Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T06:43:46.852Z Has data issue: false hasContentIssue false
  • English
  • Français

Developed liquid film passing a trailing edge under the action of gravity and capillarity

Published online by Cambridge University Press:  10 July 2018

B. Scheichl Open the ORCID record for B. Scheichl [Opens in a new window] ,
R. I. Bowles  and
G. Pasias
B. Scheichl*
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Faculty of Mechanical Engineering, Technische Universität Wien, Tower BA/E322, Getreidemarkt 9, 1060 Vienna, Austria AC2T research GmbH (Austrian Excellence Center for Tribology), Viktor-Kaplan-Straße 2/C, 2700 Wiener Neustadt, Austria
R. I. Bowles
Affiliation:
Department of Mathematics, Faculty of Mathematical and Physical Sciences, University College London (UCL), Gower Street, London WC1E 6BT, UK
G. Pasias
Affiliation:
Department of Mathematics, Faculty of Mathematical and Physical Sciences, University College London (UCL), Gower Street, London WC1E 6BT, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the asymptotic structure of a steady developed viscous thin film passing the sharp trailing edge of a horizontally aligned flat plate under the weak action of gravity acting vertically and surface tension. The surprisingly rich details of the flow in the immediate vicinity of the trailing edge are elucidated both analytically and numerically. As a central innovation, we demonstrate how streamline curvature serves to regularise the edge singularity apparent on larger scales via generic viscous–inviscid interaction. This is shown to be provoked by weak disturbances of accordingly strong exponential downstream growth, which we trace from the virtual origin of the flow towards the trailing edge. They represent a prototype of the precursor to free interaction in the most general sense, which, interestingly, has not attracted due attention previously. Moreover, we delineate how an increased effect of gravity involves marginally choked flow at the edge.

JFM classification

Boundary Layers: Boundary Layers Interfacial Flows (free surface): Thin films Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018 , pp. 924 - 953
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

Balmforth, N. J. 1999 Shear instability in shallow water. J. Fluid Mech. 387, 97–127.Google Scholar
Bowles, R. I.1990 Applications of nonlinear viscous-inviscid interactions in liquid layer flows and transonic boundary layer transition. PhD thesis, Department of Mathematics, University of London.Google Scholar
Bowles, R. I. & Smith, F. T. 1992 The standing hydraulic jump: theory, computations and comparisons with experiments. J. Fluid Mech. 242, 145–168.Google Scholar
Brotherton-Ratcliffe, R. V. & Smith, F. T. 1989 Viscous effects can destabilize linear and nonlinear water waves. Theor. Comput. Fluid Dyn. 1 (1), 21–39.Google Scholar
Brown, S. N., Stewartson, K. & Williams, P. G. 1976 On expansive free interactions in boundary layers. Proc. R. Soc. Edin. A 74 (21), 271–283.Google Scholar
Duchesne, A., Lebon, L. & Limat, L. 2014 Constant Froude number in a circular hydraulic jump and its implication on the jump radius selection. Europhys. Lett. 107 (5), 54002.Google Scholar
Gajjar, J. & Smith, F. T. 1983 On hypersonic self-induced separation, hydraulic jumps and boundary layers with algebraic growth. Mathematika 30 (1), 77–93.Google Scholar
Higuera, F. J. 1994 The hydraulic jump in a viscous laminar flow. J. Fluid Mech. 274, 69–92.Google Scholar
Higuera, F. J. & Liñán, A. 1993 Choking conditions for nonuniform viscous flow. Phys. Fluids A 5 (3), 568–570.Google Scholar
Keller, J. B. & Weitz, M. L. 1957 Upward ‘falling’ jets and surface tension. J. Fluid Mech. 2 (2), 201–203.Google Scholar
Kluwick, A., Cox, E. A., Exner, A. & Grinschgl, Ch. 2010 On the internal structure of weakly nonlinear bores in laminar high Reynolds number flow. Acta Mechanica 210 (1–2), 135–157.Google Scholar
Khayat, R. E. 2016 Slipping free jet flow near channel exit at moderate Reynolds numbers for large slip length. J. Fluid Mech. 793, 667–708.Google Scholar
Khayat, R. E. 2017 Initial development of a free surface wall jet at moderate Reynolds number. J. Fluid Mech. 826, 235–269.Google Scholar
Reyhner, T. A. & Flügge-Lotz, I. 1968 The interaction of shock wave with a laminar boundary layer. Intl J. Non-Linear Mech. 3 (2), 173–199.Google Scholar
Rojas, N. & Tirapegui, E. 2015 Harmonic solutions for polygonal hydraulic jumps in thin fluid films. J. Fluid Mech. 780, 99–119.Google Scholar
Rothmayer, A. P. 1989 The viscous flow through symmetric collapsible channels. Mathematika 36 (1), 153–181.Google Scholar
Rothmayer, A. P. & Levine, H. A. 1991 The viscous flow through a slightly distorted flexible tube. Theor. Comput. Fluid Dyn. 2 (4), 193–210.Google Scholar
Scheichl, B. & Bowles, R. I. 2017 On transcritical states in viscous flow passing the edge of a horizontal plate. Proc. Appl. Maths Mech. 17 (1), 663–664.Google Scholar
Scheichl, B. & Kluwick, A. 2018 Laminar spread of a circular liquid jet impinging axially onto a rotating disc. J. Fluid Mech. (submitted in revised form).Google Scholar
Smith, F. T. 1977 Upstream interactions in channel flows. J. Fluid Mech. 79 (4), 631–655.Google Scholar
Smith, F. T. & Duck, P. W. 1977 Separation of jets or thermal boundary layers from a wall. Q. J. Mech. Appl. Maths 30 (2), 143–156.Google Scholar
The Numerical Algorithms Group (NAG)2017 NAG C Library Mark 26.1, Oxford, UK (www.nag.com).Google Scholar
Watson, E. J. 1964 The radial spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20 (3), 481–499.Google Scholar