Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T16:02:07.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-Reynolds-number diffusion-driven flow around a horizontal cylinder

Published online by Cambridge University Press:  24 July 2017

Anis A. M. Alias  and
Michael A. Page Open the ORCID record for Michael A. Page [Opens in a new window]
Anis A. M. Alias
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
Michael A. Page*
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Diffusion-driven flow occurs when any insulated sloping surface is in contact with a quiescent stratified and viscous fluid. This startling fluid motion is generally very slow, and is caused by a hydrostatic pressure imbalance due to bending of isotherms near the surface. In contrast to previous studies of the phenomenon, the low-Reynolds-number case is considered here, for which the induced steady motion is influenced by viscous and density diffusion over a much larger length scale than the size of the insulated object. The relevant linear equations of steady two-dimensional motion in a linearly stratified fluid are solved for a circular cylinder using a matched two-region approach that yields analytical solutions for the streamfunction and the density variations both close to and far from the object. The exact analytical expressions for the solutions in the ‘outer-flow region’ are new, and after matching enable accurate solutions to be evaluated easily at any point. Similar qualitative behaviour is expected under similar conditions near isolated objects of other shapes, including for a sphere. Implications for multiple objects are also discussed.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 1035 - 1055
Check for updates
Copyright
© 2017 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

Alias, A. A. M.2015 Viscous stratified flow past an object. PhD thesis, Monash University.Google Scholar
Allshouse, M. R., Barad, M. F. & Peacock, T. 2010 Propulsion generated by diffusion-driven flow. Nat. Phys. 6, 516–519.CrossRefGoogle Scholar
Ardekani, A. M. & Stocker, R. 2010 Stratlets: low Reynolds number point-force solutions in a stratified fluid. Phys. Rev. Lett. 105 (8), 084502.Google Scholar
Baidulov, V. G., Matyushin, P. V. & Chashechkin, Y. D. 2005 Structure of a diffusion-induced flow near a sphere in a continuously stratified fluid. Dokl. Phys. 50 (4), 195–199.CrossRefGoogle Scholar
Baidulov, V. G., Matyushin, P. V. & Chashechkin, Y. D. 2007 Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid. Fluid Dyn. 42 (2), 255–267.Google Scholar
Candelier, F., Mehaddi, R. & Vauquelin, O. 2014 The history force on a small particle in a linearly stratified fluid. J. Fluid Mech. 749, 184–200.CrossRefGoogle Scholar
Dell, R. W. & Pratt, L. J. 2015 Diffusive boundary layers over varying topography. J. Fluid Mech. 769, 635–653.CrossRefGoogle Scholar
Jones, D. S. 1966 Generalised Functions. McGraw-Hill.Google Scholar
Koh, R. C. Y. 1966 Viscous stratified flow towards a sink. J. Fluid Mech. 24 (3), 555–575.CrossRefGoogle Scholar
Lighthill, M. J. 1958 Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.CrossRefGoogle Scholar
List, E. J. 1971 Laminar momentum jets in a stratified fluid. J. Fluid Mech. 45 (3), 561–574.CrossRefGoogle Scholar
Moore, D. W. & Saffman, P. G. 1969 The structure of free vertical shear layers in a rotating fluid and the motion produced by a slowly rising body. Phil. Trans. R. Soc. Lond. A 264 (1156), 597–634.Google Scholar
Page, M. A. 2011a Combined diffusion-driven and convective flow in a tilted square container. Phys. Fluids 23 (5), 056602.Google Scholar
Page, M. A. 2011b Steady diffusion-driven flow in a tilted square container. Q. J. Mech. Appl. Maths 64 (3), 319–348.Google Scholar
Page, M. A. & Johnson, E. R. 2008 On steady linear diffusion-driven flow. J. Fluid Mech. 606, 433–443.Google Scholar
Page, M. A. & Johnson, E. R. 2009 Steady nonlinear diffusion-driven flow. J. Fluid Mech. 629, 299–309.Google Scholar
Phillips, O. M. 1970 On flows induced by diffusion in a stably stratified fluid. Deep-Sea Res. Oceanogr. Abstracts 17 (3), 435–443.Google Scholar
Quon, C. 1976 Diffusively induced boundary layers in a tilted square cavity: a numerical study. J. Comput. Phys. 22 (4), 459–485.Google Scholar
Quon, C. 1983 Convection induced by insulated boundaries in a square. Phys. Fluids 26 (3), 632–637.Google Scholar
Quon, C. 1989 Cross-sectional convection induced by an insulated boundary in a cylinder. J. Fluid Mech. 202, 201–215.Google Scholar
Spiegel, M., Lipschutz, S. & Liu, J. 2009 Mathematical Handbook of Formulas and Tables, 3rd edn. McGraw Hill.Google Scholar
Tritton, D. J. 1988 Physical Fluid Dynamics, 2nd edn. Oxford University Press.Google Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics, annotated edn. Parabolic.Google Scholar
Wagner, G. L., Young, W. R. & Lauga, E. 2014 Mixing by microorganisms in stratified fluids. J. Mar. Res. 72, 47–72.Google Scholar
Woods, A. W. 1991 Boundary-driven mixing. J. Fluid Mech. 226, 625–654.Google Scholar
Wunsch, C. 1970 On oceanic boundary mixing. Deep-Sea Res. Oceanogr. Abstracts 17 (2), 293–301.Google Scholar
Zagumennyi, I. V. & Chashechkin, Y. D. 2013a Diffusion induced flows on a strip: Theoretical, numerical and laboratory modeling. Procedia IUTAM 8, 257–266.Google Scholar
Zagumennyi, I. V. & Chashechkin, Y. D. 2013b Fine structure of unsteady diffusion-induced flow over a fixed plate. Fluid Dyn. 48 (3), 374–388.Google Scholar
Zagumennyi, I. V. & Chashechkin, Y. D. 2013c The structure of convective flows driven by density variations in a continuously stratified fluid. Phys. Scr. 2013, 014034.Google Scholar
Zvirin, Y. & Chadwick, R. S. 1975 Settling of an axially symmetric body in a viscous stratified fluid. Intl J. Multiphase Flow 1, 743–752.CrossRefGoogle Scholar