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Stability and breakup of liquid threads and annular layers in a corrugated tube with zero base flow. (English) Zbl 1350.35158
Author’s abstract: We study the stability and breakup behavior of an axisymmetric liquid thread surrounded by another viscous fluid layer through a long wave approximation. The two fluids are immiscible and confined in a concentrically placed cylindrical tube and there is no base flow. The effect of the tube wall corrugation is taken into account in the model, which allows the access of the interaction between the wall shape and the thread interfacial dynamics. The linearized system is studied by Floquet theory due to the presence of nonconstant coefficients in the equation and the spectrum is computed numerically via the Fourier-Floquet-Hill method. The resulting features agree qualitatively with those obtained based on a lubrication model in the thin annulus limit, where the short wave disturbances that would be stabilizing due to the capillarity in the absence of wall corrugation, can excite some unstable long waves. Those results from the linear theory are also confirmed numerically by direct numerical simulation on the evolution equations. Meanwhile, a transition on the dominant modes from the one for a tube without corrugation to the one with a wall shape included is found. Finally the drop formation is shown in the nonlinear regime when the pinch off of the core thread is obtained. The annular film drainage regime is also approached slowly along with pinching when the tube wall is close to the thread interface. In addition, our results demonstrate the possibility of suppressing pinching, depending on the averaged annular layer thickness and the variation in tube radius.
35Q35 PDEs in connection with fluid mechanics
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
76D45 Capillarity (surface tension) for incompressible viscous fluids
76T99 Multiphase and multicomponent flows
76D08 Lubrication theory
Full Text: DOI
[1] B. Ambravaneswaran, E. D. Wilkes, and O. A. Basaran, Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops, Phys. Fluids, 14 (2002), pp. 2606–2621. · Zbl 1185.76030
[2] R. W. Aul and W. L. Olbricht, Stability of a thin annular film in pressure-driven, low Reynolds-number flow through a capillary, J. Fluid Mech., 215 (1990), pp. 585–599.
[3] N. J. Balmforth and S. Mandre, Dynamics of roll waves, J. Fluid Mech., 514 (2004), pp. 1–33. · Zbl 1067.76009
[4] N. Bermond, A. R. Thiam, and J. Bibette, Decompressing emulsion droplets favors coalescence, Phys. Rev. Lett., 100 (2008), 024501.
[5] P. J. Blennerhassett and A. P. Bossom, The linear stability of high-frequency oscillatory flow in a channel, J. Fluid Mech., 556 (2006), pp. 1–25. · Zbl 1094.76020
[6] D. B. Bogy, Drop formation in a circular liquid jets, Annu. Rev. Fluid Mech., 11 (1979), pp. 207–228.
[7] M. Booty, D. T. Papageorgiou, M. Siegel, and Q. Wang, Long-wave equations and direct numerical simulations for the breakup of a viscous fluid thread surrounded by an immiscible viscous fluid, IMA J. Appl. Math., 78 (2013), pp. 851–867. · Zbl 1282.76073
[8] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford, 1961.
[9] A. U. Chen, P. K. Notz, and O. A. Basaran, Computational and experimental analysis of pinch-off and scaling, Phys. Rev. Lett, 88 (2002), 174501.
[10] K. Chen, R. Bai, and D. D. Joseph, Lubricated pipelining. Part 3. Stability of core-annular flow in vertical tubes, J. Fluid Mech., 214 (1990), pp. 251–286. · Zbl 0709.76671
[11] K. Chen and D. D. Joseph, Long waves and lubrication theories for core annular flow, Phys. Fluids A, 3 (1991), pp. 2672–2679. · Zbl 0746.76025
[12] Y.-J. Chen and P. H. Steen, Dynamics of inviscid capillary breakup: Collapse and pinchoff of a film bridge, J. Fluid Mech., 341 (1997), pp. 245–267. · Zbl 0892.76010
[13] G. F. Christopher and S. L. Anna, Microfluidic methods for generating continuous droplet streams, J. Phys. D., 40 (2007), pp. R319–R336.
[14] I. Cohen, M. P. Brenner, J. Eggers, and S. R. Nagel, Two fluid drop snap-off problem: Experiments and theory, Phys. Rev. Lett., 83 (1999), pp. 1147–1150.
[15] R. V. Craster, O. Matar, and D. T. Papageorgiou, Pinchoff and satellite formation in surfactant covered viscous threads, Phys. Fluids, 14 (2002), pp. 1364–1376. · Zbl 1185.76093
[16] R. V. Craster, O. Matar, and D. T. Papageorgiou, On compound threads with large viscosity contrast, J. Fluid Mech., 533 (2005), pp. 95–124. · Zbl 1074.76013
[17] R. V. Craster and O. Matar, On viscous beads flowing down a vertical fibre, J. Fluid Mech., 553 (2006), pp. 85–105. · Zbl 1087.76030
[18] C. G. Dassor, J. A. Deiber, and A. E. Cassano, Slow two-phase flow through a sinusoidal channel, Int. J. Multiph. Flow, 10 (1984), pp. 181–193. · Zbl 0539.76110
[19] R. F. Day, E. J. Hinch, and J. R. Lister, Self-similarity capillary pinchoff of an inviscid fluid, Phys. Rev. Lett., 80 (1998), pp. 704–707.
[20] B. Deconinck and J. N. Kutz, Computing spectra of linear operators using the Floquet-Fourier-Hill method, J. Comput. Phys., 219 (2006), pp. 296–321. · Zbl 1105.65119
[21] G. F. Dietze and C. Ruyer-Quil, Films in narrow tubes, J. Fluid Mech., 762 (2015), pp. 68–109.
[22] J. Eggers and T. F. Dupont, Drop formation in a one-dimensional approximation of the Navier-Stokes equations, J. Fluid Mech., 262 (1994), pp. 205–221. · Zbl 0804.76027
[23] J. Eggers and E. Villermaux, Physics of liquid jet, Rep. Progr. Phys., 71 (2008), 036601.
[24] J. Eggers, Universal pinching of 3D axisymmetric free surface flow, Phys. Rev. Lett, 71 (1993), pp. 3458–3460.
[25] J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Rev. Modern Phys., 69 (1997), pp. 865–930. · Zbl 1205.37092
[26] A. L. Frenkel, A. J. Babchin, B. G. Levich, T. Shlang, and G. I. Sivashinsky, Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability, J. Colloid Interface Sci., 115 (1987), pp. 225–233.
[27] P. A. Gauglitz and C. J. Radke, The dynamics of liquid film breakup in constricted cylindrical capillaries, J. Colloid Interface Sci., 134 (1990), pp. 14–40.
[28] E. Georgiou, C. Maldarellii, , D. T. Papageorgiou, and D. S. Rumschitzki, An asymptotic theory for the linear stability of a core-annular flow in the thin annular limit, J. Fluid Mech., 243 (1992), pp. 653–677. · Zbl 0753.76058
[29] S. L. Goren, The instability of an annular thread of fluid, J. Fluid Mech., 12 (1962), pp. 309–319. · Zbl 0105.39602
[30] S. L. Goren, The shape of a thread of liquid undergoing break-up, J. Colloid Sci, 19 (1964), pp. 81–86.
[31] J. B. Grotberg, Respiratory fluid mechanics, Phys. Fluids, 23 (2011), 021310.
[32] J. G. Hagedorn, N. S. Martys, and J. F. Douglas, Breakup of a fluid thread in a confined geometry: Droplet-plug transition, perturbation sensitivity, and kinetic stabilization with confinement, Phys. Rev. E (3), 69 (2004), 056312.
[33] P. S. Hammond, Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe, J. Fluid Mech., 137 (1983), pp. 363–384. · Zbl 0571.76046
[34] C. E. Hickox, Instability due to viscosity and density stratification in axisymmetric pipe flow, Phys. Fluids, 14 (1971), pp. 251–262. · Zbl 0216.52703
[35] K. J. Humphry, A. Ajdari, A. Fernandez-Nieves, H. A. Stone, and D. A. Weitz, Suppression of instabilities in multiphase flow by geometric confinement, Phys. Rev. E (3), 79 (2009), 56310.
[36] H. H. Hu and D. D. Joseph, Lubricated pipelines: Stability of core-annular flow. Part 2, J. Fluid Mech., 205 (1989), pp. 359–396.
[37] H. H. Hu, T. Lundgren, and D. D. Joseph, Stability of core-annular flow with small viscosity ratio, Phys. Fluids A, 2 (1990), pp. 1945–1954. · Zbl 0716.76069
[38] D. D. Joseph, R. Bai, K. P. Chen, and Y. Y. Renardy, Core-annular flows, Annu. Rev. Fluid Mech., 29 (1997), pp. 65–90.
[39] S. Kalliadasis and H.-C. Chang, Drop formation during coating of vertical fibres, J. Fluid Mech., 261 (1994), pp. 135–168. · Zbl 0818.76021
[40] P. Keast and P. H. Muir, Algorithm 688 EPDCOL – a more efficient PDECOL code, ACM Trans. Math. Software, 17 (1991), pp. 153–166. · Zbl 0900.65270
[41] V. Kerchman, Strongly nonlinear interfacial dynamics in core-annular flows, J. Fluid Mech., 290 (1995), pp. 131–166. · Zbl 0855.76028
[42] C. Kouris and J. Tsamopoulos, Concentric core-annular flow in a periodically constricted circular tube. Part 1. Steady-state, linear stability and energy analysis, J. Fluid Mech., 432 (2001), pp. 31–68. · Zbl 1012.76028
[43] C. Kouris and J. Tsamopoulos, Dynamics of axisymmetric core-annular flow in a straight tube. I. The more viscous fluid in the core, bamboo waves, Phys. Fluids, 13 (2001), pp. 841–858. · Zbl 1184.76302
[44] C. Kouris and J. Tsamopoulos, Dynamics of the axisymmetric core-annular flow. II. The less viscous fluid in the core, saw tooth waves, Phys. Fluids, 14 (2002), pp. 1011–1029.
[45] T. A. Kowalewski, On the separation of droplets from a liquid jet, Fluid Dyn. Res., 17 (1996), pp. 121–145.
[46] H. C. Lee, Drop formation in liquid jets, IBM J. Res. Develop., 18 (1971), pp. 364–369.
[47] D. Leppinen and J. R. Lister, Capillary pinch-off in inviscid fluids, Phys. Fluids, 15 (2003), pp. 568–578. · Zbl 1185.76225
[48] R. Levy, D. B. Hill, M. G. Forest, and J. B. Grotberg, Pulmonary fluid flow challenges for experimental and mathematical modeling, Integrative Comparative Biol., 54 (2014), pp. 985–1000.
[49] D. R. Link, S. L. Anna, D. A. Weitz, and H. A. Stone, Geometrically mediated breakup of drops in microfluidic devices, Phys. Rev. Lett., 92 (2004), 054503.
[50] J. R. Lister and H. A. Stone, Capillary breakup of a viscous thread surrounded by another viscous fluid, Phys. Fluids, 10 (1998), pp. 2758–2764. · Zbl 1185.76548
[51] J. R. Lister, J. M. Rallison, A. A. King, L. J. Cummings, and O. E. Jensen, Capillary drainage of an annular film: The dynamics of collars and lobes, J. Fluid Mech., 552 (2006), pp. 311–343. · Zbl 1151.76376
[52] G. H. McKinley and A. Tripathi, How to extract the Newtonian viscosity from capillary breakup measurements in a filament rheometer, J. Rheol., 44 (2000), pp. 653–670.
[53] M. Muradoglu and A. D. Kayaalp, An auxiliary grid method for computations of multiphase flows in complex geometries, J. Comput. Phys., 214 (2006), pp. 858–877. · Zbl 1136.76410
[54] L. A. Newhouse and C. Pozrikidis, The capillary instability of annular layers and liquid threads, J. Fluid Mech., 242 (1992), pp. 193–209.
[55] U. Olgac, A. D. Kayaalp, and M. Muradoglu, Buoyancy-driven motion and breakup of viscous drops in constricted capillaries, Int. J. Multiph. Flow, 32 (2006), pp. 1055–1071. · Zbl 1136.76597
[56] D. T. Papageorgiou, C. Maldarelli, and D. S. Rumschitzki, Nonlinear interfacial stability of core-annular film flows, Phys. Fluids A, 2 (1990), pp. 340–352. · Zbl 0704.76060
[57] D. T. Papageorgiou, Stability of unsteady viscous flow in a curved pipe, J. Fluid Mech., 182 (1987), pp. 209–233. · Zbl 0651.76016
[58] D. T. Papageorgiou, Analytical description of the breakup of liquid jets, J. Fluid Mech., 301 (1995), pp. 109–132. · Zbl 0864.76012
[59] D. T. Papageorgiou, On the breakup of viscous liquid threads, Phys. Fluids, 7 (1995), pp. 1529–1544. · Zbl 1023.76526
[60] C. Pozrikidis, Capillary instability and breakup of a viscous thread, J. Engrg. Math, 36 (1999), pp. 255–275. · Zbl 0953.76030
[61] L. Preziosi, K. Chen, and D. D. Joseph, Lubricated pipelines: Stability of flow, J. Fluid Mech., 201 (1989), pp. 323–356. · Zbl 0683.76039
[62] T. C. Ransokoff, P. A. Gauglitz, and C. J. Radke, Snap-off of gas bubbles in smoothly constricted noncircular capillaries, AIChE J., 33 (1987), pp. 753–765.
[63] L. Rayleigh, On the instability of jets, Proc. Lond. Math. Soc., 10 (1878), pp. 4–13. · JFM 11.0685.01
[64] M. Renardy, Some comments on the surface-tension driven breakup (or lack of it) of viscoelastic jets, J. Non-Newtonian Fluid Mech., 51 (1994), pp. 97–102.
[65] M. Renardy, A numerical study of the asymptotic evolution and breakup of newtonian and viscoelastic jets, J. Non-Newton. Fluid Mech., 59 (1995), pp. 267–282.
[66] A. Rothert, R. Richter, and I. Rehberg, Formation of a drop: Viscosity dependence of three flow regimes, New J. Phys, 5 (2003), 59.
[67] S. Saprykin, R. J. Koopmans, and S. Kalliadasis, Free-surface thin-film flows over topography: Influence of inertia and viscoelasticity, J. Fluid Mech., 578 (2007), pp. 271–293. · Zbl 1112.76006
[68] X. D. Shi, M. P. Brenner, and S. R. Nage, A cascade of structure in a drop falling from a faucet, Science, 265 (1994), pp. 219–222. · Zbl 1226.76011
[69] A. Sierou and J. R. Lister, Self-similar solutions for viscous capillary pinch-off, J. Fluid Mech., 497 (2003), pp. 381–403. · Zbl 1065.76052
[70] M. E. Timmermans and J. R. Lister, The effect of surfactant on the stability of a liquid thread, J. Fluid Mech., 459 (2002), pp. 289–306. · Zbl 1031.76020
[71] M. Tjahjadi, H. A. Stone, and J. M. Otino, Satellite and subsatellite formation in capillary breakup, J. Fluid Mech., 243 (1992), pp. 297–317.
[72] S. Tomotika, On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 150 (1935), pp. 322–337. · JFM 61.1539.01
[73] A. S. Utada, E. Lorenceau, D. R. Link, P. D. Kaplan, H. A. Stone, and D. A. Weitz, Monodisperse double emulsions generated from a microcapillary device, Science, 308 (2005), pp. 537–541.
[74] Q. Wang, Breakup of a viscous poorly conducting liquid thread subject to a radial electric field at zero reynolds number, Phys. Fluids, 24 (2012), 102102.
[75] Q. Wang, Capillary instability of a viscous liquid thread in a cylindrical tube, Phys. Fluids, 25 (2013), 112104. · Zbl 1320.76040
[76] H.-H. Wei and D. S. Rumschitzki, The linear stability of a core-annular flow in an asymptotically corrugated tube, J. Fluid Mech., 466 (2002), pp. 113–147. · Zbl 1069.76023
[77] H.-H. Wei and D. S. Rumschitzki, The weakly nonlinear interfacial stability of a core-annular flow in a corrugated tube, J. Fluid Mech., 466 (2002), pp. 149–177. · Zbl 1062.76020
[78] E. D. Wilkes, S. D. Phillips, and O. A. Basaran, Computational and experimental analysis of dynamics of drop formation, Phys. Fluids, 11 (1999), pp. 3577–3598. · Zbl 1149.76585
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