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ON INTERACTION BETWEEN FALLING BODIES AND THE SURROUNDING FLUID

Part of: Dynamics of a system of particles, including celestial mechanics Thin bodies, structures Biological fluid mechanics Dynamics of a rigid body and of multi-body systems Coupling of solid mechanics with other effects

Published online by Cambridge University Press:  10 December 2009

Frank T. Smith  and
Andrew S. Ellis
Frank T. Smith
Affiliation:
Department of Mathematics, UCL, Gower Street, London WC1E 6BT, U.K. (email: frank@math.ucl.ac.uk)
Andrew S. Ellis
Affiliation:
Department of Mathematics, UCL, Gower Street, London WC1E 6BT, U.K.
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Abstract

Interactions between a finite number of bodies and the surrounding fluid, in a channel for instance, are investigated theoretically. In the planar model here the bodies or modelled grains are thin solid bodies free to move in a nearly parallel formation within a quasi-inviscid fluid. The investigation involves numerical and analytical studies and comparisons. The three main features that appear are a linear instability about a state of uniform motion, a clashing of the bodies (or of a body with a side wall) within a finite scaled time when nonlinear interaction takes effect, and a continuum-limit description of the body–fluid interaction holding for the case of many bodies.

MSC classification

Secondary: 76Z05: Physiological flows 74K35: Thin films 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 70E55: Dynamics of multibody systems
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 140 - 168
Copyright
Copyright © University College London 2010

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References

[1]Ahn, H., Brennen, C. E. and Sabersky, R. H., Analysis of the fully developed chute flow of granular materials. J. Appl. Mech. 59 (1992), 109.CrossRefGoogle Scholar
[2]Ardekani, A. M., Rangel, R. H. and Joseph, D. D., Motion of a sphere normal to a wall in a second-order fluid. J. Fluid Mech. 587 (2007), 163–172.CrossRefGoogle Scholar
[3]Andrew, M., David, M., deVeber, G. and Brooker, L. A., Arterial thromboembolic complications in paediatric patients. Thromb. Haemost. 78(1) (1997), 715–725.Google ScholarPubMed
[4]Argentina, A. and Mahadevan, L., Fluid-flow-induced flutter of a flag. Proc. Natl. Acad. Sci. 102(6) (2005), 1829–1834.CrossRefGoogle ScholarPubMed
[5]Babyn, P. S., Gahunia, H. K. and Massicotte, P., Pulmonary thromboembolism in children. Pediatr. Radiol. 35(3) (2005), 258–274.CrossRefGoogle ScholarPubMed
[6]Baker, W. F., Diagnosis of deep venous thrombosis and pulmonary embolism. Med. Clin. North Am. 82(3) (1998), 459–476.CrossRefGoogle ScholarPubMed
[7]Belmonte, A., Eisenberg, H. and Moses, E., From flutter to tumble: inertial drag and Froude similarity in falling paper. Phys. Rev. Lett. 81 (1998), 345.CrossRefGoogle Scholar
[8]Bowles, R. G. A. and Smith, , Lifting multi-blade flows with interaction. J. Fluid Mech. 415 (2000), 203–226.CrossRefGoogle Scholar
[9]Bowles, R. I., Dennis, S. C. R., Purvis, R. and Smith, F. T., Multi-branching flows from one mother tube to many daughters or to a network. Philos. Trans. R. Soc. A 363 (2005), 1045–1055.CrossRefGoogle ScholarPubMed
[10]Bowles, R. I., Ovenden, N. C. and Smith, F. T., Multi-branching 3D flow with substantial changes in vessel shapes. J. Fluid Mech. 614 (2008), 329–354.CrossRefGoogle Scholar
[11]Campbell, C. S., Rapid granular flows. Annu. Rev. Fluid Mech. 22 (1990), 57.CrossRefGoogle Scholar
[12]Comer, J. K., Kleinstreuer, C. and Kim, C. S., Flow structures and particle deposition patterns in double-bifurcation airway models. Part 2. Aerosol transport and deposition. J. Fluid Mech. 435 (2001), 55–80.CrossRefGoogle Scholar
[13]Eames, I., Hunt, J. C. R. and Belcher, S. E., Inviscid mean flow through and around groups of bodies. J. Fluid Mech. 515 (2004), 371.CrossRefGoogle Scholar
[14]Ehrentraut, H. and Chrzanowska, A., Induced anisotropy in rapid flows of nonspherical granular materials. In Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformations, Vol. 11 (eds K. Hutter and N. Kirchner), Springer (Berlin, 2003), 343–364.CrossRefGoogle Scholar
[15]Ellis, A. S., Modelling chute delivery of grains in a food-sorting process. PhD Thesis, University of London, 2007.Google Scholar
[16]Ellis, A. S. and Smith, F. T., A continuum model for a chute flow of grains. SIAM J. Appl. Math. 69(2) (2008), 305–329.CrossRefGoogle Scholar
[17]Fortes, A. F., Joseph, D. D. and Lundgren, T. S., Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177 (1987), 467–483.CrossRefGoogle Scholar
[18]Gaver, D. P., Halpern, D. and Jensen, O. E., Surfactant and airway liquid flows. In Lung Surfactant and Disorder: Lung Biology in Health and Disease (ed. K. Nag), Taylor and Francis (Boca Raton, 2005), 201.Google Scholar
[19]Gray, J. M. N. T. and Hutter, K., Pattern formation in granular media. Contin. Mech. Thermodyn. 9 (1997), 341.CrossRefGoogle Scholar
[20]Guazzelli, E., Sedimentation of small particles: how can such a simple problem be so difficult? C. R. Mécanique 334 (2006), 539–544.Google Scholar
[21]Hodges, S. R., Jensen, O. E. and Rallison, J. M., The motion of a viscous drop through a cylindrical tube. J. Fluid Mech. 501 (2004), 279–301.CrossRefGoogle Scholar
[22]Huppert, H. E., Quantitative modelling of granular suspension flows. Philos. Trans. R. Soc. Lond. A 356 (1998), 2471.CrossRefGoogle Scholar
[23]Huang, P. Y., Feng, J. and Joseph, D. D., The turning couples on an elliptic particle settling in a vertical channel. J. Fluid Mech. 271 (1994), 1–16.CrossRefGoogle Scholar
[24]Iguchi, Y. and Kimura, K., A case of brain embolism during catheter embolisation of head arteriovenous malformation. What is the mechanism of stroke? J. Neurol. Neurosurg. Psychiatry 78 (2007), 81.CrossRefGoogle ScholarPubMed
[25]Jenkins, J. T. and Savage, S. B., A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130 (1983), 187.CrossRefGoogle Scholar
[26]Jenkins, J. T., Boundary conditions for rapid granular flows: flat, frictional walls. J. Appl. Mech. 59 (1992), 120.CrossRefGoogle Scholar
[27]Jia, L.-B., Li, F., Yin, X.-Z. and Yin, X.-Y., Coupling modes between two flapping filaments. J. Appl. Mech. 581 (2007), 199–220.Google Scholar
[28]Jones, M. A. and Smith, F. T., Fluid motion for car undertrays in ground effect. J. Engrg. Math. 45 (2003), 309–334.Google Scholar
[29]Kadanoff, L. P., Built upon sand: theoretical ideas inspired by granular flows. Rev. Modern Phys. 71 (1999), 435.CrossRefGoogle Scholar
[30]Koch, D. L. and Hill, R. J., Inertial effects in suspension and porous-media flows. Annu. Rev. Fluid Mech. 33 (2001), 619.CrossRefGoogle Scholar
[31]Korobkin, A. A., Impact of two bodies one of which is covered by a thin layer of liquid. J. Fluid Mech. 300 (1995), 43–58.CrossRefGoogle Scholar
[32]Korobkin, A. A. and Ohkusu, M., Impact of two circular plates one of which is floating on a thin layer of liquid. J. Engrg. Math. 50 (2004), 343–358.CrossRefGoogle Scholar
[33]Louge, M. Y., Computer simulations of rapid granular flows of spheres interacting with a flat frictional boundary. Phys. Fluids 6 (1994), 2253.CrossRefGoogle Scholar
[34]Magnaudet, J. and Eames, I., Dynamics of high Re bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32 (2000), 659–708.CrossRefGoogle Scholar
[35]Monetti, R., Hurd, A. and Kenkre, V. M., Simulations for dynamics of granular mixtures in a rotating drum. Granular Matter. 3 (2001), 113.Google Scholar
[36]Newman, J. N., Analysis of small-aspect-ratio lifting surfaces in ground effect. J. Fluid. Mech. 117 (1982), 305–314.CrossRefGoogle Scholar
[37]Ovenden, N. C., Smith, F. T. and Wu, G.-X., Effects of nonsymmetry in a branching flow network. J. Engrg. Math. 63 (2008), 213–239.CrossRefGoogle Scholar
[38]Pancholi, K., Stride, E. and Edirisinghe, M., Dynamics of bubble formation in highly viscous liquids. Langmuir 24 (2008), 4388–4393.CrossRefGoogle ScholarPubMed
[39]Purvis, R. and Smith, F. T., Planar flow past two or more blades in ground effect. Q. J. Mech. Appl. Math. 57(1) (2004), 137–160.CrossRefGoogle Scholar
[40]Rajchenbach, J., Granular flows. Adv. Phys. 49 (2000), 229.CrossRefGoogle Scholar
[41]Schonberg, J. A. and Hinch, E. J., Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech. 203 (1989), 517–524.CrossRefGoogle Scholar
[42]Secomb, T. W., Skalak, R., Özkaya, N. and Gross, J. F., Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163 (1986), 405–423.Google Scholar
[43]Smith, F. T. and Timoshin, S. N., Blade-wake interactions and rotary boundary layers. Proc. R. Soc. A 452 (1996a), 1301–1329.Google Scholar
[44]Smith, F. T. and Timoshin, S. N., Planar flows past thin multi-blade configurations. J. Fluid Mech. 324 (1996b), 355–377.CrossRefGoogle Scholar
[45]Smith, F. T. and Jones, M. A., One-to-few and one-to-many branching tube flows. J. Fluid Mech. 423 (2000), 1–31.CrossRefGoogle Scholar
[46]Smith, F. T. and Jones, M. A., AVM modelling by multi-branching tube flow: large flow rates and dual solutions. Math. Medicine Biology 20 (2003), 183–204.CrossRefGoogle ScholarPubMed
[47]Smith, F. T., Li, L. and Wu, G.-X., Air cushioning with a lubrication/inviscid balance. J. Fluid Mech. 482 (2003), 291–318.CrossRefGoogle Scholar
[48]Smith, F. T., Ovenden, N. C. and Purvis, R., Industrial and biomedical applications. Solid Mech. Appl. 129(5) (2006), 291–300.Google Scholar
[49]Stone, H. A., Stroock, A. D. and Ajdari, A., Engineering flows in small devices: micro-fluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36 (2004), 381–411.CrossRefGoogle Scholar
[50]Tuck, E. O., A nonlinear unsteady one-dimensional theory for wings in extreme ground effect. J. Fluid Mech. 98 (1980), 33–47.CrossRefGoogle Scholar
[51]Tuck, E. O. and Bentwich, T. M., Sliding sheets: lubrication with comparable viscous and inertia forces. J. Fluid Mech. 41 (1983), 769–792.Google Scholar
[52]Villermaux, E. and Clanet, C., Life of a flapping liquid sheet. J. Fluid Mech. 462 (2002), 341–363.CrossRefGoogle Scholar
[53]White, A. H., Mathematical modelling of the embolisation process in the treatment of arteriovenous malformations. PhD Thesis, University of London, 2008.Google Scholar
[54]Willetts, B., Aeolian and fluvial grain transport. Philos. Trans. R. Soc. Lond. A 356 (1998), 2471.Google Scholar
[55]Wilson, H. J. and Davis, R. H., The viscosity of a dilute suspension of rough spheres. J. Fluid Mech. 421 (2000), 339.CrossRefGoogle Scholar
[56]Wilson, H. J. and Davis, R. H., Shear stress of a monolayer of rough spheres. J. Fluid Mech. 452 (2002), 425.CrossRefGoogle Scholar
[57]Xu, J., Maxey, M. R. and Karniadakis, G. E., Numerical simulation of turbulent drag reduction using micro-bubbles. J. Fluid Mech. 468 (2002), 271–281.CrossRefGoogle Scholar
[58]Yang, B. H., Wang, J., Joseph, D. D., Hu, H. H., Pan, T. W. and Glowinski, R., Migration of a sphere in a tube flow. J. Fluid Mech. 540 (2005), 109–131.CrossRefGoogle Scholar
[59]Yih, C.-S., Fluid mechanics of colliding plates. Phys. Fluids 17 (1974), 1936–1940.CrossRefGoogle Scholar