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arxiv: 2606.12570 · v1 · pith:SJB4VNH4new · submitted 2026-06-10 · ⚛️ physics.flu-dyn

Hydrodynamic Resistance on Oscillating Planar Interfacial Bodies

Ian Ho , Ajay Harishankar Kumar , Daniel M. Harris This is my paper

Pith reviewed 2026-06-27 08:00 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords hydrodynamic resistanceoscillatory Stokes layerinterfacial bodiesadded massdamping coefficientsWomersley numberfloating bodiesunsteady hydrodynamics
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0 comments X

The pith

At high Womersley number and small amplitude, an oscillatory Stokes boundary layer approximates the flow under laterally oscillating planar bodies on an air-water interface and supplies the leading-order hydrodynamic resistance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to show that a simple boundary-layer model captures the resistance felt by floating planar bodies that oscillate sideways on an air-water surface. Scaling analysis indicates that when the Womersley number is large and the amplitude is small the unsteady flow under the body collapses to an oscillatory Stokes layer. Experiments that drive the bodies magnetically and record their steady-state amplitude and phase confirm that the extracted added-mass and damping values line up with the layer prediction once surface deformation stays small. The same framework also reproduces the transient growth of the layer at startup through a history integral and supplies a practical way to measure unsteady interface forces.

Core claim

Scaling arguments indicate that at high Womersley number and small oscillation amplitude the flow beneath the body can be approximated by an oscillatory Stokes boundary layer, yielding a leading-order description of the hydrodynamic resistance. Frequency-response measurements with magnetic actuation extract added mass and damping coefficients that remain consistent with this description in the limit of small interfacial deformation. Startup transients are likewise captured by a history integral that follows the development of the oscillatory boundary layer.

What carries the argument

Oscillatory Stokes boundary layer that supplies the leading-order hydrodynamic resistance under the oscillating body.

If this is right

  • Added mass and damping coefficients are set by the properties of the oscillatory Stokes layer.
  • Steady-state amplitude and phase data directly yield those coefficients across ranges of frequency, mass, size, and planar shape.
  • A history integral constructed from the same layer accurately forecasts the transient response at startup.
  • The method provides a repeatable experimental route to unsteady hydrodynamic forces at fluid interfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same boundary-layer reduction may apply to other small-amplitude lateral motions at interfaces, such as in floating sensors or microscale devices.
  • If the small-deformation condition is met, the extracted coefficients could guide design of platforms that must resist or exploit oscillatory drag at a free surface.
  • Checking the model against bodies with modest curvature would test how far the strictly planar assumption can be relaxed before wave or contact-line corrections appear.

Load-bearing premise

Interfacial deformation must remain small enough that surface-wave effects and contact-line motion leave the planar boundary-layer flow unchanged.

What would settle it

If added-mass and damping values extracted at larger amplitudes where surface waves become visible deviate systematically from the Stokes-layer formulas, the leading-order approximation does not hold.

Figures

Figures reproduced from arXiv: 2606.12570 by Ajay Harishankar Kumar, Daniel M. Harris, Ian Ho.

Figure 1
Figure 1. Figure 1: FIG. 1. Magnetic-drive platform for measuring unsteady hydrodynamic forces at an air–water interface. (a) CAD rendering of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Frequency response of a magnetically driven slider at an air–water interface. (a) Normalized response amplitude, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Frequency response for sliders of various shapes, sizes, and masses. (a,b) Frequency sweeps of the normalized amplitude [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Added-mass and damping coefficients inferred from the measured response amplitude and phase lag. (a) Added-mass [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We study the unsteady dynamics of floating planar bodies undergoing lateral oscillations along an air-water interface. Scaling arguments indicate that at high Womersley number and small oscillation amplitude the flow beneath the body can be approximated by an oscillatory Stokes boundary layer, yielding a leading-order description of the hydrodynamic resistance. Using magnetic actuation, we drive the interfacial bodies harmonically and measure the amplitude response and phase lag in steady state over a range of frequencies, masses, sizes, and shapes. This frequency-response framework enables direct extraction of effective added mass and damping coefficients, which we find to be consistent with oscillatory boundary-layer theory in the limit of small interfacial deformation. The transient behavior during startup is also shown to be accurately predicted by a history integral that captures the development of the oscillatory boundary layer beneath the body. This work also establishes a simple experimental platform for quantifying unsteady hydrodynamic forces at fluid interfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the unsteady lateral oscillations of floating planar bodies at an air-water interface. Scaling arguments show that at high Womersley number and small amplitude the flow beneath the body reduces to an oscillatory Stokes boundary layer, supplying a leading-order prediction for hydrodynamic resistance (added mass and damping). Experiments using magnetic actuation measure steady-state amplitude and phase response over ranges of frequency, mass, size and shape; coefficients extracted from the frequency-response data are reported to agree with the boundary-layer theory in the small-deformation limit. A history-integral model is also shown to capture the transient startup of the boundary layer.

Significance. If the small-deformation limit is verified, the work supplies a simple, experimentally validated leading-order description of unsteady interfacial forces together with a practical platform for measuring them. The independent extraction of coefficients from frequency-response data (rather than direct force measurement) and the agreement with scaling across multiple parameters constitute clear strengths.

major comments (2)
  1. [Abstract] Abstract and scaling argument: the planar oscillatory Stokes boundary-layer approximation is invoked only when interfacial deformation remains negligible relative to the boundary-layer thickness, yet no quantitative threshold (e.g., surface elevation ≪ δ = √(2ν/ω) or relative to body size) is supplied and no direct measurement or bound on surface elevation is reported to confirm that the limit holds across the tested parameter range.
  2. [Experimental results] Experimental results: the reported consistency between measured amplitude/phase and the boundary-layer prediction rests on the small-deformation regime, but the manuscript provides no separate verification (e.g., surface profilometry or contact-line imaging) that wave or contact-line effects remained negligible; without this check the agreement cannot be unambiguously attributed to the Stokes-layer model.
minor comments (2)
  1. [Scaling arguments] Clarify the precise definition of the Womersley number used and its relation to the oscillation frequency and body length scale.
  2. [Results] Add explicit error bars or uncertainty estimates on the extracted added-mass and damping coefficients in the frequency-response plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which highlight important aspects of the small-deformation regime. We address each major comment below with proposed revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and scaling argument: the planar oscillatory Stokes boundary-layer approximation is invoked only when interfacial deformation remains negligible relative to the boundary-layer thickness, yet no quantitative threshold (e.g., surface elevation ≪ δ = √(2ν/ω) or relative to body size) is supplied and no direct measurement or bound on surface elevation is reported to confirm that the limit holds across the tested parameter range.

    Authors: We agree that an explicit quantitative threshold would strengthen the presentation. In the revised manuscript we will add a scaling estimate demonstrating that the maximum surface elevation (inferred from the small-amplitude driving and the observed response) remains ≪ δ across the reported frequency and amplitude ranges. Although direct profilometry was not performed, the data collapse onto the Stokes-layer prediction for multiple body sizes, masses, and frequencies (which vary δ) provides indirect confirmation that the regime holds; we will include this discussion and the resulting bound in the text. revision: yes

  2. Referee: [Experimental results] Experimental results: the reported consistency between measured amplitude/phase and the boundary-layer prediction rests on the small-deformation regime, but the manuscript provides no separate verification (e.g., surface profilometry or contact-line imaging) that wave or contact-line effects remained negligible; without this check the agreement cannot be unambiguously attributed to the Stokes-layer model.

    Authors: The referee correctly identifies the absence of direct imaging. We will revise the experimental section to state explicitly that the precise match to the predicted frequency dependence (including the 1/√ω scaling of damping) over the tested range, together with the lack of systematic deviations at lower frequencies where waves might appear, constitutes indirect evidence that contact-line and wave contributions remained sub-dominant. We cannot add new profilometry data, but the text revision will clarify the evidential basis and note the limitation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard external scaling

full rationale

The paper's leading-order description rests on scaling arguments for the oscillatory Stokes boundary layer at high Womersley number, a standard result in unsteady viscous flow that is not derived from or fitted to the present data. Experiments supply independent frequency-response measurements from which added-mass and damping coefficients are extracted and then compared to the external theory; no equation or coefficient is shown to reduce to a fit of the target quantity by construction. No self-citations are invoked as load-bearing uniqueness theorems, and the transient history-integral prediction is likewise drawn from established boundary-layer analysis rather than the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard fluid-mechanics assumption that the flow can be treated as an incompressible Newtonian fluid with a no-slip condition at the body surface, plus the domain assumption that surface deformation is negligible at the stated limits.

axioms (2)
  • standard math Incompressible Newtonian fluid obeying the no-slip condition at the solid-fluid interface
    Invoked implicitly in the oscillatory Stokes boundary-layer scaling argument.
  • domain assumption Interfacial deformation remains small enough that the body can be treated as planar with negligible surface-wave coupling
    Stated as the limit condition under which the leading-order resistance description holds.

pith-pipeline@v0.9.1-grok · 5679 in / 1348 out tokens · 20624 ms · 2026-06-27T08:00:03.369417+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

81 extracted references

  1. [1]

    O. M. Faltinsen,Hydrodynamics of High-Speed Marine Vehicles(Cambridge University Press, 2005)

    2005

  2. [2]

    M. J. Barratt, The wave drag of a hovercraft, J. Fluid Mech.22, 39 (1965)

    1965

  3. [3]

    J. N. Newman,Marine Hydrodynamics(MIT Press, Cambridge, MA, 1977)

    1977

  4. [4]

    J. R. Morison, M. P. O’Brien, J. W. Johnson, and S. A. Schaaf, The force exerted by surface waves on piles, J. Pet. Technol. 2, 149 (1950)

    1950

  5. [5]

    Sarpkaya,Wave Forces on Offshore Structures(Cambridge University Press, Cambridge, UK, 2010)

    T. Sarpkaya,Wave Forces on Offshore Structures(Cambridge University Press, Cambridge, UK, 2010)

    2010

  6. [6]

    J. W. M. Bush and D. L. Hu, Walking on water: Biolocomotion at the interface, Annu. Rev. Fluid Mech.38, 339 (2006)

    2006

  7. [7]

    D. L. Hu and J. W. M. Bush, The hydrodynamics of water-walking arthropods, J. Fluid Mech.644, 5 (2010)

    2010

  8. [8]

    R. B. Suter, O. Rosenberg, S. Loeb, H. Wildman, and J. H. Long, Jr., Locomotion on the water surface: Propulsive mechanisms of the fisher spider Dolomedes triton, J. Exp. Biol.200, 2523 (1997)

    1997

  9. [9]

    Mukundarajan, T

    H. Mukundarajan, T. C. Bardon, D. H. Kim, and M. Prakash, Surface tension dominates insect flight on fluid interfaces, J. Exp. Biol.219, 752 (2016)

    2016

  10. [10]

    S. T. Hsieh and G. V. Lauder, Running on water: Three-dimensional force generation by basilisk lizards, Proc. Natl. Acad. Sci. U.S.A.101, 16784 (2004)

    2004

  11. [11]

    Roh and M

    C. Roh and M. Gharib, Honeybees use their wings for water surface locomotion, Proc. Natl. Acad. Sci. U.S.A.116, 24446 (2019)

    2019

  12. [12]

    Jung, Ground effect on undulation and pumping near surfaces, Integr

    S. Jung, Ground effect on undulation and pumping near surfaces, Integr. Comp. Biol. , icag026 (2026)

    2026

  13. [13]

    I. Ho, G. Pucci, A. U. Oza, and D. M. Harris, Capillary surfers: Wave-driven particles at a vibrating fluid interface, Phys. Rev. Fluids8, L112001 (2023)

    2023

  14. [14]

    A. U. Oza, G. Pucci, I. Ho, and D. M. Harris, Theoretical modeling of capillary surfer interactions on a vibrating fluid bath, Phys. Rev. Fluids8, 114001 (2023)

    2023

  15. [15]

    D. M. Harris and J.-W. Barotta, Propulsion and interaction of wave-propelled interfacial particles, Phys. Rev. Fluids10, 100503 (2025)

    2025

  16. [16]

    Barotta, G

    J.-W. Barotta, G. Pucci, E. Silver, A. Hooshanginejad, and D. M. Harris, Synchronization of wave-propelled capillary spinners, Phys. Rev. E111, 035105 (2025). 11

    2025

  17. [17]

    Sungar, J

    N. Sungar, J. Sharpe, L. Ijzerman, and J.-W. Barotta, Synchronization and self-assembly of free capillary spinners, Phys. Rev. E111, 035104 (2025)

    2025

  18. [18]

    Z. Izri, M. N. van der Linden, S. Michelin, and O. Dauchot, Self-propulsion of pure water droplets by spontaneous Marangoni-stress-driven motion, Phys. Rev. Lett.113, 248302 (2014)

    2014

  19. [19]

    Bormashenko, Y

    E. Bormashenko, Y. Bormashenko, R. Grynyov, H. Aharoni, G. Whyman, and B. P. Binks, Self-propulsion of liquid marbles: Leidenfrost-like levitation driven by Marangoni flow, J. Phys. Chem. C119, 9910 (2015)

    2015

  20. [20]

    E. Rhee, R. Hunt, S. J. Thomson, and D. M. Harris, SurferBot: A wave-propelled aquatic vibrobot, Bioinspir. Biomim. 17, 055001 (2022)

    2022

  21. [21]

    G. P. Benham, O. Devauchelle, and S. J. Thomson, On wave-driven propulsion, J. Fluid Mech.987, A44 (2024)

    2024

  22. [22]

    O’Donovan, M

    D. O’Donovan, M. D. Bustamante, O. Devauchelle, and G. P. Benham, Achieving optimal locomotion using self-generated waves, J. Fluid Mech.1029, A4 (2026)

    2026

  23. [23]

    S. W. Tarr, J. S. Brunner, D. Soto, and D. I. Goldman, Probing hydrodynamic fluctuation-induced forces with an oscillating robot, Phys. Rev. Lett.132, 084001 (2024)

    2024

  24. [24]

    Hartmann, M

    F. Hartmann, M. Baskaran, G. Raynaud, M. Benbedda, K. Mulleners, and H. Shea, Highly agile flat swimming robot, Sci. Robot.10, eadr0721 (2025)

    2025

  25. [25]

    Y. Chen, N. Doshi, B. Goldberg, H. Wang, and R. J. Wood, Controllable water surface to underwater transition through electrowetting in a hybrid terrestrial-aquatic microrobot, Nat. Commun.9, 2495 (2018)

    2018

  26. [26]

    Y. S. Song and M. Sitti, Surface-tension-driven biologically inspired water strider robots: Theory and experiments, IEEE Trans. Robot.23, 578 (2007)

    2007

  27. [27]

    R. Hunt, Z. Zhao, E. Silver, J. Yan, Y. Bazilevs, and D. M. Harris, Drag on a partially immersed sphere at the capillary scale, Phys. Rev. Fluids8, 084003 (2023)

    2023

  28. [28]

    D¨ orr, S

    A. D¨ orr, S. Hardt, H. Masoud, and H. A. Stone, Drag and diffusion coefficients of a spherical particle attached to a fluid–fluid interface, J. Fluid Mech.790, 607 (2016)

    2016

  29. [29]

    Z. Zhou, P. M. Vlahovska, and M. J. Miksis, Drag force on spherical particles trapped at a liquid interface, Phys. Rev. Fluids7, 124001 (2022)

    2022

  30. [30]

    J. T. Petkov, N. D. Denkov, K. D. Danov, O. D. Velev, R. Aust, and F. Durst, Measurement of the drag coefficient of spherical particles attached to fluid interfaces, J. Colloid Interface Sci.172, 147 (1995)

    1995

  31. [31]

    Ally and A

    J. Ally and A. Amirfazli, Magnetophoretic measurement of the drag force on partially immersed microparticles at air–liquid interfaces, Colloids Surf. A: Physicochem. Eng. Asp.360, 120 (2010)

    2010

  32. [32]

    Loudet, M

    J.-C. Loudet, M. Qiu, J. Hemauer, and J. J. Feng, Drag force on a particle straddling a fluid interface: Influence of interfacial deformations, Eur. Phys. J. E43, 13 (2020)

    2020

  33. [33]

    Le Merrer, C

    M. Le Merrer, C. Clanet, D. Qu´ er´ e, E. Rapha¨ el, and F. Chevy, Wave drag on floating bodies, Proc. Natl. Acad. Sci. U.S.A. 108, 15064 (2011)

    2011

  34. [34]

    K. D. Danov, R. Aust, F. Durst, and U. Lange, Influence of the surface viscosity on the hydrodynamic resistance and surface diffusivity of a large Brownian particle, J. Colloid Interface Sci.175, 36 (1995)

    1995

  35. [35]

    K. D. Danov, R. Dimova, and B. Pouligny, Viscous drag of a solid sphere straddling a spherical or flat surface, Phys. Fluids 12, 2711 (2000)

    2000

  36. [36]

    Pozrikidis, Particle motion near and inside an interface, J

    C. Pozrikidis, Particle motion near and inside an interface, J. Fluid Mech.575, 333 (2007)

    2007

  37. [37]

    A. Dani, G. Keiser, M. Yeganeh, and C. Maldarelli, Hydrodynamics of particles at an oil–water interface, Langmuir31, 13290 (2015)

    2015

  38. [38]

    Kamoliddinov, I

    F. Kamoliddinov, I. Vakarelski, and S. Thoroddsen, Hydrodynamic regimes and drag on horizontally pulled floating spheres, Phys. Fluids33, 093308 (2021)

    2021

  39. [39]

    Pucci, I

    G. Pucci, I. Ho, and D. M. Harris, Friction on water sliders, Sci. Rep.9, 4095 (2019)

    2019

  40. [40]

    J. D. Anderson, Ludwig Prandtl’s boundary layer, Phys. Today58, 42 (2005)

    2005

  41. [41]

    G. G. Stokes,Mathematical and Physical Papers, Vol. 3 (Cambridge University Press, 1922)

    1922

  42. [42]

    Rayleigh, LXXXII

    L. Rayleigh, LXXXII. on the motion of solid bodies through viscous liquid, Philos. Mag.21, 697 (1911)

    1911

  43. [43]

    Schlichting and K

    H. Schlichting and K. Gersten,Boundary-Layer Theory(Springer, 2016)

    2016

  44. [44]

    Panton, The transient for Stokes’s oscillating plate: A solution in terms of tabulated functions, J

    R. Panton, The transient for Stokes’s oscillating plate: A solution in terms of tabulated functions, J. Fluid Mech.31, 819 (1968)

    1968

  45. [45]

    Liu and I.-C

    C.-M. Liu and I.-C. Liu, A note on the transient solution of Stokes’ second problem with arbitrary initial phase, J. Mech. 22, 349 (2006)

    2006

  46. [46]

    M. E. Erdogan, A note on an unsteady flow of a viscous fluid due to an oscillating plane wall, Int. J. Non-Linear Mech. 35, 1 (2000)

    2000

  47. [47]

    K. W. Li and A. C. Marfatia, Stokes second problem for the cylinder, J. Basic Eng.93, 326 (1971)

    1971

  48. [48]

    Rivero, F

    M. Rivero, F. Garz´ on, J. Nunez, and A. Figueroa, Study of the flow induced by circular cylinder performing torsional oscillation, Eur. J. Mech. B/Fluids78, 245 (2019)

    2019

  49. [49]

    Song and M

    Y. Song and M. J. Rau, Viscous fluid flow inside an oscillating cylinder and its extension to Stokes’ second problem, Phys. Fluids32, 043603 (2020)

    2020

  50. [50]

    Von Kerczek and S

    C. Von Kerczek and S. H. Davis, Linear stability theory of oscillatory Stokes layers, J. Fluid Mech.62, 753 (1974)

    1974

  51. [51]

    Hall, The linear stability of flat Stokes layers, Proc

    P. Hall, The linear stability of flat Stokes layers, Proc. R. Soc. Lond. A359, 151 (1978)

    1978

  52. [52]

    P. J. Blennerhassett and A. P. Bassom, The linear stability of flat Stokes layers, J. Fluid Mech.464, 393 (2002)

    2002

  53. [53]

    Wang, The flow field induced by an oscillating sphere, J

    C.-Y. Wang, The flow field induced by an oscillating sphere, J. Sound Vib.2, 257 (1965)

    1965

  54. [54]

    Riley, On a sphere oscillating in a viscous fluid, Q

    N. Riley, On a sphere oscillating in a viscous fluid, Q. J. Mech. Appl. Math.19, 461 (1966)

    1966

  55. [55]

    M. S. Longuet-Higgins, Mass transport in water waves, Philos. Trans. R. Soc. Lond. A245, 535 (1953). 12

    1953

  56. [56]

    M. T. Hehner, D. Gatti, and J. Kriegseis, Stokes-layer formation under absence of moving parts—a novel oscillatory plasma actuator design for turbulent drag reduction, Phys. Fluids31, 051701 (2019)

    2019

  57. [57]

    Pal and S

    D. Pal and S. Chakraborty, Fluid flow induced by periodic temperature oscillation over a flat plate: Comparisons with the classical Stokes problems, Phys. Fluids27, 053602 (2015)

    2015

  58. [58]

    Ishfaq, W

    N. Ishfaq, W. A. Khan, and Z. H. Khan, The Stokes’ second problem for nanofluids, J. King Saud Univ. Sci.31, 61 (2019)

    2019

  59. [59]

    Miller and L

    C. Miller and L. Scriven, The oscillations of a fluid droplet immersed in another fluid, J. Fluid Mech.32, 417 (1968)

    1968

  60. [60]

    Plateau,Statique Exp´ erimentale et Th´ eorique des Liquides Soumis aux Seules Forces Mol´ eculaires, Vol

    J. Plateau,Statique Exp´ erimentale et Th´ eorique des Liquides Soumis aux Seules Forces Mol´ eculaires, Vol. 2 (Gauthier- Villars, 1873)

  61. [61]

    Marangoni, The principle of the surface viscosity of liquids established by Mr

    C. Marangoni, The principle of the surface viscosity of liquids established by Mr. J. Plateau, Nuovo Cim.5, 239 (1872)

  62. [62]

    Manikantan and T

    H. Manikantan and T. M. Squires, Surfactant dynamics: Hidden variables controlling fluid flows, J. Fluid Mech.892, P1 (2020)

    2020

  63. [63]

    Fitzgibbon, E

    S. Fitzgibbon, E. S. G. Shaqfeh, G. G. Fuller, and T. W. Walker, Scaling analysis and mathematical theory of the interfacial stress rheometer, J. Rheol.58, 999 (2014)

    2014

  64. [64]

    Braunreuther, M

    M. Braunreuther, M. Liegeois, J. V. Fahy, and G. G. Fuller, Nondestructive rheological measurements of biomaterials with a magnetic microwire rheometer, J. Rheol.67, 579 (2023)

    2023

  65. [65]

    A. Shih, S. J. Chung, O. B. Shende, S. E. Herwald, A. M. Vezeridis, and G. G. Fuller, Viscoelastic measurements of abscess fluids using a magnetic stress rheometer, Phys. Fluids36, 113603 (2024)

    2024

  66. [66]

    Z. A. Zell, V. Mansard, J. Wright, K. Kim, S. Q. Choi, and T. M. Squires, Linear and nonlinear microrheometry of small samples and interfaces using microfabricated probes, J. Rheol.60, 141 (2016)

    2016

  67. [67]

    D. M. Harris, J. Quintela, V. Prost, P.-T. Brun, and J. W. M. Bush, Visualization of hydrodynamic pilot-wave phenomena, J. Vis.20, 13 (2017)

    2017

  68. [68]

    D. J. Griffiths,Introduction to Electrodynamics(Cambridge University Press, 2023)

    2023

  69. [69]

    I. Ho, G. Pucci, and D. M. Harris, Direct measurement of capillary attraction between floating disks, Phys. Rev. Lett. 123, 254502 (2019)

    2019

  70. [70]

    Rapha¨ el and P.-G

    E. Rapha¨ el and P.-G. de Gennes, Capillary gravity waves caused by a moving disturbance: Wave resistance, Phys. Rev. E 53, 3448 (1996)

    1996

  71. [71]

    Closa, A

    F. Closa, A. D. Chepelianskii, and E. Rapha¨ el, Capillary-gravity waves generated by a sudden object motion, Phys. Fluids 22, 052107 (2010)

    2010

  72. [72]

    J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol.127, 553 (1955)

    1955

  73. [73]

    R. C. Ackerberg and J. H. Phillips, The unsteady laminar boundary layer on a semi-infinite flat plate due to small fluctuations in the magnitude of the free-stream velocity, J. Fluid Mech.51, 137 (1972)

    1972

  74. [74]

    Rott and M

    N. Rott and M. L. Rosenzweig, On the response of the laminar boundary layer to small fluctuations of the free-stream velocity, J. Aeronaut. Sci.27, 741 (1960)

    1960

  75. [75]

    M. J. Lighthill, The response of laminar skin friction and heat transfer to fluctuations in the stream velocity, Proc. R. Soc. Lond. Ser. A-Math. Phys. Sci.224, 1 (1954)

    1954

  76. [76]

    Ai and K

    L. Ai and K. Vafai, An investigation of Stokes’ second problem for non-Newtonian fluids, Numer. Heat Transf. A47, 955 (2005)

    2005

  77. [77]

    Fetecau, M

    C. Fetecau, M. Jamil, C. Fetecau, and I. Siddique, A note on the second problem of Stokes for Maxwell fluids, Int. J. Non-Linear Mech.44, 1085 (2009)

    2009

  78. [78]

    Foffano, J

    G. Foffano, J. S. Lintuvuori, K. Stratford, M. E. Cates, and D. Marenduzzo, Colloids in active fluids: Anomalous mi- crorheology and negative drag, Phys. Rev. Lett.109, 028103 (2012)

    2012

  79. [79]

    Kneˇ zevi´ c, L

    M. Kneˇ zevi´ c, L. E. Avil´ es Podgurski, and H. Stark, Oscillatory active microrheology of active suspensions, Sci. Rep.11, 22706 (2021)

    2021

  80. [80]

    P. A. R¨ uhs, L. B¨ oni, G. G. Fuller, R. F. Inglis, and P. Fischer, In-situ quantification of the interfacial rheological response of bacterial biofilms to environmental stimuli, PLoS One8, e78524 (2013)

    2013

Showing first 80 references.