×

A neutrally stable shell in a Stokes flow: a rotational Taylor’s sheet. (English) Zbl 1472.76038

Summary: In a seminal paper published in 1951, Taylor studied the interactions between a viscous fluid and an immersed flat sheet which is subjected to a travelling wave of transversal displacement. The net reaction of the fluid over the sheet turned out to be a force in the direction of the wave phase-speed. This effect is a key mechanism for the swimming of micro-organisms in viscous fluids. Here, we study the interaction between a viscous fluid and a special class of nonlinear morphing shells. We consider pre-stressed shells showing a one-dimensional set of neutrally stable equilibria with almost cylindrical configurations. Their shape can be effectively controlled through embedded active materials, generating a large-amplitude shape-wave associated with precession of the axis of maximal curvature. We show that this shape-wave constitutes the rotational analogue of a Taylor’s sheet, where the translational swimming velocity is replaced by an angular velocity. Despite the net force acting on the shell vanishes, the resultant torque does not. A similar mechanism can be used to manoeuver in viscous fluids.

MSC:

76D25 Wakes and jets
76U05 General theory of rotating fluids
76T10 Liquid-gas two-phase flows, bubbly flows

Software:

Gmsh; PETSc; FEniCS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Taylor G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447-461. (doi:10.1098/rspa.1951.0218) · Zbl 0043.40302 · doi:10.1098/rspa.1951.0218
[2] Lauga E, Powers TR. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601. (doi:10.1088/0034-4885/72/9/096601) · doi:10.1088/0034-4885/72/9/096601
[3] Childress S. 1981 Mechanics of swimming and flying. Cambridge Studies in Mathematical Biology. Cambridge, UK: Cambridge University Press. · Zbl 0499.76118
[4] Dreyfus R, Baudry J, Roper ML, Fermigier M, Stone HA, Bibette J. 2005 Microscopic artificial swimmers. Nature 437, 862-865. (doi:10.1038/nature04090) · Zbl 1132.76062 · doi:10.1038/nature04090
[5] Alouges F, DeSimone A, Giraldi L, Or Y, Wiezel O. 2019 Energy-optimal strokes for multi-link microswimmers: Purcell’s loops and Taylor’s waves reconciled. New J. Phys. 21, 043050. (doi:10.1088/1367-2630/ab1142) · doi:10.1088/1367-2630/ab1142
[6] Cicconofri G, DeSimone A. 2016 Motion planning and motility maps for flagellar microswimmers. Eur. Phys. J. E 39, 72. (doi:10.1140/epje/i2016-16072-y) · doi:10.1140/epje/i2016-16072-y
[7] Sauzade M, Elfring GJ, Lauga E. 2011 Taylor’s swimming sheet: analysis and improvement of the perturbation series. Physica D 240, 1567-1573. (doi:10.1016/j.physd.2011.06.023) · Zbl 1431.76164 · doi:10.1016/j.physd.2011.06.023
[8] Lauga E. 2007 Propulsion in a viscoelastic fluid. Phys. Fluids 19, 083104. (doi:10.1063/1.2751388) · Zbl 1182.76430 · doi:10.1063/1.2751388
[9] Katz DF. 1974 On the propulsion of micro-organisms near solid boundaries. J. Fluid Mech. 64, 33-49. (doi:10.1017/S0022112074001984) · Zbl 0297.76089 · doi:10.1017/S0022112074001984
[10] Taylor G. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214, 158-183. (doi:10.1098/rspa.1952.0159) · Zbl 0047.43901 · doi:10.1098/rspa.1952.0159
[11] Setter E, Bucher I, Haber S. 2012 Low-Reynolds-number swimmer utilizing surface traveling waves: analytical and experimental study. Phys. Rev. E 85, 066304. (doi:10.1103/PhysRevE.85.066304) · doi:10.1103/PhysRevE.85.066304
[12] Li L, Spagnolie SE. 2014 Swimming and pumping of rigid helical bodies in viscous fluids. Phys. Fluids 26, 041901. (doi:10.1063/1.4871084) · doi:10.1063/1.4871084
[13] Dasgupta M, Liu B, Fu HC, Berhanu M, Breuer KS, Powers TR, Kudrolli A. 2013 Speed of a swimming sheet in newtonian and viscoelastic fluids. Phys. Rev. E 87, 013015. (doi:10.1103/PhysRevE.87.013015) · doi:10.1103/PhysRevE.87.013015
[14] Brunetti M, Vidoli S, Vincenti A. 2018 Bistability of orthotropic shells with clamped boundary conditions: an analysis by the polar method. Compos. Struct. 194, 388-397. (doi:10.1016/j.compstruct.2018.04.009) · doi:10.1016/j.compstruct.2018.04.009
[15] DeSimone A. 2018 Spontaneous bending of pre-stretched bilayers. Meccanica 53, 511-518. (doi:10.1007/s11012-017-0732-z) · Zbl 1446.74157 · doi:10.1007/s11012-017-0732-z
[16] Guest S, Pellegrino S. 2006 Analytical models for bistable cylindrical shells. Proc. R. Soc. A 462, 839-854. (doi:10.1098/rspa.2005.1598) · Zbl 1149.74358 · doi:10.1098/rspa.2005.1598
[17] Hamouche W, Maurini C, Vincenti A, Vidoli S. 2016 Basic criteria to design and produce multistable shells. Meccanica 51, 2305-2320. (doi:10.1007/s11012-016-0375-5) · doi:10.1007/s11012-016-0375-5
[18] Hamouche W, Maurini C, Vidoli S, Vincenti A. 2017 Multi-parameter actuation of a neutrally stable shell: a flexible gear-less motor. Proc. R. Soc. A 473, 20170364. (doi:10.1098/rspa.2017.0364) · Zbl 1404.74096 · doi:10.1098/rspa.2017.0364
[19] Seffen K, McMahon R. 2007 Heating of a uniform wafer disk. Int. J. Mech. Sci. 49, 230-238. (doi:10.1016/j.ijmecsci.2006.08.003) · doi:10.1016/j.ijmecsci.2006.08.003
[20] Wittrick W, Myers D, Blunden W. 1953 Stability of a bimetallic disk. Q. J. Mech. Appl. Mech. 6, 15. (doi:10.1093/qjmam/6.1.15) · Zbl 0051.41103 · doi:10.1093/qjmam/6.1.15
[21] Lighthill J. 1976 Flagellar hydrodynamics. SIAM Rev. 18, 161-230. (doi:10.1137/1018040) · Zbl 0366.76099 · doi:10.1137/1018040
[22] Geuzaine C, Remacle J-F. 2009 Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309-1331. (doi:10.1002/nme.2579) · Zbl 1176.74181 · doi:10.1002/nme.2579
[23] Alnæs MS et al. 2015 The FEniCS project version 1.5. Archive Numer. Softw. 3, 9-23.
[24] Balay S et al. 2019 PETSc users manual. Technical Report ANL-95/11 - Revision 3.11, Argonne National Laboratory. See https://www.mcs.anl.gov/petsc.
[25] Allendes A, Otárola E, Salgado AJ. 2019 A posteriori error estimates for the Stokes problem with singular sources. Comput. Methods Appl. Mech. Eng. 345, 1007-1032. (doi:10.1016/j.cma.2018.11.004) · Zbl 1440.65171 · doi:10.1016/j.cma.2018.11.004
[26] Happel J, Brenner H. 1983 Low Reynolds number hydrodynamics. Dordrecht, The Netherlands: Martinus Nijhoff Publishers. · Zbl 0612.76032
[27] Lamb H. 1932 Hydrodynamics. Cambridge, UK: Cambridge University Press. · JFM 58.1298.04
[28] Plaza A, Carey G. 2000 Local refinement of simplicial grids based on the skeleton. Appl. Numer. Math. 32, 195-218. (doi:10.1016/S0168-9274(99)00022-7) · Zbl 0940.65142 · doi:10.1016/S0168-9274(99)00022-7
[29] Pozrikidis C. 1992 Boundary integral and singularity methods for linearized viscous flow. Cambridge, UK: Cambridge University Press. · Zbl 0772.76005
[30] Purcell EM. 1977 Life at low Reynolds numbers. Am. J. Phys. 45, 3-11. (doi:10.1119/1.10903) · doi:10.1119/1.10903
[31] Djellouli A, Marmottant P, Djeridi H, Quilliet C, Coupier G. 2017 Buckling instability causes inertial thrust for spherical swimmers at all scales. Phys. Rev. Lett. 119, 224501. (doi:10.1103/PhysRevLett.119.224501) · doi:10.1103/PhysRevLett.119.224501
[32] Hale JS, Brunetti M, Bordas S, Maurini C. 2018 Simple and extensible plate and shell finite element models through automatic code generation tools. Comput. Struct. 209, 163-181. (doi:10.1016/j.compstruc.2018.08.001) · doi:10.1016/j.compstruc.2018.08.001
[33] Cha Y, Chae W, Kim H, Walcott H, Peterson S, Porfiri M. 2016 Energy harvesting from a piezoelectric biomimetic fish tail. Renew. Energy 86, 449-458. (doi:10.1016/j.renene.2015.07.077) · doi:10.1016/j.renene.2015.07.077
[34] Michelin S, Doaré O. 2013 Energy harvesting efficiency of piezoelectric flags in axial flows. J. Fluid Mech. 714, 489-504. (doi:10.1017/jfm.2012.494) · Zbl 1284.74034 · doi:10.1017/jfm.2012.494
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.