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A 1D model for a closed rigid filament in Stokes flow. (English) Zbl 1171.76010

Summary: We adapt an existing asymptotic method to set up a one-dimensional model for the fall of a closed filament in an infinite fluid in the Stokes regime. Starting from the single-layer integral representation of the fluid velocity around the filament, we get, for a very slender filament, a Fredholm integral equation on the filament centerline. From this equation, we can compute the drag and force acting on the filament, and consequently the resistance matrix. The integral equation is discretized with a collocation method. The study of a scalar model problem yields existence and uniqueness results together with an error estimate for the discretization scheme. Then, we compare the resistance matrix of thin ideal knots obtained from the discretization of the present model by a boundary element method; numerical convergence results and a good agreement of both methods validate our model.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M15 Boundary element methods applied to problems in fluid mechanics
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References:

[1] Bary N. K., A Treatise on Trigonometric Series 1 (1964) · Zbl 0129.28002
[2] DOI: 10.1017/S0022112075000614 · Zbl 0309.76016 · doi:10.1017/S0022112075000614
[3] DOI: 10.1007/978-1-4612-1382-6 · doi:10.1007/978-1-4612-1382-6
[4] DOI: 10.1007/BF01694049 · Zbl 0483.76033 · doi:10.1007/BF01694049
[5] Galdi G., On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications 1 (2002) · Zbl 1230.76016
[6] DOI: 10.1216/jiea/1020282207 · Zbl 0985.76022 · doi:10.1216/jiea/1020282207
[7] DOI: 10.1017/S0022112004001284 · Zbl 1065.76048 · doi:10.1017/S0022112004001284
[8] DOI: 10.1007/BF00948292 · Zbl 0626.45018 · doi:10.1007/BF00948292
[9] DOI: 10.1007/978-3-0348-9215-5 · doi:10.1007/978-3-0348-9215-5
[10] DOI: 10.1017/S0022112080000687 · Zbl 0447.76037 · doi:10.1017/S0022112080000687
[11] DOI: 10.1017/S0022112076000475 · Zbl 0377.76036 · doi:10.1017/S0022112076000475
[12] Ladyzhenskaya O. A., The Mathematical Theory of Viscous Incompressible Flow (1969) · Zbl 0184.52603
[13] DOI: 10.1017/S0022112096008889 · Zbl 0894.76006 · doi:10.1017/S0022112096008889
[14] DOI: 10.1080/10586458.2003.10504499 · Zbl 1073.57003 · doi:10.1080/10586458.2003.10504499
[15] Reed M., Methods of Modern Mathematical Physics (1980) · Zbl 0459.46001
[16] Riesz F., Functional Analysis (1990)
[17] DOI: 10.1063/1.1670977 · doi:10.1063/1.1670977
[18] DOI: 10.1007/s004660050507 · Zbl 0984.76071 · doi:10.1007/s004660050507
[19] DOI: 10.1016/S0167-2789(00)00131-7 · Zbl 1049.76016 · doi:10.1016/S0167-2789(00)00131-7
[20] DOI: 10.1017/S0022112070001908 · Zbl 0216.52304 · doi:10.1017/S0022112070001908
[21] DOI: 10.1016/j.jcp.2003.10.017 · Zbl 1115.76413 · doi:10.1016/j.jcp.2003.10.017
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