Stability of liquid jets immersed in another liquid

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Stability of liquid jets immersed in another liquid Part of the CONEX project „Emulsions

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 11

Contents • Introduction – break-up of submerged jets in emulsification • Description of jet

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 22

Introduction – Jet instability and break-up in another viscous liquid Modes of drop formation

Introduction – Jet instability and break-up in another viscous liquid Modes of drop formation Dripping Jetting Transition dripping – jetting jet vcont = vdisp = drip 0. 39 m/s 0. 36 m/s 0. 18 m/s Institute of Fluid Mechanics and Heat Transfer jet Transition nomogram drip 0. 49 m/s 0. 46 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. Conex Mid-Term Meeting, Warsaw, October 2004 33

Description of jet dynamics Basic equations of motion (u – r-velocity, w – z-velocity)

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 44

Elimination of pressure and linearization Eliminating the pressure from the momentum equations yields with

Elimination of pressure and linearization Eliminating the pressure from the momentum equations yields with the differential operator Linearization: neglect products of velocities and their derivatives Final equation for the stream function reads Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 55

Solutions of the differential equation This differential equation is satisfied by functions 1 and

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 l 2=k 2+i / General solution of the linearised equation Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 66

Inner and outer solutions and boundary conditions Inner and outer solutions are specified from

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 inner where l´ 2 = k 2+i / ´, outer where l 2 = k 2+i / , Boundary conditions Velocities at the interface equal in the two sub-systems u´|r=a = u|r=a Continuity of tangential stress w´|r=a = w|r=a Jump of radial stress by surface tension where Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 77

Determinantal dispersion relation from boundary conditions The boundary conditions lead to the following dispersion

Determinantal dispersion relation from boundary conditions The boundary conditions lead to the following dispersion relation with the functions F 1 through F 4 reading Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 88

Specialisation for low inertial effects Dispersion relation for neglected densities and ´ with the

Specialisation for low inertial effects Dispersion relation for neglected densities and ´ with the functions G 1, G 2, and G 4 reading Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 99

Graph of special dispersion relation for low inertia Dispersion relation for low inertia and

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 2 a, since ka|opt = 0. 568. The dispersion relation is where • Drop size is Dd=2. 024 2 a. • Cut-off wavelength unchanged against the Rayleigh case of jet with ´=0 in a vacuum. 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. Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 1010

Comparison with Taylor’s experiment Flow situation: jet of lubricating oil in syrup Syrup Oil

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 -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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 1111

Problems with applications of the Tomotika results • Undisturbed relative motion of the two

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 1212

Remedy – dispersion relation from generalized approach Introduce into conservation the equations Continuity r-momentum

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 1313

Conservation equations for the disturbances The disturbance approach with non-parallel flow (U≠ 0) yields

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 1414

Summary, conclusions and further work • Instability of jets in another liquid is described

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 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 2004 1515