Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T07:24:47.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermocapillary instabilities in a horizontal liquid layer under partial basal slip

Published online by Cambridge University Press:  20 September 2018

Katarzyna N. Kowal Open the ORCID record for Katarzyna N. Kowal [Opens in a new window] ,
Stephen H. Davis  and
Peter W. Voorhees
Katarzyna N. Kowal*
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Trinity College, University of Cambridge, Cambridge CB2 1TQ, UK
Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Peter W. Voorhees
Affiliation:
Department of Materials Science and Engineering, Northwestern University, 2225 Campus Drive, Evanston, IL 60208, USA
*
†Email address for correspondence: katarzyna.kowal@northwestern.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the onset of three-dimensional hydrothermal waves in a low-capillary-number liquid layer of arbitrary depth, bounded by a free liquid–gas interface from above and a partial slip, rigid surface from below. A selection of two- and three-dimensional hydrothermal waves, longitudinal rolls and longitudinal travelling waves, form the preferred mode of instability, which depends intricately on the magnitude of the basal slip. Partial slip is destabilizing for all modes of instability. Specifically, the minimal Marangoni number required for the onset of instability follows $M_{m}\sim a(\unicode[STIX]{x1D6FD}^{-1}+b)^{-c}$ for each mode, where $a,b,c>0$ and $\unicode[STIX]{x1D6FD}^{-1}$ is the slip parameter. In the limit of free slip, longitudinal travelling waves disappear in favour of longitudinal rolls. With increasing slip, it is common for two-dimensional hydrothermal waves to exchange stability in favour of longitudinal rolls and oblique hydrothermal waves. Two types of oblique hydrothermal waves appear under partial slip, which exchange stability with increasing slip. The oblique mode that is preferred under no slip persists and remains near longitudinal for small slip parameters.

JFM classification

Convection: Convection Interfacial Flows (free surface): Interfacial Flows (free surface) Convection: Marangoni convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 839 - 859
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carpenter, B. M. & Homsy, G. M. 1990 High Marangoni number convection in a square cavity. Part II. Phys. Fluids A 2 (2), 137–149.Google Scholar
Chen, J.-C. & Hwu, F.-S. 1993 Oscillatory thermocapillary flow in a rectangular cavity. Intl J. Heat Mass Transfer 36 (15), 3743–3749.Google Scholar
Chin, S. Y., Poh, Y. C., Kohler, A.-C., Compton, J. T., Hsu, L. L., Lau, K. M., Kim, S., Lee, B. W., Lee, F. Y. & Sia, S. K. 2017 Additive manufacturing of hydrogel-based materials for next-generation implantable medical devices. Sci. Robot. 2 (2), eaah6451.Google Scholar
Cowley, S. J. & Davis, S. H. 1983 Viscous thermocapillary convection at high Marangoni number. J. Fluid Mech. 135, 175–188.Google Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403–435.Google Scholar
Gibson, I., Rosen, D. & Stucker, B. 2015 Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping and Direct Digital Manufacturing. Springer.Google Scholar
Hamed, M. & Floryan, J. M. 2000 Marangoni convection. Part 1. A cavity with differentially heated sidewalls. J. Fluid Mech. 405, 79–110.Google Scholar
He, Y., Xue, G. & Fu, J. 2014 Fabrication of low cost soft tissue prostheses with the desktop 3D printer. Sci. Rep. 4, 6973.Google Scholar
Hof, B., Juel, A., Zhao, L., Henry, D., Ben Hadid, H. & Mullin, T. 2004 On the onset of oscillatory convection in molten gallium. J. Fluid Mech. 515, 391–413.Google Scholar
Hofmann, D. C., Roberts, S., Otis, R., Kolodziejska, J., Dillon, R. P., Suh, J.-O., Shapiro, A. A., Liu, Z.-K. & Borgonia, J.-P. 2014 Developing gradient metal alloys through radial deposition additive manufacturing. Sci. Rep. 4, 5357.Google Scholar
Juel, A., Mullin, T., Ben Hadid, H. & Henry, D. 2001 Three-dimensional free convection in molten gallium. J. Fluid Mech. 436, 267–281.Google Scholar
Khairallah, S. A., Anderson, A. T., Rubenchik, A. & King, W. E. 2016 Laser powder-bed fusion additive manufacturing: physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones. Acta Mater. 108, 36–45.Google Scholar
Kowal, K. N., Altieri, A. L. & Davis, S. H. 2017a Strongly nonlinear theory of rapid solidification near absolute stability. Phys. Rev. E 96, 042801.Google Scholar
Kowal, K. N., Davis, S. H. & Voorhees, P. W. 2017b Instabilities in rapid directional solidification under weak flow. Phys. Rev. E 96, 062802.Google Scholar
Matthews, M. J., Guss, G., Khairallah, S. A., Rubenchik, A. M., Depond, P. J. & King, W. E. 2016 Denudation of metal powder layers in laser powder bed fusion processes. Acta Mater. 114, 33–42.Google Scholar
Miller, J. S., Stevens, K. R., Yang, M. T., Baker, B. M., Nguyen, D.-H. T., Cohen, D. M., Toro, E., Chen, A. A., Galie, P. A., Yu, X., Chaturvedi, R., Bhatia, S. N. & Chen, C. S. 2012 Rapid casting of patterned vascular networks for perfusable engineered three-dimensional tissues. Nat. Mater. 11, 768–774.Google Scholar
Mukherjee, T., Zuback, J. S., De, A. & DebRoy, T. 2016 Printability of alloys for additive manufacturing. Sci. Rep. 6, 19717.Google Scholar
Mundrane, M., Xu, J. & Zebib, A. 1995 Thermocapillary convection in a rectangular cavity with a deformable interface. Adv. Space Res. 16 (7), 41–53.Google Scholar
Murphy, S. V. & Atala, A. 2014 3D bioprinting of tissues and organs. Nat. Biotechnol. 32, 773–785.Google Scholar
Nepomnyashchy, A. A., Simanovskii, I. B. & Braverman, L. M. 2001 Stability of thermocapillary flows with inclined temperature gradient. J. Fluid Mech. 442, 141–155.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Saenz, P. J., Valluri, P., Sefiane, K., Karapetsas, G. & Matar, O. K. 2013 Linear and nonlinear stability of hydrothermal waves in planar liquid layers driven by thermocapillarity. Phys. Fluids 25, 094101.Google Scholar
Sames, W. J., List, F. A., Pannala, S., Dehoff, R. R. & Babu, S. S. 2016 The metallurgy and processing science of metal additive manufacturing. Intl Mater. Rev. 61 (5), 315–360.Google Scholar
Sen, A. K. & Davis, S. H. 1982 Steady thermocapillary flows in two-dimensional slots. J. Fluid Mech. 121, 163–186.Google Scholar
Shevtsova, V. M., Nepomnyashchy, A. A. & Legros, J. C. 2003 Thermocapillary-buoyancy convection in a shallow cavity heated from the side. Phys. Rev. E 67, 066308.Google Scholar
Smith, M. K. & Davis, S. H. 1983a Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 114–144.Google Scholar
Smith, M. K. & Davis, S. H. 1983b Instabilities of dynamic thermocapillary liquid layers. Part 2. Surface-wave instabilities. J. Fluid Mech. 132, 145–162.Google Scholar
Takashima, M. 1981 Surface tension driven instability in a horizontal liquid layer with a deformable free surface. I. Stationary convection. J. Phys. Soc. Japan 50 (8), 2745–2750.Google Scholar
Wegst, U. G. K., Bai, H., Saiz, E., Tomsia, A. P. & Ritchie, R. O. 2015 Bioinspired structural materials. Nat. Mater. 14 (1), 23–36.Google Scholar
Xu, J. & Zebib, A. 1998 Oscillatory two- and three-dimensional thermocapillary convection. J. Fluid Mech. 364, 187–209.Google Scholar
Yung, W. K. C., Sun, B., Meng, Z., Huang, J., Jin, Y., Choy, H. S., Cai, Z., Li, G., Ho, C. L., Yang, J. & Wong, W. Y. 2016 Additive and photochemical manufacturing of copper. Sci. Rep. 6, 39584.Google Scholar
Zebib, A., Homsy, G. M. & Meiburg, E. 1985 High Marangoni number convection in a square cavity. Phys. Fluids 28 (12), 3467–3476.Google Scholar
Zheng, Y., He, Z., Gao, Y. & Liu, J. 2013 Direct desktop printed-circuits-on-paper flexible electronics. Sci. Rep. 3, 1786.Google Scholar