)
•
Limiting case of vanishing outer viscosity (Rayleigh,
1892) is contained in the solution
•
Cut

off wave number for instability remains unchanged
against the Rayleigh (1879) case of an inviscid jet in a
vacuum
•
Further work should lead to a description of jet instability
with relative motion against the host medium. This will
increase the value of the cut

off wave number
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