Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-23T11:42:15.867Z Has data issue: false hasContentIssue false
  • English
  • Français

Centrifugally forced Rayleigh–Taylor instability

Published online by Cambridge University Press:  08 August 2018

M. M. Scase Open the ORCID record for M. M. Scase [Opens in a new window]  and
R. J. A. Hill
M. M. Scase*
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
R. J. A. Hill
Affiliation:
School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
*
†Email address for correspondence: matthew.scase@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the effect of high rotation rates on two liquid layers that initially form concentric cylinders, centred on the axis of rotation. The configuration may be thought of as a fluid–fluid centrifuge. There are two types of perturbation to the interface that may be considered, an azimuthal perturbation around the circumference of the interface and a varicose perturbation in the axial direction along the length of the interface. It is the first of these types of perturbation that we consider here, and so the flow may be considered essentially two-dimensional, taking place in a circular domain. A linear stability analysis is carried out on a perturbation to the hydrostatic background state and a fourth-order Orr–Sommerfeld-like equation that governs the system is derived. We consider the dynamics of systems of stable and unstable configurations, inviscid and viscous fluids, immiscible fluid layers with surface tension and miscible fluid layers that may have some initial diffusion of density. In the most simple case of two layers of inviscid fluid separated by a sharp interface with no surface tension acting, we show that the effects of the curvature of the interface and the confinement of the system may be characterised by a modified Atwood number. The classical Atwood number is recovered in the limit of high azimuthal wavenumber, or the outer fluid layer being unconfined. Theoretical predictions are compared with numerical experiments and the agreement is shown to be good. We do not restrict our analysis to equal volume fluid layers and so our results also have applications in coating and lubrication problems in rapidly rotating systems and machinery.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 852 , 10 October 2018 , pp. 543 - 577
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

Alvarez-Lacalle, E., Ortín, J. & Casademunt, J. 2004 Low viscosity contrast fingering in a rotating Hele-Shaw cell. Phys. Fluids 16 (4), 908–924.Google Scholar
Baldwin, K. A., Scase, M. M. & Hill, R. J. A. 2015 The inhibition of the Rayleigh–Taylor instability by rotation. Sci. Rep. 5, 11706.Google Scholar
Boffetta, G., Mazzino, A. & Musacchio, S. 2016 Rotating Rayleigh–Taylor turbulence. Phys. Rev. Fluids 1, 054405.Google Scholar
Carnevale, G. F., Orlandi, P., Zhou, Y. & Kloosterziel, R. C. 2002 Rotational suppression of Rayleigh–Taylor instability. J. Fluid Mech. 457, 181–190.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Deshpande, S. S., Anumolu, L. & Trujillo, M. F. 2012 Evaluating the performance of the two-phase flow solver interFoam. Comput. Sci. Disc. 5, 014016.Google Scholar
Driscoll, T. A., Hale, N. & Trefethen, L. N.(Eds) 2014 Chebfun Guide. Pafnuty Publications.Google Scholar
Hide, R. 1956 The character of the equilibrium of a heavy, viscous, incompressible, rotating fluid of variable density II. Two special cases. Q. J. Mech. Appl. Maths 9, 35–50.Google Scholar
Hocking, L. M. & Michael, D. H. The stability of a column of rotating liquid. Mathematika 6, 25–32.Google Scholar
Issa, R. I. 1986 Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comput. Phys. 62, 40–65.Google Scholar
van Leer, B. 1974 Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14 (4), 361–370.Google Scholar
Patzek, T. W., Basaran, O. A., Benner, R. E. & Scriven, L. E. Nonlinear oscillations of two-dimensional rotating inviscid drops. J. Comput. Phys. 116, 3–25.Google Scholar
Peng, J. & Zhu, K.-Q. Linear instability of two-fluid Taylor–Couette flow in the presence of surfactant. J. Fluid Mech. 651, 357–385.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. R. Math. Soc. 14, 170–177.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Scase, M. M., Baldwin, K. A. & Hill, R. J. A. 2017a Magnetically induced rotating Rayleigh–Taylor instability. J. Vis. Exp. 121, e55088.Google Scholar
Scase, M. M., Baldwin, K. A. & Hill, R. J. A. 2017b Rotating Rayleigh–Taylor instability. Phys. Rev. Fluids 2, 024801.Google Scholar
Schwartz, L. W. 1989 Instability and fingering in a rotating Hele-Shaw cell or porous medium. Phys. Fluids A 1, 167–169.Google Scholar
Tao, J. J., He, X. T., Ye, W. H. & Busse, F. H. 2013 Nonlinear Rayleigh–Taylor instability of rotating inviscid fluids. Phys. Rev. E 87, 013001.Google Scholar
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213–218.Google Scholar
Taylor, G. I. 1950 The instability of fluid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201, 192–196.Google Scholar
Weller, H. G., Tabor, G., Jasak, H. & Fureby, C. 1998 A tensorial approach to computational continuum mechanics using object-oriented techniques. J. Comp. Phys. 12 (6), 620–631.Google Scholar
Zhou, Y. 2007 Unification and extension of the similarity scaling criteria and mixing transition for studying astrophysics using high energy density laboratory experiments or numerical simulations. Phys. Plasmas 14, 082701.Google Scholar
Zhou, Y. 2017a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720–722, 1–136.Google Scholar
Zhou, Y. 2017b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723–725, 1–160.Google Scholar