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Linear stability analysis of capillary instabilities for concentric cylindrical shells

Published online by Cambridge University Press:  19 August 2011

X. Liang*
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
D. S. Deng
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
J.-C. Nave
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada
Steven G. Johnson
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Email address for correspondence:


Motivated by complex multi-fluid geometries currently being explored in fibre-device manufacturing, we study capillary instabilities in concentric cylindrical flows of fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier–Stokes problem. Generalizing previous work by Tomotika (), Stone & Brenner (, equal viscosities) and others, we present a full linear stability analysis of the growth modes and rates, reducing the system to a linear generalized eigenproblem in the Stokes case. Furthermore, we demonstrate by Plateau-style geometrical arguments that only axisymmetric instabilities need be considered. We show that the case is already sufficient to obtain several interesting phenomena: limiting cases of thin shells or low shell viscosity that reduce to problems, and a system with competing breakup processes at very different length scales. The latter is demonstrated with full three-dimensional Stokes-flow simulations. Many cases remain to be explored, and as a first step we discuss two illustrative cases, an alternating-layer structure and a geometry with a continuously varying viscosity.

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1. Abouraddy, A., Bayindir, M., Benoit, G., Hart, S., Kuriki, K., Orf, N., Shapira, O., Sorin, F., Temelkuran, B. & Fink, Y. 2007 Towards multimaterial multifunctional fibres that see, hear, sense and communicate. Nat. Mater. 6 (5), 336347.Google Scholar
2. Abramowitz, M. & Stegun, I. A. 1992 Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. Dover.Google Scholar
3. Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J. D., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S. & Sorensen, D. 1999 LAPACK Users’ Guide, 3rd edn. SIAM.Google Scholar
4. Andrew, A. L., Chu, K. E. & Lancaster, P. 1995 On the numerical solution of nonlinear eigenvalue problems. Computing 55 (2), 91111.Google Scholar
5. Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. & Van der Vorst, H. 1994 Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM.Google Scholar
6. Batchelor, G. K. 1973 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
7. Buchak, P. 2010, Private communication.Google Scholar
8. Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
9. Chang, Y. C., Hou, T. Y., Merriman, B. & Osher, S. 1996 A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys. 124 (2), 449464.Google Scholar
10. Chauhan, A., Maldarelli, C., Papageorgiou, D. T. & Rumschitzki, D. S. 2000 Temporal instability of compound threads and jets. J. Fluid Mech. 420 (1), 125.Google Scholar
11. Cohen, I., Brenner, M. P., Eggers, J. & Nagel, S. R. 1999 Two fluid drop snap-off problem: experiments and theory. Phys. Rev. Lett. 83 (6), 11471150.Google Scholar
12. Crowdy, D. 2002 On a class of geometry-driven free boundary problems. SIAM J. Appl. Math. 62, 945964.Google Scholar
13. Crowdy, D. G. 2003 Compressible bubbles in Stokes flow. J. Fluid Mech. 476, 345356.Google Scholar
14. Demmel, J. W. & Kagstrom, B. 1987 Computing stable eigendecompositions of matrix pencils. Linear Algebr. Applics. 88–89, 139186.Google Scholar
15. Deng, D. S., Orf, N. D., Abouraddy, A. F., Stolyarov, A. M., Joannopoulos, J. D., Stone, H. A. & Fink, Y. 2008 In-fiber semiconductor filament arrays. Nano Lett. 8 (12), 42654269.Google Scholar
16. Deng, D. S., Nave, J.-C., Liang, X., Johnson, S. G. & Fink, Y. 2011 Exploration of in-fiber nanostructures from capillary instability. Opt. Express (in press).Google Scholar
17. Deng, D. S., Orf, N. D., Danto, S., Abouraddy, A. F., Joannopoulo, J. D. & Fink, Y. 2010 Processing and properties of centimeter-long, in-fiber, crystalline-selenium filaments. Appl. Phys. Lett. 96, 023102.Google Scholar
18. Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
19. Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71 (21), 34583460.Google Scholar
20. Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.Google Scholar
21. Egusa, S., Wang, Z., Chocat, N., Ruff, Z. M., Stolyarov, A. M., Shemuly, D., Sorin, F., Rakich, P. T., Joannopoulos, J. D. & Fink, Y. 2010 Multimaterial piezoelectric fibres. Nat. Mater. 9, 643648.Google Scholar
22. Guillaume, P. 1999 Nonlinear eigenproblems. SIAM J. Matrix Anal. Appl. 20, 575595.Google Scholar
23. Gunawan, A. Y., Molenaar, J. & van de Ven, A. A. F. 2002 In-phase and out-of-phase break-up of two immersed liquid threads under influence of surface tension. Eur. J. Mech. B Fluids 21 (4), 399412.Google Scholar
24. Gunawan, A. Y., Molenaar, J. & van de Ven, A. A. F. 2004 Break-up of a set of liquid threads under influence of surface tension. J. Engng Math. 50 (1), 2549.Google Scholar
25. Hart, S. D., Maskaly, G. R., Temelkuran, B., Prideaux, P. H., Joannopoulos, J. D. & Fink, Y. 2002 External reflection from omnidirectional dielectric mirror fibers. Science 296, 511513.Google Scholar
26. Hopper, R. W. 1991 Plane Stokes flow driven by capillarity on a free surface. Part II. Further developments. J. Fluid Mech. 230, 355364.Google Scholar
27. Kinoshita, C., Teng, H. & Masutani, S. 1994 A study of the instability of liquid jets and comparison with Tomotika’s analysis. Intl J. Multiphase Flow 20 (3), 523533.Google Scholar
28. Kuiken, H. K. 1990 Viscous sintering: the surface-tension-driven flow of a liquid form under the influence of curvature gradients at its surface. J. Fluid Mech. 214, 503515.Google Scholar
29. Kundu, P. K. & Cohen, I. M. 2007 Fluid Mechanics. Academic.Google Scholar
30. Kuriki, K., Shapira, O., Hart, S. D., Benoit, B., Kuriki, Y., Viens, J. F., Bayindir, M., Joannopoulos, J. D. & Fink, Y. 2004 Hollow multilayer photonic bandgap fibers for NIR applications. Opt. Express 12, 15101517.Google Scholar
31. Liao, B. S., Bai, Z. J., Lee, L. Q. & Ko, K. 2010 Nonlinear Rayleigh–Ritz iterative method for solving large scale nonlinear eigenvalue problems. Taiwanese J. Math. 14 (3A), 869883.Google Scholar
32. Lin, S. P. 2003 Breakup of Liquid Sheets and Jets. Cambridge University Press.Google Scholar
33. Lister, J. R. & Stone, H. A. 1998 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 10 (11), 27582764.Google Scholar
34. Liu, X.-D., Osher, S. & Chan, T. 1994 Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115 (1), 200212.Google Scholar
35. Meister, B. J. & Scheele, G. F. 1967 Generalized solution of the Tomotika stability analysis for a cylindrical jet. AIChE J. 682688.Google Scholar
36. Ockendon, H. & Ockendon, J. R. 1995 Viscous Flow. Cambridge University Press.Google Scholar
37. Osher, S. & Fedkiw, R. 2002 Level Set Methods and Dynamic Implicit Surfaces. Springer.Google Scholar
38. Plateau, J. A. F. 1873 Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires, vol. 2. Gauthier-Villars.Google Scholar
39. Pone, E., Dubois, C., Gu, N., Gao, Y., Dupuis, A., Boismenu, F., Lacroix, S. & Skorobogatiy, M. 2006 Drawing of the hollow all-polymer Bragg fibers. Opt. Express 14 (13), 58385852.Google Scholar
40. Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 7197.Google Scholar
41. Rayleigh, Lord 1892 On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. 34 (207), 145154.Google Scholar
42. Ruhe, A. 2006 Rational Krylov for large nonlinear eigenproblems. In Applied Parallel Computing, pp. 357363. Springer.Google Scholar
43. Shah, R. K., Shum, H. C., Rowat, A. C., Lee, D., Agresti, J. J., Utada, A. S., Chu, L.-Y., Kim, J.-W., Fernandez-Nieves, A., Martinez, C. J. & Weitz, D. A. 2008 Designer emulsions using microfluidics. Mater. Today 11 (4), 1827.Google Scholar
44. Shu, C. W. & Osher, S. 1989 Efficient implementation of essentially non-oscillatory shock-capturing schemes. Part II. J. Comput. Phys. 83, 3278.Google Scholar
45. Sorin, F., Abouraddy, A., Orf, N., Shapira, O., Viens, J., Arnold, J., Joannopoulos, J. & Fink, Y. 2007 Multimaterial photodetecting fibers: a geometric and structural study. Adv. Mater. 19, 38723877.Google Scholar
46. Sterling, A. M. & Sleicher, C. A. 1975 The instability of capillary jets. J. Fluid Mech. 68, 477495.Google Scholar
47. Stone, H. A. & Brenner, M. P. 1996 Note on the capillary thread instability for fluids of equal viscosities. J. Fluid Mech. 318, 373374.Google Scholar
48. Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146159.Google Scholar
49. Tanveer, S. & Vasconcelos, G. L. 1995 Time-evolving bubbles in two-dimensional Stokes flow. J. Fluid Mech. 301, 325344.Google Scholar
50. Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. (A) Math. Phys. Sci. 150 (870), 322337.Google Scholar
51. Trefethen, L. N. & Bau, D. 1997 Numerical Linear Algebra. SIAM.Google Scholar
52. Utada, A. S., Lorenceau, E., Link, D. R., Kaplan, P. D., Stone, H. A. & Weitz, D. A. 2005 Monodisperse double emulsions generated from a microcapillary device. Science 308 (5721), 537541.Google Scholar
53. Voss, H. 2007 A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems. Comput. Struct. 85 (17–18), 12841292.Google Scholar