1. 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 , Warsaw

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. 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 = m/s m/s m/s m/s vdisp = m/s m/s
C. Cramer, P. Fischer, E.J. Windhab: Drop formation in a co-flowing ambient fluid. Chem. Eng. Sci. 59 (2004),

4. 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

5. Elimination of pressure and linearization
Eliminating the pressure from the momentum 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

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
where i = 1, 2
and obtain the amplitude functions
where l2=k2+i/
General solution of the linearised equation

7. Inner and outer solutions and boundary conditions
Inner and outer solutions are specified from the general solution by excluding Bessel functions diverging for r→0 and for r→, respectively
where l´2 = k2+i/´,
inner
outer
where l2 = k2+i/,
Boundary conditions
u´|r=a = u|r=a
Velocities at the interface equal in the two sub-systems
Continuity of tangential stress
w´|r=a = w|r=a
Jump of radial stress by surface tension
where

8. Determinantal dispersion relation from boundary conditions
The boundary conditions lead to the following dispersion relation
with the functions F1 through F4 reading

9. Specialisation for low inertial effects
Dispersion relation for neglected densities  and ´
with the functions G1, G2, and G4 reading

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 =
Drop size is Dd=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),

11. Comparison with Taylor’s 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 = ka|opt = 0.568
Measurements on photographs by Taylor yield a = mm,  = mm → ka = 0.495
Deviation of -13% → Tomotika claims satisfactory agreement

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),
Inviscid host medium allows for top-hat velocity profiles
Analytical derivation of dispersion relation is therefore possible

13. 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

14. 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

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