Stability of liquid jets immersed in another liquid

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Oct 24, 2013 (3 years and 10 months ago)

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1

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

2

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

3

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.

4

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

5

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

6

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

7

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

8

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

9

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

10

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.

11

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

12

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

13

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

14

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

15

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