Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-18T18:48:01.830Z Has data issue: false hasContentIssue false

Instability of stationary unbounded stratified fluid

Published online by Cambridge University Press:  26 April 2006

G. K. Batchelor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
J. M. Nitsche
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

Abstract

Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.

Type
Research Article
Copyright
© 1991 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

Batchelor, G. K. 1988 A new theory of the instability of a uniform fluidized bed. J. Fluid Mech. 193, 75110.Google Scholar
Batchelor, G. K. 1991 The formation of bubbles in fluidized beds. Proc. of Symposium honoring John W. Miles on his 70th birthday. Scripps Institution of Oceanography, Reference Series 91.
Bellman, R. & Pennington, R. H. 1954 Effects of surface tension and viscosity on Taylor instability. Q. Appl. Maths 12, 151162.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Dongarra, J. J. & Moler, C. B. 1984 EISPACK – a package for solving matrix eigenvalue problems. Argonne National Laboratory, Math. & Comp. Sci. Div., Tech. Memo.
Gribov, V. N. & Gurevich, L. E. 1957 On the theory of the stability of a layer located at a superadiabatic temperature gradient in a gravitational field. Sov. Phys. JETP 4, 720729.Google Scholar
Ince, E. L. 1944 Ordinary Differential Equations. Dover.
Jackson, R. 1963 The mechanics of fluidized beds. I The stability of the state of uniform fluidization. Trans. Inst. Chem. Engrs 41, 1321.Google Scholar
Kynch, G. J. 1952 A theory of sedimentation. Trans. Faraday Soc. 48, 166176.Google Scholar
Matthews, P. C. 1988 A model for the onset of penetrative convection. J. Fluid Mech. 188, 571583.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I.. Proc. R. Soc. Lond. A 201, 192196.Google Scholar
Whittaker, E. T. & Watson, G. N. 1915 A Course of Modern Analysis. Cambridge University Press.
Yih, C.-S. 1959 Thermal instability of viscous fluids. Q. Appl. Maths 17, 2542.Google Scholar