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

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2 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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3 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October Introduction – Jet instability and break-up in another viscous liquid Modes of drop formation DrippingJettingTransition 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),

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4 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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5 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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6 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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7 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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8 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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9 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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10 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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 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),

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11 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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 = Measurements on photographs by Taylor yield a = mm, = mm → ka = Deviation of -13% → Tomotika claims satisfactory agreement Oil Syrup

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

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13 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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14 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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15 Institute of Fluid Mechanics and Heat Transfer Conex Mid-Term Meeting, Warsaw, October 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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