Stability of liquid jets

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Stability of liquid jets immersed in another
liquid
Part of the CONEX project
Emulsions with Nanoparticles for New Materials
Univ.-Prof. Dr. Günter Brenn Ass.-Prof. Dr.
Helfried Steiner
Conex mid-term meeting, Oct. 28 to 30 2004, Warsaw
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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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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 vcont 0.39 m/s 0.36
m/s 0.49 m/s 0.46 m/svdisp
0.18 m/s 0.03 m/s
C. Cramer, P. Fischer, E.J. WindhabDrop
formation in a co-flowing ambient fluid. Chem.
Eng. Sci. 59 (2004), 3045-3058.
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Description of jet dynamics
Basic equations of motion(u r-velocity, w
z-velocity)
Continuity
r-momentum
z-momentum
For solution introduce the disturbance stream
function to satisfy continuity
Definition of stream function
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Elimination of pressure and linearization
Eliminating the pressure from themomentum
equations yields
with the differential operator
Linearization neglect products of velocities and
products of velocities and their derivatives
Final equation for the stream function reads
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Solutions of the differential equation
This differential equation is satisfied by
functions ?1 and ?2which are solutions of the
two following equations
We make the ansatz for wavelike solutions of the
form
where i 1, 2
and obtain the amplitude functions
where l2k2i?/?
General solution of the linearised equation
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Inner and outer solutions and boundary conditions
Inner and outer solutions are specified from the
general solutionby excluding Bessel functions
diverging for r?0 and for r??, respectively
where l2 k2i?/?,
inner
where l2 k2i?/?,
outer
Boundary conditions
ura ura
Velocities at the interface equal in the two
sub-systems
wra wra
Continuity of tangential stress
Jump of radial stress by surface tension
where
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Determinantal dispersion relation from boundary
conditions
The boundary conditions lead to the following
dispersion relation
with the functions F1 through F4 reading
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Specialisation for low inertial effects
Dispersion relation for neglected densities ? and
?
with the functions G1, G2, and G4 reading
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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 kaopt 0.568.
• Drop size is Dd2.024 ? 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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Comparison with Taylors experiment
Flow situation jet of lubricating oil in syrup
Syrup
Oil
Syrup
Dynamic viscosity ratio ?/?0.91
Calculation of the function (1-x2)?(x) yields the
maximum at ka kaopt 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
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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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Remedy dispersion relation from generalized
approach
Introduce into conservation the equations
Continuity
r-momentum
z-momentum
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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Conservation equations for the disturbances
The disturbance approach with non-parallel flow
(U?0) yields
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

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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 (Taylors
  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