Inertial drag on a sphere settling in a stratified fluid. (English) Zbl 1415.76691
Summary: We compute the drag force on a sphere settling slowly in a quiescent, linearly stratified fluid. Stratification can significantly enhance the drag experienced by the settling particle. The magnitude of this effect depends on whether fluid-density transport around the settling particle is due to diffusion, to advection by the disturbance flow caused by the particle or due to both. It therefore matters how efficiently the fluid disturbance is convected away from the particle by fluid-inertial terms. When these terms dominate, the Oseen drag force must be recovered. We compute by perturbation theory how the Oseen drag is modified by diffusion and stratification. Our results are in good agreement with recent direct numerical simulation studies of the problem.
MSC:
| 76T20 | Suspensions |
| 76D50 | Stratification effects in viscous fluids |
| 76V05 | Reaction effects in flows |
References:
| [1] | Abaid, N.; Adalsteinsson, D.; Agyapong, A.; Mclaughlin, R. M., An internal splash: levitation of falling spheres in stratified fluids, Phys. Fluids, 16, 5, 1567-1580, (2004) · Zbl 1186.76012 · doi:10.1063/1.1687685 |
| [2] | Alias, A. A.; Page, M. A., Low-Reynolds-number diffusion-driven flow around a horizontal cylinder, J. Fluid Mech., 825, 1035-1055, (2017) · Zbl 1374.86006 · doi:10.1017/jfm.2017.427 |
| [3] | Ardekani, A. M.; Doostmohammadi, A.; Desai, N., Transport of particles, drops, and small organisms in density stratified fluids, Phys. Rev. Fluids, 2, 10, (2017) · doi:10.1103/PhysRevFluids.2.100503 |
| [4] | Ardekani, A. M.; Stocker, R., Stratlets: low Reynolds number point-force solutions in a stratified fluid, Phys. Rev. Lett., 105, (2010) · doi:10.1103/PhysRevLett.105.084502 |
| [5] | Bergström, B.; Strömberg, J.-O., Behavioural differences in relation to pycnoclines during vertical migration of the euphausiids meganyctiphanes norvegica (m. sars) and thysanoessa raschii (m. sars), J. Plankton Res., 19, 2, 255-261, (1997) · doi:10.1093/plankt/19.2.255 |
| [6] | Blanchette, F.; Shapiro, A. M., Drops settling in sharp stratification with and without marangoni effects, Phys. Fluids, 24, 4, (2012) · doi:10.1063/1.4704790 |
| [7] | Bloomfield, L. J.; Kerr, R. C., Turbulent fountains in a stratified fluid, J. Fluid Mech., 358, 335-356, (1998) · Zbl 0945.76533 · doi:10.1017/S0022112097008252 |
| [8] | Camassa, R.; Falcon, C.; Lin, J.; Mclaughlin, R. M.; Parker, R., Prolonged residence times for particles settling through stratified miscible fluids in the stokes regime, Phys. Fluids, 21, 3, (2009) · Zbl 1183.76124 · doi:10.1063/1.3094922 |
| [9] | Candelier, F.; Mehaddi, R.; Vauquelin, O. |
| [10] | Candelier, F.; Mehaddi, R.; Vauquelin, O., The history force on a small particle in a linearly stratified fluid, J. Fluid Mech., 749, 183-200, (2014) · doi:10.1017/jfm.2014.219 |
| [11] | Candelier, F.; Mehlig, B.; Magnaudet, J. |
| [12] | Chadwick, R. S.; Zvirin, Y., The effect of ambient density stratification on the oseen drag of small spherical particles, Israel J. Technol., 12, 262-267, (1974) · Zbl 0304.76043 |
| [13] | Chadwick, R. S.; Zvirin, Y., Slow viscous flow of an incompressible stratified fluid past a sphere, J. Fluid Mech., 66, 377-383, (1974) · Zbl 0294.76076 · doi:10.1017/S0022112074000255 |
| [14] | De Pietro, M.; Van Hinsberg, M. A. T.; Biferale, L.; Clercx, H. J. H.; Perlekar, P.; Toschi, F., Clustering of vertically constrained passive particles in homogeneous isotropic turbulence, Phys. Rev. E, 91, (2015) |
| [15] | Doostmohammadi, A.; Dabiri, S.; Ardekani, A. M., A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratified fluid, J. Fluid Mech., 750, 5-32, (2014) · doi:10.1017/jfm.2014.243 |
| [16] | Doostmohammadi, A.; Stocker, R.; Ardekani, A. M., Low-Reynolds-number swimming at pycnoclines, Proc. Natl Acad. Sci. USA, 109, 10, 3856-3861, (2012) · doi:10.1073/pnas.1116210109 |
| [17] | Durham, W. M.; Climent, E.; Barry, M.; De Lillo, F.; Boffetta, G.; Cencini, M.; Stocker, R., Turbulence drives microscale patches of motile phytoplankton, Nat. Commun., 4, 2148, 1-7, (2013) · doi:10.1038/ncomms3148 |
| [18] | Franks, P. J. S.; Jaffe, J. S., Microscale variability in the distributions of large fluorescent particles observed in situ with a planar laser imaging fluorometer, J. Mar. Sys., 69, 254-270, (2008) · doi:10.1016/j.jmarsys.2006.03.027 |
| [19] | Gray, D. D.; Giorgini, A., The validity of the Boussinesq approximation for liquids and gases, Intl J. Heat Mass Transfer, 19, 5, 545-551, (1976) · Zbl 0328.76066 · doi:10.1016/0017-9310(76)90168-X |
| [20] | Guasto, J. S.; Rusconi, R.; Stocker, R., Fluid mechanics of planktonic microorganisms, Annu. Rev. Fluid Mech., 44, 373-400, (2012) · Zbl 1358.76086 · doi:10.1146/annurev-fluid-120710-101156 |
| [21] | Gustavsson, K.; Berglund, F.; Jonsson, P. R.; Mehlig, B., Preferential sampling and small-scale clustering of gyrotactic microswimmers in turbulence, Phys. Rev. Lett., 116, (2016) · doi:10.1103/PhysRevLett.116.108104 |
| [22] | Happel, J.; Brenner, H., Low Reynolds Number Hydrodynamics, (1983), Kluwer Academic Publishers |
| [23] | Jacobson, M. Z., Fundamentals of Atmospheric Modeling, (2005), Cambridge University Press · Zbl 1094.76001 · doi:10.1017/CBO9781139165389 |
| [24] | Jephson, T.; Carlsson, P., Species-and stratification-dependent diel vertical migration behaviour of three dinoflagellate species in a laboratory study, J. Plankton Res., 31, 11, 1353-1362, (2009) · doi:10.1093/plankt/fbp078 |
| [25] | Jonsson, P., Vertical distributions of planktonic ciliates - an experimental analysis of swimming behaviour, Mar. Ecol. Prog. Ser., 52, 39-53, (1989) · doi:10.3354/meps052039 |
| [26] | Kessler, J. O., Hydrodynamic focusing of motile algal cells, Nature, 313, 218-220, (1985) · doi:10.1038/313218a0 |
| [27] | LeelőSsy, Á.; Molnár, F.; Izsák, F.; Havasi, Á.; Lagzi, I.; Mészáros, R., Dispersion modeling of air pollutants in the atmosphere: a review, Centr. Eur. J. Geosci., 6, 257-278, (2014) |
| [28] | Linden, P. F., The fluid mechanics of natural ventilation, Annu. Rev. Fluid Mech., 31, 1, 201-238, (1999) · doi:10.1146/annurev.fluid.31.1.201 |
| [29] | Lovalenti, P. M.; Brady, J. F., The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number, Phys. Fluids, 5, 9, 2104-2116, (1993) · Zbl 0782.76033 · doi:10.1063/1.858550 |
| [30] | Maxey, M. R.; Riley, J. J., Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids, 26, 883-889, (1983) · Zbl 0538.76031 · doi:10.1063/1.864230 |
| [31] | Maxworthy, T.; Leilich, J.; Simpson, J. E.; Meiburg, E. H., The propagation of a gravity current into a linearly stratified fluid, J. Fluid Mech., 453, 371-394, (2002) · Zbl 0987.76504 · doi:10.1017/S0022112001007054 |
| [32] | Mcdougall, T. J., Bubble plumes in stratified environments, J. Fluid Mech., 85, 4, 655-672, (1978) · Zbl 0374.76093 · doi:10.1017/S0022112078000841 |
| [33] | Mehaddi, R.; Candelier, F.; Vauquelin, O., Naturally bounded plumes, J. Fluid Mech., 717, 472-483, (2013) · Zbl 1284.76200 · doi:10.1017/jfm.2012.587 |
| [34] | Mehaddi, R.; Vauquelin, O.; Candelier, F., Analytical solutions for turbulent boussinesq fountains in a linearly stratified environment, J. Fluid Mech., 691, 487-497, (2012) · Zbl 1241.76290 · doi:10.1017/jfm.2011.487 |
| [35] | Meibohm, J.; Candelier, F.; Rosen, T.; Einarsson, J.; Lundell, F.; Mehlig, B., Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number, Phys. Rev. Fluids, 1, 8, (2016) · doi:10.1103/PhysRevFluids.1.084203 |
| [36] | Mowbray, D. E.; Rarity, B. S. H., A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid, J. Fluid Mech., 28, 1-16, (1967) · doi:10.1017/S0022112067001867 |
| [37] | Oseen, C. W., Über die Stokes’sche Formel und über eine verwandte Aufgabe in der Hydrodynamik, Ark. Mat. Astron. Fys., 6, 143-152, (1910) · JFM 41.0832.02 |
| [38] | Pierson, J.-L.; Magnaudet, J., Inertial settling of a sphere through an interface. Part 2. Sphere and tail dynamics, J. Fluid Mech., 835, 808-851, (2018) · doi:10.1017/jfm.2017.748 |
| [39] | Proudman, I.; Pearson, J. R. A., Expansions at small Reynolds numbers for the flow past a sphere and circular cylinder, J. Fluid Mech., 22, 2, 385-400, (1957) · Zbl 0077.39103 |
| [40] | Roberts, A. M., Geotaxis in motile micro-organisms, J. Exp. Biol., 53, 687-699, (1970) |
| [41] | Saffman, P. G., The lift on a small sphere in a slow shear flow, J. Fluid Mech., 22, 2, 385-400, (1965) · Zbl 0218.76043 · doi:10.1017/S0022112065000824 |
| [42] | Salazar, A., On thermal diffusivity, Eur. J. Phys., 24, 4, 351-358, (2003) · Zbl 1049.80001 · doi:10.1088/0143-0807/24/4/353 |
| [43] | Sozza, A.; De Lillo, F.; Musacchio, S.; Boffetta, G., Large-scale confinement and small-scale clustering of floating particles in stratified turbulence, Phys. Rev. Fluids, 1, 5, (2016) · doi:10.1103/PhysRevFluids.1.052401 |
| [44] | Srdić-Mitrović, A. N.; Mohamed, N. A.; Fernando, H. J. S., Gravitational settling of particles through density interfaces, J. Fluid Mech., 381, 175-198, (1999) · Zbl 0939.76503 · doi:10.1017/S0022112098003590 |
| [45] | Torres, C. R.; Hanazaki, H.; Ochoa, J.; Castillo, J.; Van Woert, M., Flow past a sphere moving vertically in a stratified diffusive fluid, J. Fluid Mech., 417, 211-236, (2000) · Zbl 0971.76023 · doi:10.1017/S0022112000001002 |
| [46] | Turner, J. S., Buoyancy Effects in Fluids, (1979), Cambridge University Press · Zbl 0443.76091 |
| [47] | Van Aartrijk, M.; Clercx, H. J. H., Vertical dispersion of light inertial particles in stably stratified turbulence: the influence of the basset force, Phys. Fluids, 22, 1, (2010) · Zbl 1183.76536 · doi:10.1063/1.3291678 |
| [48] | Wang, S.; Ardekani, A. M., Inertial squirmer, Phys. Fluids, 24, 10, (2012) |
| [49] | Wang, S.; Ardekani, A. M., Unsteady swimming of small organisms, J. Fluid Mech., 702, 286-297, (2012) · Zbl 1248.76171 · doi:10.1017/jfm.2012.177 |
| [50] | Woods, A. W., Turbulent plumes in nature, Annu. Rev. Fluid Mech., 42, 391-412, (2010) · doi:10.1146/annurev-fluid-121108-145430 |
| [51] | Yick, K. Y.; Torres, C. R.; Peacock, T.; Stocker, R., Enhanced drag of a sphere settling in a stratified fluid at small Reynolds numbers, J. Fluid Mech., 632, 49-68, (2009) · Zbl 1183.76058 · doi:10.1017/S0022112009007332 |
| [52] | Zhang, J.; Mercier, M.; Magnaudet, J., Wake of a vertically moving sphere in a linearly stratified fluid, 16th Eur. Turbulence Conf., Stockholm, Sweden, (2017) |
| [53] | Zvirin, Y.; Chadwick, R. S., Settling of an axially symmetric body in a viscous stratified fluid, Intl J. Multiphase Flow, 1, 743-752, (1975) · Zbl 0354.76075 · doi:10.1016/0301-9322(75)90033-6 |
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