Abstract
A model describing the ciliary driven flow and motion of suspended particles in downstream suspension feeders is developed. The quasi-steady Stokes equations for creeping flow are solved numerically in an unbounded fluid domain around cylindrical bodies using a boundary integral formulation. The time-dependent flow is approximated with a continuous sequence of steady state creeping flow fields, where metachronously beating ciliary bands are modelled by linear combinations of singularity solutions to the Stokes equations. Generally, the computed flow fields can be divided into an unsteady region close to the driving ciliary bands and a steady region covering the remaining fluid domain. The size of the unsteady region appears to be comparable to the metachronal wavelength of the ciliary band. A systematic investigation is performed of trajectories of infinitely small (fluid) particles in the simulated unsteady ciliary driven flow. A fraction of particles appear to follow trajectories, that resemble experimentally observed particle capture events in the downstream feeding system of the polycheate Sabella penicillus, indicating that particles can be captured by ciliary systems without mechanical contact between particle and cilia. A local capture efficiency is defined and its value computed for various values of beat frequencies and other parameters. The results indicate that the simulated particle capture process is most effective when the flow field oscillates within timescales comparable to transit timescales of suspended particles passing the unsteady region near the ciliary bands. However, the computed retention efficiencies are found to be much lower than those obtained experimentally.
Similar content being viewed by others
References
Aref, H. and S. Balachandar (1986). Chaotic advection in a stokes flow. Phys. Fluids 29, 3515–3521.
Blake, J. R. (1972). A model for the micro-structure in ciliated organisms. J. Fluid Mech. 55, 1–23.
Blake, J. R. (1973). A finite model for ciliated micro-organisms. J. BioMech. 6, 133–140.
Blake, J. R. and C. R. Fulford (1984). Mechanics of muco-ciliary transport. Phys. Chem. Hydrodyn. 5, 401–411.
Blake, J. R., N. Liron and G. K. Aldis (1982). Flow pattern around ciliated micro organisms and ciliated ducts. J. Theo. Biol. 98, 127–141.
Brennen, C. and H. Winet (1977). Fluid mechanics of propulsion by cilia and flagella. Ann. Rev. Fluid. Mech. 9, 339–398.
Fehlberg, E. (1970). Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnungmit Schrittweiten-Kontrolle und ihre Anvendung auf Wärmeleitungsprobleme. Computing 6, 61–71.
Goren, S. L. and M. E. O’Niell (1971). On the hyfrodynamic resistance to a particle of a dilute suspension when in the neighbourhood of a large obstacle. Chem. Engng. Sci. 26, 325–338.
Gueron, S. and K. L. Gurewich (1998). Computation of the internal forces in cilia: application to ciliary motion, the effect of viscosity, and the cilia interactions. Biophys. J. 74, 1658–1676.
Gueron, S. and N. Liron (1992). Ciliary motion modeling, and dynamic multicilia interactions. Biophys. J. 63, 1045–1085.
Happel, J. and H. Brenner (1965). Low Reynolds Number Hydrodynamics, Prentice Hall Inc.
Hiramoto, Y. (1974). Mechanics of the ciliary movement, in Cilia and flagella, M. A. Sleigh (Ed.), Academic Press, pp. 177–198.
Jørgensen, C. B. (1982). Fluid mechanics of the mussel gill: the lateral cilia. Mar. Biol. 70, 275–281.
Jørgensen, C. B. (1989). Water processing in ciliary feeders, with special reference to the bivalve filter pump. Comp. Biochem. Physio. 94A, 383–394.
Jørgensen, C. B., T. Kjøerboe, F. Møhlenberg and H. U. Riisgård (1984). Ciliary and mucus-net filter feeding, with special reference to fluid mechanical characteristics. Mar. Ecol. Prog. Ser. 15, 183–292.
Jørgensen, C. B., P. S. Larsen and H. U. Riisgård (1990). Effects of temperature on the mussel pump. Mar. Ecol. Prog. Ser. 64, 89–97.
Kim, S. and S. Karrila (1991). Microhydrodynamics, Butterworth-Heinemann.
LaBarbara, M. (1984). Feeding currents and particle capture mechanisms in suspension feeding animals. Amer. Zool. 24, 71–84.
Liron, N. (1978). Fluid transport by cilia between parallel plates. J. Fluid Mech. 86, 705–726.
Liron, N. (1984). Stokeslet arrays in a pipe and their applications to ciliary transport. J. Fluid Mech. 143, 173–195.
Liron, N. (1996). Stokes flow due to infinite arrays of stokeslets in three dimensions. J. Eng. Math. 30, 267–297.
Liron, N. and S. Mochon (1976). The discrete-cilia approach to propulsion of ciliated microorganisms. J. Fluid Mech. 75, 593–607.
Mayer, S. (1994a). Particle capture in the crown of the ciliary suspension feeding polycheate sabella penicillus: videotape recordings and interpretations. Mar. Biol. 119, 571–582.
Mayer, S. (1994b). Particle motion in unsteady three-dimensional flow at low Reynolds numbers, PhD thesis, Technical University of Denmark.
Nielsen, C. (1987). Structure and function of metazoan ciliary bands and their phylogenetic significance. Acta. Zool. 68, 205–262.
Nielsen, U. o. C. C (1999). Zoological museum. Pers. communication.
Odquist, F. K. G. (1930). Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten. Math. Z. 32, 329–375.
Ottino, J. M. (1990). Mixing, chaotic advection, and turbulence. Ann. Rev. Fluid Mech. 22, 207–254.
Power, H. and G. Miranda (1987). Second kind integral equation formulation of Stokes flow past a particle of arbitrary shape. SIAM J. Appl. Math. 47, 689–698.
Riisgård, H. U. and N. M. Ivarsson (1990). The crown filament pump of the suspension-feeding polychaete sabella penicillus: filtration, effects of temperature, and energy cost. Mar. Ecol. Prog. Ser. 62, 249–257.
Shimeta, J. (1993). Diffusional encounter of submicrometer particles and small cells by suspension feeders. Limnol Oceanogr. 38, 456–465.
Shimeta, J. and P. A. Jumars (1991). Physical mechanisms and the rate of particle capture by suspension feeders. Oceanogr. Mar. Biol. Annu. Rev. 29, 191–257.
Sleigh, M. A. (1974). Cilia and Flagella, Academic Press.
Spielmann, L. A. (1977). Particle capture from low-speed laminar flows. Ann Rev. Fluid Mech. 9, 197–319.
Spielmann, L. A. and S. L. Goren (1970). Capture of small particles by London forces from low-speed liquid flows. Environ. Sci. Tech. 4, 135–140.
Weinbaum, S., P. Ganatos and Z. Yan (1990). Numerical multipole and boundary integral equation techniques in stokes flow. Annu. Rev. Fluid Mech. 22, 275–316.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mayer, S. Numerical simulation of flow fields and particle trajectories in ciliary suspension feeding. Bull. Math. Biol. 62, 1035–1059 (2000). https://doi.org/10.1006/bulm.2000.0190
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.2000.0190