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2024 | OriginalPaper | Chapter

3. Theoretical Methods

Author : José María Montanero

Published in: Tip Streaming of Simple and Complex Fluids

Publisher: Springer Nature Switzerland

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Abstract

This chapter briefly describes the theoretical approaches commonly used to gain insight into the tip streaming phenomenon. These approaches allow one to solve or at least obtain information from the governing equations presented in the previous chapter.
We discuss the main characteristics of direct numerical simulations and global and local stability analyses. The differences between these two last approaches are explained to emphasize the importance of the global stability analysis for tip streaming. Concepts such as global modes, asymptotic stability, and short-term response are also discussed.
The temporal and spatiotemporal local stability analyses provide information on the behavior of the jets emitted in the microjetting mode of the tip streaming configurations. This chapter closes with some results obtained from those analyses and directly related to those configurations, including the effects of electric fields, surfactants, and viscoelasticity.

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Literature
1.
Ferziger JH, Peric M (2013) Computational methods for fluid dynamics. Springer
2.
Popinet S (2003) Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J Comput Phys 190:572–600MathSciNetCrossRef
3.
Karniadakis G, Sherwin SJ (1999) Spectral/hp element methods for CFD. Oxford University Press
4.
Canuto C, Hussaini MY, Quarteroni A (2007) Spectral methods: fundamentals in single domains. Springer
5.
Boyd JP (1989) Chebyshev and Fourier spectral methods. Springer
6.
Khorrami MR (1989) Application of spectral collocation techniques to the stability of swirling flows. J Comput Phys 81:206–229CrossRef
7.
Higuera FJ (2006) Stationary viscosity-dominated electrified capillary jets. J Fluid Mech 558:143–152CrossRef
8.
Gamero-Castaño M, Magnani M (2019) Numerical simulation of electrospraying in the cone-jet mode. J Fluid Mech 859:247–267MathSciNetCrossRef
9.
Hirt CW, Nichols BD (1981) Volume of Fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201–225CrossRef
10.
Chang YC, Hou TY, Merriman B, Osher S (1996) A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J Comput Phys 124:449–464MathSciNetCrossRef
11.
12.
Verfurth R (1994) A posteriori error estimation and adaptive mesh-refinement techniques. J Comput Appl Math 50:67–83MathSciNetCrossRef
13.
Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100:335–354MathSciNetCrossRef
14.
Zahoor R, Belsak G, Bajt S, Sarler B (2018) Simulation of liquid micro-jet in free expanding high-speed co-flowing gas streams. Microfluidics and Nanofluidics 22:87CrossRef
15.
Mu K, Qiao R, Guo J, Yang C, Wu Y, Si T (2021) Parametric study on stability and morphology of liquid cone in flow focusing. Int J Multiphase Flow 135(103):507MathSciNet
16.
Donea J, Huerta A, Ponthot JP, Rodríguez-Ferrán A (2004) Arbitrary Lagrangian–Eulerian methods. In: Encyclopedia of computational mechanics. Wiley, pp 413–437
17.
Anthony CR et al (2023) Sharp interface methods for simulation and analysis of free surface flows with singularities: breakup and coalescence. Annu Rev Fluid Mech 55:707–747CrossRef
18.
Collins RT, Sambath K, Harris MT, Basaran OA (2013) Universal scaling laws for the disintegration of electrified drops. Proc Nat Acad Sci 110:4905–4910CrossRef
19.
Thompson JF, Warsi ZUA (1982) Boundary-fitted coordinate systems for numerical solution of partial differential equations-a review. J Comput Phys 47:1–108MathSciNetCrossRef
20.
Thompson JF, Warsi ZUA, Mastin CW (1985) Numerical grid generation: foundations and applications. Elsevier
21.
Herrada MA, Montanero JM (2016) A numerical method to study the dynamics of capillary fluid systems. J Comput Phys 306:137–147MathSciNetCrossRef
22.
Dimakopoulos Y, Tsamopoulos J (2003) A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations. J Comput Phys 192:494–522CrossRef
23.
Herrada MA, Ponce-Torres A, Rubio M, Eggers J, Montanero JM (2022) Stability and tip streaming of a surfactant-loaded drop in an extensional flow: influence of surface viscosity. J Fluid Mech 934:A26
24.
Chomaz J (2005) Global instabilities in spatially developing flows. Annu Rev Fluid Mech 37:357–392CrossRef
25.
26.
Cruz-Mazo F, Herrada MA, Gañán-Calvo AM, Montanero JM (2017) Global stability of axisymmetric flow focusing. J Fluid Mech 832:329–344MathSciNetCrossRef
27.
Ponce-Torres A, Rebollo-Muñoz N, Herrada MA, Gañán-Calvo AM, Montanero JM (2018) The steady cone-jet mode of electrospraying close to the minimum volume stability limit. J Fluid Mech 857:142–172MathSciNetCrossRef
28.
Blanco-Trejo S, Herrada MA, Gañán-Calvo AM, Montanero JM (2019) Electrospray cone-jet mode for weakly viscoelastic liquids. Phys Rev E 100(043):114
29.
Cabezas MG, Rubio M, Rebollo-Muñoz N, Herrada MA, Montanero JM (2021) Global stability analysis of axisymmetric liquid-liquid flow focusing. J Fluid Mech 909:A10MathSciNetCrossRef
30.
López M, Cabezas MG, Montanero JM, Herrada MA (2022) On the hydrodynamic focusing for producing microemulsions via tip streaming. J Fluid Mech 934:A47MathSciNetCrossRef
31.
Gordillo JM, Sevilla A, Campo-Cortés F (2014) Global stability of stretched jets: conditions for the generation of monodisperse micro-emulsions using coflows. J Fluid Mech 738:335–357CrossRef
32.
López-Herrera J, Herrada M, Gamero-Castaño M, Gañán-Calvo AM (2020) A numerical simulation of coaxial electrosprays. J Fluid Mech 885:A15MathSciNetCrossRef
33.
Montanero JM, Ponce-Torres A (2020) Review on the dynamics of isothermal liquid bridges. Appl Mech Rev 72(010):803
34.
Beroz J, Hart AJ, Bush JM (2019) Stability limit of electrified droplets. Phys Rev Lett 122(244):501
35.
Rubio M, Rodríguez-Díaz P, López-Herrera JM, Herrada MA, Gañán-Calvo AM, Montanero JM (2023) The role of charge relaxation in electrified tip streaming. Phys Fluids 35(017):131
36.
Augello L, Fani A, Gallaire F (2018) The influence of the entry region on the instability of a coflowing injector device. J Phys Condens Matter 30(284):003
37.
Tammisola O, Lundell F, Soderberg LD (2012) Surface tension-induced global instability of planar jets and wakes. J Fluid Mech 713:632–658MathSciNetCrossRef
38.
Rubio M, Montanero JM, Eggers J, Herrada MA (2024) Stable production of fluid jets with vanishing diameters via tip streaming. J Flui Mech 893: A4
39.
40.
de Luca L, Costa M, Caramiello C (2002) Energy growth of initial perturbations in two-dimensional gravitational jets. Phys Fluids 14:289–299MathSciNetCrossRef
41.
Hwang H, Moin P, Hack MJP (2021) A mechanism for the amplification of interface distortions on liquid. J Fluid Mech 911:A51MathSciNetCrossRef
42.
Danaila I, Dusek J, Anselmet F (1998) Nonlinear dynamics at a Hopf bifurcation with axisymmetry breaking in a jet. Phys Rev E 57:R3695–R3698CrossRef
43.
Sauter US, Buggisch HW (2005) Stability of initially slow viscous jets driven by gravity. J Fluid Mech 533:237–257MathSciNetCrossRef
44.
Rubio-Rubio M, Sevilla A, Gordillo JM (2013) On the thinnest steady threads obtained by gravitational stretching of capillary jets. J Fluid Mech 729:471–483CrossRef
45.
Martínez-Calvo A, Rubio-Rubio M, Sevilla A (2018) The nonlinear states of viscous capillary jets confined in the axial direction. J Fluid Mech 834:335–358CrossRef
46.
Schmidt S, Tammisola O, Lesshafft L, Oberleithner K (2021) Global stability and nonlinear dynamics of wake flows with a two-fluid interface. J Fluid Mech 919:A96MathSciNetCrossRef
47.
Javadi A, Eggers J, Bonn D, Habibi M, Ribe NM (2013) Delayed capillary breakup of falling viscous jets. Phys Rev Lett 110(144):501
48.
Huerre P, Monkewitz PA (1990) Local and global instabilites in spatially developing flows. Annu Rev Fluid Mech 22:473–537CrossRef
49.
Lin SP (2003) Breakup of liquid sheets and jets. Cambridge University Press, New YorkCrossRef
50.
Guillot P, Colin A, Utada AS, Ajdari A (2007) Stability of a jet in confined pressure-driven biphasic flows at low Reynolds numbers. Phys Rev Lett 99(104):502
51.
Gañán-Calvo AM, Herrada MA, Garstecki P (2006) Bubbling in unbounded coflowing liquids. Phys Rev Lett 96(124):504
52.
de Luca L (1999) Experimental investigation of the global instability of plane sheet flows. J Fluid Mech 399:355–376CrossRef
53.
Dizes SL (1997) Global modes in falling capillary jets. Eur J Mech B/Fluids 16:761–778MathSciNet
54.
Si T, Li F, Yin XY, Yin XZ (2009) Modes in flow focusing and instability of coaxial liquid-gas jets. J Fluid Mech 629:1–23CrossRef
55.
Vega EJ, Montanero JM, Herrada MA, Gañán-Calvo AM (2010) Global and local instability of flow focusing: the influence of the geometry. Phys Fluids 22(064):105
56.
Briggs RJ (1964) Electron-stream interaction with plasmas. MIT Press, CambridgeCrossRef
57.
Montanero JM, Gañán-Calvo AM (2008) Stability of coflowing capillary jets under non-axisymmetric perturbations. Phys Rev E 77(046):301
58.
van Saarloos W (1987) Dynamical velocity selection: marginal stability. Phys Rev Lett 58:2571–2574CrossRef
59.
Montanero JM, Gañán-Calvo AM (2008) Viscoelastic effects on the jetting-dripping transition in co-flowing capillary jets. J Fluid Mech 610:249–260MathSciNetCrossRef
60.
Cabezas MG, Herrada MA, Montanero JM (2019) Stability of a jet moving in a rectangular microchannel. Phys Rev E 100(053):104MathSciNet
61.
Montanero JM, Gañán-Calvo AM (2020) Dripping, jetting and tip streaming. Rep Prog Phys 83(097):001MathSciNet
62.
Ismail AS, Yao J, Xia HH, Stark JPW (2018) Breakup length of electrified liquid jets: scaling laws and applications. Phys Rev Appl 10(064):010
63.
Gañán-Calvo AM, Chapman HN, Heymann M, Wiedorn MO, Knoska J, Gañán-Riesco B, López-Herrera JM, Cruz-Mazo F, Herrada MA, Montanero JM, Bajt S (2021) The natural breakup length of a steady capillary jet: application to serial femtosecond crystallography. Crystals 11:990CrossRef
64.
Umemura A (2016) Self-destabilising loop of a low-speed water jet emanating from an orifice in microgravity. J Fluid Mech 25:146–180MathSciNetCrossRef
65.
Castro-Hernández E, Gundabala V, Fernández-Nieves A, Gordillo JM (2009) Scaling the drop size in coflow experiments. New J Phys 11(075):021
66.
Keller JB, Rubinov SI, Tu YO (1973) Spatial instability of a jet. Phys Fluids 16:2052–2055CrossRef
67.
Rayleigh L (1878) On the instability of jets. Proc London Math Soc s1-10:4–13
68.
Tomotika S (1935) On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc R Soc Lond 150:322–337
69.
Eggers J, Villermaux E (2008) Physics of liquid jets. Rep Prog Phys 71(036):601
70.
Hickox CE (1971) Instability due to density and viscosity stratification in an axisymmetric pipe flow. Phys Fluids 14:251–262CrossRef
71.
Guillot P, Colin A, Ajdari A (2008) Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries. Phys Rev E 78(016):307
72.
Janssen PJA, Meijer HEH, Anderson PD (2012) Stability and breakup of confined threads. Phys Fluids 24(012):102
73.
Kashid MN, Kowalinski W, Renken A, Baldyga J, Kiwi-Minsker L (2012) Analytical method to predict two-phase flow pattern in horizontal micro-capillaries. Chem Eng Sci 74:219–232CrossRef
74.
Sanz A, Meseguer J (1985) One-dimensional linear analysis of the compound jet. J Fluid Mech 159:55–68CrossRef
75.
Chauhan A, Maldarelli C, Papageorgiou DT, Rumschitzk DS (2000) Temporal instability of compound threads and jets. J Fluid Mech 420:1–25CrossRef
76.
Lee SY, Snider C, Park K, Robinson JP (2007) Compound jet instability in a microchannel for mononuclear compound drop formation. J Micromech Microeng 17:1558CrossRef
77.
Herrada MA, Montanero JM, Ferrera C, Gañán-Calvo AM (2010) Analysis of the dripping-jetting transition in compound capillary jets. J Fluid Mech 649:523–536CrossRef
78.
Gordillo JM, Pérez-Saborid M, Gañán-Calvo AM (2001) Linear stability of co-flowing liquid-gas jets. J Fluid Mech 448:23–51MathSciNetCrossRef
79.
Gañán-Calvo AM, Herrada MA, Montanero JM (2014) How does a shear boundary layer affect the stability of a capillary jet? Phys Fluids 26(061):701
80.
Lasheras JC, Hopfinger EJ (2000) Liquid jet instability and atomization in a coaxial gas stream. Annu Rev Fluid Mech 32:275–308CrossRef
81.
Gordillo JM, Pérez-Saborid M (2005) Aerodynamic effects in the break-up of liquid jets: on the first wind-induced break-up regime. J Fluid Mech 541:1–20CrossRef
82.
Acero AJ, Ferrera C, Montanero JM, Ga\(\tilde{\text{n}}\)án-Calvo AM, (2012) Focusing liquid microjets with nozzles. J Micromech Microeng 22(065):011
83.
Rubio M, Sadek SH, Gañán-Calvo AM, Montanero JM (2021) Diameter and charge of the first droplet emitted in electrospray. Phys Fluids 33(032):002
84.
Herrada MA, Ferrera C, Montanero JM, Gañán-Calvo AM (2010) Absolute lateral instability in capillary coflowing jets. Phys Fluids 22(064):104
85.
Gañán-Calvo AM (1998) Generation of steady liquid microthreads and micron-sized monodisperse sprays in gas streams. Phys Rev Lett 80:285–288CrossRef
86.
Saville DA (1971) Electrohydrodynamic stability: effects of charge relaxation at the interface of a liquid jet. J Fluid Mech 48:815–827CrossRef
87.
88.
89.
Xie L, Yang L, Qin L, Fu Q (2017) Temporal instability of charged viscoelastic liquid jets under an axial electric field. Eur J Mech/Fluids 66:60–70MathSciNetCrossRef
90.
Carroll CP, Joo YL (2008) Axisymmetric instabilities of electrically driven viscoelastic jets. J Non-Newtonian Fluid Mech 153:130–148CrossRef
91.
Hohman MM, Shin M, Rutledge G, Brenner MP (2001) Electrospinning and electrically forced jets. I. Stability theory. Phys Fluids 13:2201–2220
92.
Yang W, Duan H, Li C, Deng W (2014) Crossover of varicose and whipping instabilities in electrified microjets. Phys Rev Lett 112(054):501
93.
Guerrero J, Rivero J, Gundabala VR, Perez-Saborid M, Fernandez-Nieves A (2014) Whipping of electrified liquid jets. Proc Natl Acad Sci 111:13,763–13,767
94.
Li F, Gañán-Calvo AM, López-Herrera JM, Yin XY, Yin XZ (2013) Absolute and convective instability of a charged viscoelastic liquid jet. J Non-Newtonian Fluid Mech 196:58–69CrossRef
95.
Kiselev P, Rosell-Llompart J (2012) Highly aligned electrospun nanofibers by elimination of the whipping motion. J Appl Polym Sci 125:2433–2441CrossRef
96.
Gañán-Calvo AM (2007) Electro-flow focusing: the high-conductivity low-viscosity limit. Phys Rev Lett 98(134):503
97.
Forbes TP, Sisco E (2014) Chemical imaging of artificial fingerprints by desorption electro-flow focusing ionization mass spectrometry. Analyst 139:2982CrossRef
98.
Huang Y, Bu N, Duan Y, Pan Y, Liu H, Yin Z, Xiong Y (2013) Electrohydrodynamic direct-writing. Nanoscale 5:12,007–12,017
99.
Jiang J, Wang X, Li W, Liu J, Liu Y, Zheng G (2018) Electrohydrodynamic direct-writing micropatterns with assisted airflow. Micromachines 9:456CrossRef
100.
Kwak S, Pozrikidis C (2001) Effect of surfactants on the instability of a liquid thread or annular layer. Part I: quiescent fluids. Int J Multiphase Flow 27:1–37CrossRef
101.
Timmermans MLE, Lister JR (2002) The effect of surfactant on the stability of a liquid thread. J Fluid Mech 459:289–306CrossRef
102.
Hansen S, Peters GWM, Meijer HEH (1999) The effect of surfactant on the stability of a fluid filament embedded in a viscous fluid. J Fluid Mech 382:331–349CrossRef
103.
Bird RB, C R, Armstrong, Hassager O (1987) Dynamics of polymeric liquids volume I: fluid mechanics; volume II: kinetic theory. Wiley, New York
104.
Middleman S (1965) Stability of a viscoelastic jet. Chem Eng Sci 20:1037–1040CrossRef
105.
Goldin M, Yerushalmi J, Pfeffer R, Shinnar R (1969) Breakup of a laminar capillary jet of a viscoelastic fluid. J Fluid Mech 38:689–711CrossRef
106.
Brenn G, Liu Z, Durst F (2000) Linear analysis of the temporal instability of axisymmetrical non-Newtonian liquid jets. Int J Multiphase Flow 26:1621–1644CrossRef
107.
Funada T, Joseph DD (2003) Viscoelastic potential flow analysis of capillary instability. J Non-Newtonian Fluid Mech 111:87–105CrossRef
108.
Ye HY, Yang LJ, Fu QF (2016) Instability of viscoelastic compound jets. Phys Fluids 28(043):101
109.
Goren S, Gottlieb M (1982) Surface-tension-driven breakup of viscoelastic liquid threads. J Fluid Mech 120:245–266MathSciNetCrossRef
110.
Mohamed AS, Herrada MA, Gañán-Calvo AM, Montanero JM (2015) Convective-to-absolute instability transition in a viscoelastic capillary jet subject to unrelaxed axial elastic tension. Phys Rev E 92(023):006MathSciNet
111.
Reneker DH, Yarin AL (2008) Electrospinning jets and polymer nanofibers. Polymer 49:2387–2425CrossRef
112.
Ponce-Torres A, Vega EJ, Castrejón-Pita AA, Montanero JM (2017) Smooth printing of viscoelastic microfilms with a flow focusing ejector. J Non-Newtonian Fluid Mech 249:1–7MathSciNetCrossRef
113.
Rubio A, Galindo F, Vega EJ, Montanero JM, Cabezas MG (2022) Viscoelastic transition in transonic flow focusing. Phys Rev Fluids 7(074):201
114.
Rubio A, Vega EJ, Gañán-Calvo AM, Montanero JM (2022) Unexpected stability of micrometer weakly viscoelastic jets. Phys Fluids 34(062):014
115.
Ponce-Torres A, Montanero JM, Vega EJ, Gañán-Calvo AM (2016) The production of viscoelastic capillary jets with gaseous flow focusing. J Non-Newtonian Fluid Mech 229:8–15MathSciNetCrossRef
116.
Gill SJ, Gavis J (1956) Tensile stress in jets of viscoelastic fluids. I. J Polym Sci 20:287–298CrossRef
117.
Alhushaybari A, Uddin J (2020) Absolute instability of free-falling viscoelastic liquid jets with surfactants. Phys Fluids 32(013):102
118.
He D, Wylie JJ (2021) Temporal instability of a viscoelastic liquid thread surrounded by another viscoelastic fluid in presence of insoluble surfactant and inertia. J Non-Newtonian Fluid Mech 288(104):468MathSciNet
119.
Li F, He D (2023) Dynamics of a surfactant-laden viscoelastic thread in the presence of surface viscosity. J Fluid Mech 966:A35MathSciNetCrossRef
Metadata
Title
Theoretical Methods
Author
José María Montanero
Copyright Year
2024
Book
Tip Streaming of Simple and Complex Fluids

Print ISBN: 978-3-031-52767-8

Electronic ISBN: 978-3-031-52768-5

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-52768-5_3

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