Skip to main content Accessibility help
×
Home

Boundary layers in Helmholtz flows

  • M. R. Moore (a1), R. Cimpeanu (a1) (a2), H. Ockendon (a1), J. R. Ockendon (a1) and J. M. Oliver (a1)...

Abstract

Recent comparisons between classical Wagner theory for the impact of two liquid droplets and direct numerical simulations in Cimpeanu & Moore (J. Fluid Mech., vol. 856, 2018, pp. 764–796) show that, in some regimes, the inviscid theory over-predicts the thickness of the root of the splash jet that forms in the impact, while also struggling to predict the angle at which the jet is emitted. The effect of capillary and viscous perturbations to Helmholtz flows was investigated in a previous study, see Moore et al. (J. Fluid Mech., vol. 742, 2014, R1). However, the paper in question ignored a term in the second-order perturbation analysis, which needs to be included in order to predict the displacement of the inviscid free boundary to lowest order. In this paper, we derive a singular integro-differential equation for the free-surface perturbations caused by viscosity in Helmholtz flows and discuss its application both in the context of Wagner theory and more generally. In particular, viscosity can induce non-monotonic behaviour in the free boundary profiles near points of maximum curvature.

Copyright

Corresponding author

Email address for correspondence: moorem@maths.ox.ac.uk

References

Hide All
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Cimpeanu, R. & Moore, M. R. 2018 Early-time jet formation in liquid–liquid impact problems: theory and simulations. J. Fluid Mech. 856, 764796.
Hooper, A. P. & Boyd, W. G. C. 1983 Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507528.
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215230.
Milne-Thomson, L. M. 1996 Theoretical Hydrodynamics. Dover.
Moore, M. R., Ockendon, H., Ockendon, J. R. & Oliver, J. M. 2014 Capillary and viscous perturbations to Helmholtz flows. J. Fluid Mech. 742, R1.
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572600.
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 58385866.
Purvis, R. & Smith, F. T. 2005 Droplet impact on water layers: post-impact analysis and computations. Phil. Trans. R. Soc. Lond. A 363, 12091221.
Wagner, H. 1932 Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12, 193215.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Boundary layers in Helmholtz flows

  • M. R. Moore (a1), R. Cimpeanu (a1) (a2), H. Ockendon (a1), J. R. Ockendon (a1) and J. M. Oliver (a1)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.