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Self-induced flow over a cylinder in a stratified fluid

Published online by Cambridge University Press:  05 June 2023

Jim Thomas Open the ORCID record for Jim Thomas [Opens in a new window]  and
Roberto Camassa
Jim Thomas*
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore 560065, India
Roberto Camassa
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, NC 27599, USA
*
†Email address for correspondence: jimthomas.edu@gmail.com
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Abstract

In this paper we study the self-induced low-Reynolds-number flow generated by a cylinder immersed in a stratified fluid. In the low Péclet limit, where the Péclet number is the ratio of the radius of the cylinder and the Phillips length scale, the flow is captured by a set of linear equations obtained by linearising the governing equations with respect to the prescribed far field conditions. We specifically focus on the low Péclet regime and develop a Green's function approach to solve the linearised equations governing the flow over the cylinder. We cross check our analytical solution against numerical solution of the nonlinear equations to obtain the range of the Péclet numbers for which the linear solution is valid. We then take advantage of the analytical solution to find explicit far-field decay rates of the flow. Our detailed analysis points out that the streamfunction and the velocity field decays algebraically in the far field. Intriguingly, this algebraic decay of the flow is much slower when compared with the exponential decay of the flow generated by a slow moving cylinder in the homogeneous Stokes regime, in the absence of stratification. Consequently, the flow generated by a cylinder in the stratified Stokes regime will have a larger domain of influence when compared with the flow generated by a cylinder in the homogeneous Stokes regime.

JFM classification

Geophysical and Geological Flows: Stratified flows Low-Reynolds-number flows: Low-Reynolds-number flows Geophysical and Geological Flows: Ocean processes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 964 , 10 June 2023 , A38
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Ardekani, A.M. & Stocker, R. 2010 Stratlets: low Reynolds number point-force solutions in a stratified fluid. Phys. Rev. Lett. 105, 084502.10.1103/PhysRevLett.105.084502CrossRefGoogle Scholar
Batchelor, G.K. 1974 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Burd, A.B. & Jackson, G.A. 2009 Particle aggregation. Annu. Rev. Mar. Sci. 1, 65–90.CrossRefGoogle ScholarPubMed
Camassa, R., Falcon, C., Lin, J., McLaughlin, R.M. & Mykins, N. 2010 A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number. J. Fluid Mech. 664, 436–465.10.1017/S0022112010003800CrossRefGoogle Scholar
Camassa, R., Harris, D.M., Hunt, R., Kilic, Z. & McLaughlin, R.M. 2019 A first-principle mechanism for particulate aggregation and self-assembly in stratified fluids. Nat. Commun. 10, 5804.10.1038/s41467-019-13643-yCrossRefGoogle ScholarPubMed
D'Asaro, E.A. 2003 Performance of autonomous Lagrangian floats. J. Atmos. Ocean. Technol. 20, 896–911.10.1175/1520-0426(2003)020<0896:POALF>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Drescher, K., Goldstein, R.E., Michel, N., Polin, M. & Tuval, I. 2010 Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett. 105, 168101.CrossRefGoogle ScholarPubMed
Durham, W.M. & Stocker, R. 2012 Thin phytoplankton layers: characteristics, mechanisms, and consequences. Annu. Rev. Mar. Sci. 4, 177–207.CrossRefGoogle ScholarPubMed
Fouxon, I. & Leshansky, A. 2014 Convective stability of turbulent Boussinesq flow in the dissipative range and flow around small particles. Phys. Rev. E 90, 053002.CrossRefGoogle ScholarPubMed
Gelfand, I.M. & Shilov, G.E. 1964 Generalized Functions. Volume I: Properties and Operations. Academic.Google Scholar
Guasto, J.S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373–400.10.1146/annurev-fluid-120710-101156CrossRefGoogle Scholar
Hanazaki, H. 1988 A numerical study of three-dimensional stratified flow past a sphere. J. Fluid Mech. 192, 393–419.10.1017/S0022112088001910CrossRefGoogle Scholar
Kisselev, A.V. 2021 Exact expansions of Hankel transforms and related integrals. Ramanujan J. 55, 349–367.CrossRefGoogle Scholar
Lamb, H. 1911 On the uniform motion of a sphere through a viscous fluid. Phil. Mag. 21, 112–121.CrossRefGoogle Scholar
Leal, L.G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Lee, H., Fouxon, I. & Lee, C. 2019 Sedimentation of a small sphere in stratified fluid. Phys. Rev. Fluids 4, 104101.CrossRefGoogle Scholar
List, E.J. 1971 Laminar momentum jets in a stratified fluid. J. Fluid Mech. 45, 561–574.10.1017/S0022112071000193CrossRefGoogle Scholar
MacIntyre, S., Alldredge, A.L. & Gottschalk, C.C. 1995 Accumulation of marine snow at density discontinuities in the water column. Limnol. Oceanogr. 40, 449–468.CrossRefGoogle Scholar
Okino, S., Akiyama, S. & Hanazaki, H. 2017 Velocity distribution around a sphere descending in a linearly stratified fluid. J. Fluid Mech. 826, 759–780.10.1017/jfm.2017.474CrossRefGoogle Scholar
Oseen, C.W. 1910 Über die stokessche formel und über die verwandte aufgabe in der hydrodynamik. Ark. Mat. Astron. Fys. 29, 143–152.Google Scholar
Phillips, O.M. 1970 On flows induced by diffusion in a stably stratified fluid. Deep Sea Res. 17, 435–443.Google Scholar
Pozrikidis, C. 2017 Fluid Dynamics: Theory, Computation, and Numerical Simulation, 3rd edn. Springer.CrossRefGoogle Scholar
Stokes, G.G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Phil. Soc. 9, 8–106.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.Google Scholar
Wunsch, C. 1970 On oceanic boundary mixing. Deep Sea Res. 17, 293–301.Google Scholar