Abstract
The processes which lead to topological reconfiguration of fluid interfaces are studied in Hele-Shaw flow. This is motivated by recent experiments of droplet forma tion through the osculation of two interfaces separating immiscible fluids. In the Hele-Shaw approximation, such a configuration reduces to interface dynamics through their representation as vortex sheets. To begin, we focus on thin fluid layers and develop an asymptotic theory which yields simplified, nonlinear equations for the local thickness. As particular examples, we consider the dynamics of gravity driven fluid jets, as well as the motion of unstably stratified fluid layers. In both cases, the bounding interfaces collide at a finite time, with an associated singularity in the fluid velocity. Some comparison is made with simulations of the full Hele-Shaw equations, and connections with experiments are discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baker, G. and Shelley, M. 1986 Boundary integral techniques for multi-connected domains.J. Comp. Phys. 64, 112.
Baker, G., and Shelley, M. 1991 On the relation between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161.
Carrier, G., Krook, M. & Pearson, C. 1966 Functions of a Complex Variable, New York: Mcgraw-Hill.
Constantin, P., Dupont, T., Goldstein, R., Kadanoff, L., Shelley, M. & Zhou, S. 1992 Droplet Separation in the Hele-Shaw Cell? submitted to Physical Review A.
Dai, W., and Shelley, M. 1992 A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble. submitted to Physics of Fluids A.
Dance, M., Goldstein, R.E., Kachru, S. & Shyamsunder, E. 1992. unpublished.
Goldstein, R.E., Mason, T. & Shyamsunder, E. 1991. unpublished.
Goldstein, R.E., Pesci, A.I. & Shelley, M.J. 1992 Interface motion and topological transitions in Hele-Shaw flow. preprint.
Goldstein, R.E. and Petrich, D. 1992 Solitons, Euler’s equation, and vortex patch dynamics. Phys. Rev. Lett. 69, 555.
Hou, T, Lowengrub, J. & Shelley, M.J. 1992. In preparation.
Kellogg, D. Foundations of Potential Theory, New York: Dover, 1929.
Kuramoto, Y. and Tsuzuki, T. 1976 Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356.
Shelley, M. 1992 A study of singularity formation in vortex sheet motion by a spectrally accurate vortex method. J. Fluid Mech., in press.
Pumir, A., Shraiman, B. & Siggia, E. 1992 Vortex morphology and Kelvin’s theorem. Phys. Rev. A 45, 5351.
Sivashinsky, G.I. 1977 Non-linear analysis of hydrodynamic instability in laminar flames. 1. Derivation of basic equations. Acta Astronaut 4, 1177.
Tryggvason, G. and Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136, 1.
Zhou, S. 1992 . Ph.D. thesis, University of Chicago, unpublished.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Shelley, M.J., Goldstein, R.E., Pesci, A.I. (1993). Topological Transitions in Hele-Shaw Flow. In: Caflisch, R.E., Papanicolaou, G.C. (eds) Singularities in Fluids, Plasmas and Optics. NATO ASI Series, vol 404. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2022-7_13
Download citation
DOI: https://doi.org/10.1007/978-94-011-2022-7_13
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4894-1
Online ISBN: 978-94-011-2022-7
eBook Packages: Springer Book Archive