Self-similar cuspidal formation by runaway thermocapillary forces in thin liquid films
Many physical systems give rise to dynamical behavior leading to cuspidal shapes which represent a singularity of the governing equation. The cusp tip often exhibits self-similarity as well, indicative of scaling symmetry invariant in time up to a change of scale. Cuspidal shapes even occur in liqui...
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Published in | New journal of physics Vol. 21; no. 1; pp. 13018 - 13031 |
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Main Authors | , |
Format | Journal Article |
Language | English |
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IOP Publishing
18.01.2019
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Abstract | Many physical systems give rise to dynamical behavior leading to cuspidal shapes which represent a singularity of the governing equation. The cusp tip often exhibits self-similarity as well, indicative of scaling symmetry invariant in time up to a change of scale. Cuspidal shapes even occur in liquid systems when the driving force for fluid elongation is sufficiently strong to overcome leveling by capillarity. In almost all cases reported in the literature, however, the moving interface is assumed to be shear-free and the operable forces orient exclusively in the direction normal to the advancing boundary. Here we focus on a system in which a slender liquid film is exposed to large thermocapillary stresses, a system previously shown to undergo a linear instability resembling microlens arrays. We demonstrate by analytic and numerical means how in the nonlinear regime runaway thermocapillary forces induce cuspidal formations terminated by a conical tip whose slope is given by an analytic relation. On a fundamental level, this finding broadens our understanding of known categories of flows that can generate cuspidal forms. More practically, the system examined here introduces a potentially novel lithographic method for one-step non-contact fabrication of cuspidal microarrays. |
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AbstractList | Many physical systems give rise to dynamical behavior leading to cuspidal shapes which represent a singularity of the governing equation. The cusp tip often exhibits self-similarity as well, indicative of scaling symmetry invariant in time up to a change of scale. Cuspidal shapes even occur in liquid systems when the driving force for fluid elongation is sufficiently strong to overcome leveling by capillarity. In almost all cases reported in the literature, however, the moving interface is assumed to be shear-free and the operable forces orient exclusively in the direction normal to the advancing boundary. Here we focus on a system in which a slender liquid film is exposed to large thermocapillary stresses, a system previously shown to undergo a linear instability resembling microlens arrays. We demonstrate by analytic and numerical means how in the nonlinear regime runaway thermocapillary forces induce cuspidal formations terminated by a conical tip whose slope is given by an analytic relation. On a fundamental level, this finding broadens our understanding of known categories of flows that can generate cuspidal forms. More practically, the system examined here introduces a potentially novel lithographic method for one-step non-contact fabrication of cuspidal microarrays. |
Author | Troian, Sandra M Zhou, Chengzhe |
Author_xml | – sequence: 1 givenname: Chengzhe surname: Zhou fullname: Zhou, Chengzhe organization: California Institute of Technology , 1200 E. California Blvd, Physics, MC 103-33, Pasadena, CA 91125, United States of America – sequence: 2 givenname: Sandra M surname: Troian fullname: Troian, Sandra M email: stroian@caltech.edu organization: California Institute of Technology , 1200 E. California Blvd., Applied Physics, MC 128-95, Pasadena, CA 91125, United States of America |
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Cites_doi | 10.1103/PhysRevLett.108.063003 10.1364/OE.24.010172 10.1103/PhysRevLett.113.054301 10.1103/PhysRevLett.106.144501 10.1103/PhysRevE.96.052308 10.1364/CLEO_SI.2011.CML3 10.1016/S0167-2789(00)00165-2 10.1103/PhysRevLett.88.074501 10.4171/IFB/88 10.1063/1.870110 10.1073/pnas.1210770110 10.1103/PhysRevLett.96.034501 10.1063/1.3685831 10.1103/PhysRevE.91.043019 10.1103/PhysRevLett.102.144501 10.1006/jcis.1993.1142 10.1103/PhysRevLett.103.114501 10.1103/PhysRevFluids.1.041902 10.1103/PhysRevLett.93.184502 10.1063/1.3484276 10.1023/A:1010714612865 10.1023/B:JOSS.0000033251.81126.af 10.1103/PhysRevLett.119.198001 10.1017/CBO9781316161692 10.1063/1.3475516 10.1006/jdeq.2001.4108 10.1103/PhysRevLett.91.054501 10.1063/1.1722742 10.1103/PhysRevE.92.023014 10.1017/S0022112088002484 10.1364/JOSAA.8.000549 10.1103/PhysRevLett.103.074501 10.1557/PROC-1179-BB08-02 10.1103/PhysRevLett.86.4290 10.1017/jfm.2017.918 10.1103/PhysRevLett.114.175501 10.1103/PhysRevLett.102.255005 10.1103/PhysRevA.84.043834 10.1103/PhysRevLett.71.3458 10.1103/PhysRevLett.106.175501 10.1063/1.870138 10.1017/CBO9780511627200 10.1039/C7ME00009J 10.1063/1.4968575 |
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References | 22 23 45 24 46 25 47 26 27 28 29 Fiedler K R (36) 2018 COMSOL Inc. (44) 30 31 10 32 11 33 12 34 13 14 15 37 16 38 17 39 18 19 Huber M (35) 2011 1 2 3 4 5 6 7 8 9 40 41 20 42 21 43 |
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Snippet | Many physical systems give rise to dynamical behavior leading to cuspidal shapes which represent a singularity of the governing equation. The cusp tip often... |
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SubjectTerms | blowup Capillarity driven singularities Elongation free surface cusps Physics runaway process Self-similarity Stability analysis thermocapillary Thermocapillary force thin film equation Thin films |
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Title | Self-similar cuspidal formation by runaway thermocapillary forces in thin liquid films |
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