Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T19:16:04.865Z Has data issue: false hasContentIssue false
  • English
  • Français

Azimuthal instability of divergent flows

Published online by Cambridge University Press:  26 April 2006

Vladimir Shtern  and
Fazle Hussain
Vladimir Shtern
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, USA
Fazle Hussain
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate a new mechanism for instability (named divergent instability), characterized by the formation of azimuthal cells, and find it to be a generic feature of three-dimensional steady axisymmetric flows of viscous incompressible fluid with radially diverging streamlines near a planar or conical surface. Four such flows are considered here: (i) Squire–Wang flow in a half-space driven by surface stresses; (ii) recirculation of fluid inside a conical meniscus; (iii) two-cell regime of free convection above a rigid cone; and (iv) Marangoni convection in a half-space induced by a point source of heat (or surfactant) placed at the liquid surface. For all these cases, bifurcation of the secondary steady solutions occurs: for each azimuthal wavenumber m = 2, 3,…, a critical Reynolds number (Re*) exists. The intent to compare with experiments led us to investigate case (iv) in more detail. The results show a non-trivial influence of the Prandtl number (Pr): instability does not occur in the range 0.05 < Pr < 1; however, outside this range, Re*(m) exists and has bounded limits as Pr tends to either zero or infinity. A nonlinear analysis shows that the primary bifurcations are supercritical and produce new stable regimes. We find that the neutral curves intersect and subcritical secondary bifurcation takes place; these suggest the presence of complex unsteady dynamics in some ranges of Re and Pr. These features agree with the experimental data of Pshenichnikov & Yatsenko (Pr = 103).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 256 , November 1993 , pp. 535 - 560
Copyright
© 1993 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

Bayley, A. G. 1988 Electrostatic Spraying of Liquids. Wiley.
Bratukhin, Yu. K. & Maurin, L. M. 1967 Thermocapillary convection in a fluid filling a half-space. J. Appl. Math. Mech. 31, 605–608.Google Scholar
Goldshtik, M. A., Hussain, F. & Shtern, V. N. 1991 Symmetry breaking in vortex-source and Jeffery-Hamel flows. J. Fluid Mech. 232, 521–566.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1990a Collapse in conical viscous flows. J. Fluid Mech. 218, 483–508.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1990b Free convection near a thermal quadrupole. Int J. Heat Mass Transfer 33, 1475–1483.Google Scholar
Guckenheimer, J. & Holmes, P. J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Hayati, I., Bayley, A. I. & Tadros, Th. F. 1986 Mechanism of stable jet formation in electrohydrodynamic atomization. Nature 319, 41–43.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions, Vol. 1. Springer.
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice Hall.
Pshenichnikov, A. F. & Yatsenko, S. S. 1974 Convective diffusion from localized source of surfactant. Hydrodynamics V, 175–181 (Scientific papers of Perm University, in Russian.)
Squire, B. 1952 Some viscous fluid flow problems. 1. Jet emerging from a hole in a plane wall. Phil. Mag. 43, 942–945.Google Scholar
Wang, C. V. 1971 Effect of spreading of material of the surface of a liquid. Intl J. Nonlinear Mech. 6, 255–262.Google Scholar
Wang, C. V. 1991 Exact solutions of the steady-state Navier–Stokes equations. Ann. Rev. Fluid Mech. 23, 159–177.Google Scholar
Yudovich, V. I. 1965 Stability of steady flows of viscous incompressible fluid. Sov. Phys. Dokl. 10(4), 293–295.Google Scholar