Pith. sign in

REVIEW 3 major objections 5 minor 79 references

Viscous adhesion in vibrated sheets: elastohydrodynamics with inertia and compressibility effects

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The maximum weight a vibrating elastic sheet can support saturates at about 7.8 times the elastohydrodynamic force scale, and fluid inertia and compressibility reduce it further.

desk verdict Strong-forcing saturation Wmax ~ 7.8 F_bv is a solid, well-validated result; the inertial corrections in Sec. 4.2 run past the model's stated validity range, so treat those numbers as provisional. read the letter →

arxiv 2509.02262 v1 pith:EQ4W5GRE submitted 2025-09-02 physics.flu-dyn

Stephane Poulain , Timo Koch , L. Mahadevan , Andreas Carlson This is my paper · ORCID
classification physics.flu-dyn MSC 76D0874F1076N10
keywords elastohydrodynamicadhesionvibratedelasticsheetslubricationtheorysqueezefilmfluidinertiacompressibilityhoveringheightcontactlessgripper
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the theory of elastohydrodynamic adhesion—a vibrating elastic sheet hovering near a wall because viscous lubrication breaks time-reversal symmetry—from weak to strong forcing. The central result is that the maximum weight a sheet can support, Gmax, grows as the square of the forcing at weak amplitudes but saturates near 7.8 times the elastohydrodynamic force scale Fbv once the forcing exceeds about 20 Fbv. The paper also computes first-order corrections from fluid inertia and compressibility, finding both add repulsive contributions that reduce the load capacity and the hovering height. These results matter for contactless grippers and soft suction-cup-like devices because they put quantitative limits on what a given actuator-plus-sheet combination can lift.

What carries the argument

The elastohydrodynamic force scale Fbv = (µωB²)^(1/3) and height scale Hbv = R²(µω/B)^(1/3) set the problem. The dynamics is governed by an inertial-lubrication equation (adapted from Rojas et al. 2010 to flow between two solids) coupled to a Kirchhoff-Love plate, with an entrance/exit pressure boundary condition at the sheet's free edge carrying a loss coefficient k = 0.5. The key analytical objects are the eigenmode-reduced evolution equation (3.2) for weak forcing and the interpolation (3.4) for the maximum supported weight, with the sheet's convexity during the cycle acting as the gatekeeper between contactless adhesion and edge contact.

What would settle it

Measure the maximum load a vibrating sheet supports as a function of the motor's force amplitude for a fixed geometry and frequency. The model predicts that Wmax saturates at about 7.8 Fbv once Fa exceeds roughly 20 Fbv; a continued rise as Fa², a peak followed by a decline, or a saturation value differing from ~7.8 Fbv would contradict the saturation formula (3.4). A separate test would be to vary the fluid pressure: the model says lowering ambient pressure weakens the compressibility-induced repulsion and thus raises Gmax.

Watch

Extended reading notes

Core claim

The paper's claim is that for a uniformly forced elastic sheet in a viscous fluid, the maximum dimensionless supported weight is Gmax = 0.137 α² / (1 + 0.0176 α²), interpolating between the weak-forcing scaling α² and a saturation at α ≳ 20 with Gmax ≈ 7.8. The saturation is set not by the fluid but by the sheet losing convexity: when the active force pulls hard enough, the sheet's edges touch the substrate during part of the cycle, and the threshold weight for that contact scales as Fbv. Fluid inertia, however, acts as a height-dependent added mass and Bernoulli-like pressure, giving a net repulsive force that lowers Gmax and causes sharp drops as successive bending modes become inaccessibl

Load-bearing premise

The first-order inertial lubrication equation (2.5), together with the edge pressure loss coefficient k = 0.5, correctly represents the repulsive inertial force for deformable gaps up to Reynolds numbers around 50; if higher-order inertia or a different loss coefficient is needed, the predicted reductions in Gmax and the sharp drops would change quantitatively.

Editorial extensions

If this is right

  • For a fixed actuator, there is an optimal bending stiffness B* ≈ 0.05 Fa^(3/2) (µω)^(-1/2); making the sheet softer beyond that point reduces rather than increases the maximum load.
  • In the strong-forcing regime the equilibrium height becomes linear in the weight, heq ≈ 0.05 G, and at maximum load the hovering height is always approximately 0.3 Hbv, independent of forcing amplitude.
  • Fluid inertia is destabilizing: in weak forcing the equilibrium Reynolds number Re_eq stays below about 1, whereas in strong forcing it can reach O(100), where the first-order inertial lubrication equation is no longer quantitatively reliable.
  • Compressibility shifts the adhesion threshold downward as an effective added weight, so experiments in air (Sq ≈ 0.006) see only a modest reduction, while higher-pressure gases or larger gaps would make the reduction significant.
  • Higher-order bending modes are excited as the sheet approaches the wall, and inertia suppresses the lowest modes one by one, which is why Gmax drops sharply at specific Reynolds numbers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the saturation formula holds, the practical design rule for a contactless gripper is to match the actuator's force scale to the sheet's elastohydrodynamic force scale: overdriving a soft sheet wastes energy and adds nothing to payload.
  • The sharp drops in Gmax at Re_bv ≈ 1/e_i² suggest that a load-capacity scan across vibration frequency could act as a modal spectrometer for thin elastic sheets, revealing which bending modes participate in the adhesion.
  • The compressibility-as-added-weight result implies that reducing ambient pressure (e.g., operating in a vacuum or low-pressure chamber) should strengthen the viscous adhesion, a testable consequence not pursued in the paper.
  • Since the model neglects solid inertia, real sheets with strong resonances might show either enhanced or suppressed adhesion near resonance; extending the analysis to I_bv > 0 would be the natural next step.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the elastohydrodynamic hovering of a thin elastic sheet vibrating near a rigid substrate, extending the authors' earlier weak-forcing asymptotic analysis to strong forcing through one-dimensional numerical simulations of a depth-integrated lubrication model coupled to Kirchhoff-Love bending. The main claims are: (i) the maximum supported weight crosses over from Gmax ~ 0.137 α^2 for weak forcing to a saturation Gmax ~ 7.8 for α ≳ 20, captured by the interpolation (3.4); (ii) at strong forcing the equilibrium height obeys heq ≈ 0.05 G; (iii) fluid inertia and compressibility each introduce repulsive contributions that reduce adhesive strength, with inertia causing sharp drops in Gmax at Reynolds numbers interpreted through the eigenmode heights e_i of the authors' previous reduced model. The paper combines asymptotic rigid-sheet analyses (Appendices D and E), numerical bifurcation diagrams, and regime maps.

Significance. If the quantitative claims hold, the paper provides a useful design rule for soft contactless grippers: the load capacity of a vibrated elastic sheet is bounded by the elastohydrodynamic force scale F_bv, and inertial/compressible corrections impose additional penalties. The viscous part of the work is convincing: the numerics reproduce the prior weak-forcing asymptotic model in Fig. 2(b), and the saturation of Gmax is a clear, reproducible numerical result. The rigid-sheet asymptotic calculations for inertia and compressibility are also valuable and appear internally consistent. The main concern is that the soft-sheet inertial results, which carry the paper's central 'inertia reduces adhesion' message, are computed in a regime where the paper itself states the model is not expected to be valid. The paper is honest about this limitation in §4.2 and §6, but the quantitative claims—especially the magnitude of the Gmax reduction and the locations of the sharp drops—are not yet secured.

major comments (3)
  1. [§4.2, Figs. 8–9, Eq. (2.5), Appendix B] The quantitative inertial results for soft sheets rely on Eq. (2.5), derived under the assumption Re ≪ 1 in Appendix A, and on the edge loss coefficient k = 0.5, which Appendix B calibrates for 'Reynolds numbers that remain small, Re < 10'. Yet the paper reaches Re_eq = O(10^2) for α = 20 in Fig. 9(c), and the sharp drops in Gmax occur precisely in that range. The paper itself states that Eq. (2.5) 'may no longer be valid' for Re_eq ≳ 50 and that full Navier-Stokes simulations or higher-order corrections are needed. Since the Bernoulli-like repulsion and the threshold locations depend on k and on neglected higher-order inertial terms, the numerical magnitude of the Gmax reduction and the quantitative locations of the drops are not established. Please either restrict the inertial soft-sheet claims to Re_eq ≲ 50, supply a validation against full Navier-Stokes or higher-order inertial lubri
  2. [§3.2.1, Eq. (3.4)] The strong-forcing saturation Gmax ≈ 7.8 and the interpolation (3.4) are central to the paper's design conclusions, but the coefficient 0.0176 is a numerical fit and the α ≫ 1 plateau is not derived analytically. The formula is presented as a general result and is used to compute the optimal bending stiffness and the design curves in Fig. 4(d). This is acceptable as a numerical finding, but the paper should state explicitly that (3.4) is a fit over 0 < α ≤ 100, not a parameter-free law, and should indicate its uncertainty. The current wording ('captured by the following interpolation') understates the degree to which the saturation law rests on simulation data.
  3. [§4.2, Figs. 8–9 and Eq. (3.2)] The sharp drops in Gmax are interpreted as happening at Re_bv ≈ 1/e_i^2, where e_i = 0.242/i^2 are characteristic heights from the authors' earlier asymptotic model (Eq. 3.2). This is not circular, because the numerical simulations are independent of the eigenmode decomposition, but the interpretation is not a test of the reduced model: the e_i are derived for the purely viscous, weak-forcing regime and are not re-derived for finite Re. Please mark this as an interpretation and clarify that the eigenmode heights are used only as a heuristic to label the drops.
minor comments (5)
  1. [Appendix C] The eigenfunctions ζ_n are said to be 'shown in figure 2(d)', but Fig. 2 has only panels (a) and (b); the eigenmodes appear in Fig. 11. Please correct the cross-reference.
  2. [§3.2.2, Eq. (3.5a)] The linear relation heq ≈ 0.05 G for α ≫ 1 is a numerical observation, not derived. Since it underpins the dimensional scaling heq ∼ W R^2/B and the contact criterion in §3.2.3, please state explicitly that the prefactor 0.05 is fitted and give the range of G/α and α over which it was verified.
  3. [§2.2, Eq. (2.5)] The statement 'to O(ε_bv^2, Re_bv, ε_bv^2 Re_bv, Re_bv Sq_bv, Sq_bv)' is confusing: the displayed equation contains only the leading-order inertial terms, and the compressible correction appears only through ρ in (2.4). Consider rewriting the order-of-accuracy statement to match the truncated expansion actually used.
  4. [General] Typos and wording: 'figure figure 7' in §4.1; 'a related investigations' in §2.6; 'centrimetric' in §6 should likely be 'centimetric'. Also, the phrase 'anecdotal observation' in §4.2 is fine but should be clearly separated from the quantitative claims.
  5. [§5.2, Fig. 10] The effective-weight relation (5.2) is shown to underestimate the compressibility effect by up to 50% at larger Sq_bv for α = 20. This is acknowledged, but the abstract and introduction present compressibility primarily as a simple 'effective weight' correction. Please soften the wording so that the quantitative scope of (5.2) is clear from the outset.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new strong-forcing saturation and inertial/compressible corrections are obtained from direct numerical solution of the governing equations and from independent external asymptotics, not from re-deriving the paper's own inputs.

full rationale

The derivation chain is self-contained in the relevant sense. The central new result, the saturation of the maximum supported weight Gmax ≈ 7.8 Fbv for strong forcing, comes from numerical solutions of the full lubrication-elastohydrodynamic system (2.6) and (3.1), as stated in §3.2.1: "We show in figure 4(a,b) bifurcation diagrams obtained by numerically solving the governing equations (2.6) and (3.1)". Equation (3.4) is explicitly an interpolation of those numerical results, not a hidden fit masquerading as a prediction: "These two asymptotic behaviors are captured by the following interpolation, which also captures well the numerical results". The weak-forcing asymptotic relation Gmax = 0.137α² is taken from the authors' previous work, but it is independently checked against the same numerical model in figure 2(b), so it is not load-bearing by self-citation alone. The inertial correction model (2.5) is derived in Appendix A from Navier-Stokes by an explicit small-Re expansion and is attributed to Rojas et al. (2010) with a derivation reproduced in the appendix; the boundary-condition coefficient k=0.5 is justified in Appendix B by matching the independent numerical/analytical work of Ramanarayanan et al. (2022), with the paper itself noting k varies at high Reynolds number. The compressibility effective-weight relation (5.2) is derived from a two-timescale asymptotic analysis in Appendix E and compared with direct numerical simulations. The interpretation of the sharp drops in Gmax at Rebv ≈ 1/e_i² uses the eigenmode heights e_i from the authors' prior asymptotic model, but the drops themselves are observed in the direct simulations and the paper shows the corresponding higher-order deformation modes from those simulations (figure 9b). This is cross-validation, not circularity. The main legitimate caveat is a correctness/validity risk, which the paper itself flags: the caption of figure 9 states "The inertial lubrication theory is not expected to be valid for Re_eq ≳ 50", and §4.2 repeats "the first-order inertial corrections to lubrication theory that we use (eq. (2.5) and appendix A) may no longer be valid" for the largest Re_eq values. That limits confidence in the quantitative Gmax drops at high Re_eq, but it is not a circular reduction of the prediction to its inputs. No step in the claimed derivation is equivalent by construction to a fitted parameter renamed as a prediction, nor is any central premise imported solely from a self-citation chain.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper's central new results rely on two fitted interpolations (0.0176 in Gmax, 0.05 in heq) and on the prior asymptotic model of the same group. The inertial and compressible corrections rest on standard but non-trivial closures (inertial lubrication, entrance loss coefficient, isothermal gas) that are not fully validated for deformable sheets at the highest Reynolds numbers studied.

free parameters (4)
  • k, loss coefficient in inflow boundary condition = 0.5
    Inflow pressure boundary condition (2.10), chosen by analogy with pipe flows and matching Ramanarayanan et al. (2022); the inertial rigid-sheet result (4.4) depends on 1 - 7k/16.
  • c_interp, coefficient in Gmax interpolation = 0.0176
    Dimensionless coefficient in Eq. (3.4), fitted to numerical Gmax data over 0 < alpha <= 100.
  • c_heq, slope of heq vs G for alpha >> 1 = 0.05
    Numerical slope of the linear relation heq ~ 0.05G observed in Fig. 4b for strong forcing.
  • collision potential parameters A and n = A = 1e-5, n = 5
    Repulsive potential f_w = (A/h)^n introduced in Sec. 2.3 to prevent numerical contact; authors verified insensitivity to A as long as A is small.
assumptions (6)
  • domain assumption The prior reduced-order model (Poulain et al. 2025), Eq. (3.2) here, with eigenmodes zeta_i, heights e_i = 0.242/i^2, and coefficients d_ij, is correct.
    Used in Sec. 3.1 for the weak-forcing comparison and in Sec. 4.2 to explain drops in Gmax at Re_bv ~ 1/e_i^2.
  • domain assumption The first-order inertial lubrication equation (2.5) is valid for arbitrary h(x,t) and Re up to O(50).
    Adapted from Rojas et al. (2010) for free surfaces to a solid-wall geometry; underlies all soft-sheet inertial results in Sec. 4.
  • domain assumption The entrance-loss boundary condition (2.10) with k = 0.5 is accurate for oscillatory inflow at small Reynolds numbers.
    Justified in Appendix B by comparing edge pressure scaling with Ramanarayanan et al. (2022); affects the computed inertial force.
  • domain assumption The isothermal ideal gas law rho = 1 + Sq p, Eq. (2.4b), holds in the gap.
    Assumed valid for small Peclet number Pe = Re_bv Pr; used throughout Sec. 5, where Re_bv is set to zero.
  • standard math The sheet deformation follows the Kirchhoff-Love pure bending model in 1D, Eq. (2.6).
    Standard thin-plate model; neglects stretching and 2D effects, which the authors state as a limitation in Sec. 6.
  • ad hoc to paper The collision potential f_w = (A/h)^n with A = 1e-5, n = 5 does not perturb contactless dynamics.
    Introduced in Sec. 2.3 to handle numerical contact; authors checked small-A insensitivity and only use it in contactless simulations as a safeguard.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Viscous adhesion in vibrated sheets: elastohydrodynamics with inertia and compressibility effects." pith.science (2026). https://pith.science/paper/EQ4W5GRE

@misc{pith2026250902262,
  author       = {Pith},
  title        = {Pith review of: Viscous adhesion in vibrated sheets: elastohydrodynamics with inertia and compressibility effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQ4W5GRE}},
  note         = {Machine review of arXiv:2509.02262}
}
read the original abstract

Inspired by recent experiments demonstrating that vibrating elastic sheets can function as seemingly contactless suction cups, we investigate the elastohydrodynamic hovering of a thin elastic sheet vibrating near a rigid substrate. Previous theoretical work suggests that the hovering height results from a balance between the active forcing that triggers the vibrations, the bending forces associated with the sheet's deformation, the viscous lubrication flow between the sheet and the substrate, and the sheet's weight. Here, we extend this analysis beyond the asymptotic regime of weak forcing and explore the regime of strong forcing through numerical simulations. We further quantify the influence of fluid inertia and compressibility on the equilibrium hovering height and the maximum load that can be supported. Both effects are found to introduce repulsive contributions to the net force on the sheet, which can significantly reduce its adhesive strength. Beyond providing insights into soft contactless grippers and swimming near surfaces, our analysis is relevant to the elastohydrodynamics of squeeze films and near-field acoustic levitation.

Figures

Figures reproduced from arXiv: 2509.02262 by the authors.

Figure 1
Figure 1. (a) An elastic sheet (radius R˜, density ˜ρs, bending rigidity B˜, Poisson’s ratio ν, thickness ˜e) immersed in a fluid (ambient density ˜ρa, ambient pressure p˜a, dynamic viscosity ˜µ) and forced periodically at its center (force F˜a, angular frequency ˜ω, radius ˜ℓ) is placed below a solid substrate with gravity pointing downward. x⊥ = (x, y) represent the horizontal coordinates. (b) When the dimensionless weight … view at source ↗
Figure 3
Figure 3. (a) Illustration of the decomposition (3.3) of the sheet’s shape into a static shape ⟨h⟩(x) (independent of time at the time-averaged steady state), a rigid-body translation h¯(t) − heq, and a time-periodic deformation hd(x, t). (b, c, d) Time-averaged shape ⟨h⟩ and the periodic deformation hd for G = 0.02, 0.04, 0.06, respectively, and α = 1, Ibv = Rebv = Sqbv = 0. These are obtained from numerical simulations once… view at source ↗
Figure 4
Figure 4. Varying active forcing α for an inertialess and incompressible system, Ibv = Rebv = Sqbv = 0. (a, b) Equilibrium height as a function of the dimensionless weight. As α increases, the scaling Gmax ∼ α 2 does not collapse the data anymore. Open symbols represent cases where contact occurs at the sheet’s edges. (c) Regime map showing the three different possibilities (adhesion with or without edge contact, and adhesion… view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: Time-averaged shape ⟨h⟩ and the periodic deformation hd for α = 20, Ibv = Rebv = Sqbv = 0, and G = 1, 2, 4 in (a, b, c), respectively. equilibrium height results from a balance between the bending force and gravity, with viscosity setting the maximum supported weight. …
Figure 6
Figure 6. Figure 6: (a, b) Sheet’s shape and pressure field over a period of vibration for α = 20 and (a) G = 4, (b) G = 8. The green arrows represent the active force schematically, periodically pushing and pulling at the center of the sheet. In (b), at t = 3π/2, the edges of the sheet t…
Figure 7
Figure 7. Figure 7: Squeeze flow of a rigid plate moving normal to a wall with a height [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: (a−b) Equilibrium height as a function of the dimensionless weight with Sqbv = Ibv = 0 and α = 1 for (a) Rebv < 200 and (b) Rebv > 200. Black lines are the stable equilibria of (3.2). (c) Phase diagram showing the accessible weights as a function of Rebv. The first equ…
Figure 9
Figure 9. Figure 9: Effect of the fluid inertia with Sqbv = Ibv = 0 and Rebv > 0 for α = 20. (a) Equilibrium height as a function of the dimensionless weight for the regime of contactless adhesion. (b) Regime maps and the associated (c) range of Reynolds number based on equilibrium height…
Figure 10
Figure 10. Figure 10: (a − b) Effect of the fluid compressibility for Rebv = Ibv = 0 and Sqbv > 0. (a, b) correspond to the weak forcing regime with α = 1 and (c, d) to the strong forcing regime with α = 20. The dashed lines in (b) and (d) are derived from (5.2) and show Gmax(Sqbv) = Gmax(…
Figure 11
Figure 11. Figure 11: Asymptotic results for α ≲ 1, Ibv = Rebv = Sqbv = 0, adapted from Poulain et al. (2025). As G and heq decrease, the sheet presents higher and higher order deformation modes. The i−th mode ζi is excited if h ≲ ei. gradient as a function of the height h from (D 1): 1 x …
Figure 12
Figure 12. Figure 12: Comparison between numerical results (symbols) and first-order [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references · 77 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    1977 Diagonally implicit R unge- K utta methods for stiff O.D.E

    Alexander, R. 1977 Diagonally implicit R unge- K utta methods for stiff O.D.E. 's . SIAM Journal on Numerical Analysis 14 (6), 1006--1021

    work page 1977

  4. [4]

    , P \'e rez, N

    Andrade, M.A.B. , P \'e rez, N. & Adamowski, J.C. 2018 Review of progress in acoustic levitation . Braz. J. Phys. 48 , 190--213

    work page 2018

  5. [5]

    , Ramos, T.S

    Andrade, M.A.B. , Ramos, T.S. , Adamowski, J.C. & Marzo, A. 2020 Contactless pick-and-place of millimetric objects using inverted near-field acoustic levitation . Appl. Phys. Lett. 116 (5)

    work page 2020

  6. [6]

    , Skotheim, J

    Argentina, M. , Skotheim, J. & Mahadevan, L. 2007 Settling and swimming of flexible fluid-lubricated foils . Phys. Rev. Lett. 99 (22), 224503

    work page 2007

  7. [7]

    , Van Ostayen, R.A.J

    Atalla, M.A. , Van Ostayen, R.A.J. , Sakes, A. & Wiertlewski, M. 2023 Incompressible squeeze-film levitation . Appl. Phys. Lett. 122 (24)

    work page 2023

  8. [8]

    & Yang, H

    Bao, M. & Yang, H. 2007 Squeeze film air damping in MEMS . Sens. Actuator A-Phys. 136 (1), 3--27

    work page 2007

Show all 79 references
  1. [9]

    1967 An Introduction to Fluid Dynamics\/

    Batchelor, G.K. 1967 An Introduction to Fluid Dynamics\/ . Cambridge University Press

    1967

  2. [10]

    , Liz \'e e, M

    Bigan, N. , Liz \'e e, M. , Pascual, M. , Nigu \`e s, A. , Bocquet, L. & Siria, A. 2024 Long range signature of liquid's inertia in nanoscale drainage flows . Soft Matter 20 (44), 8804--8811

    2024

  3. [11]

    1982 A review of added mass and fluid inertial forces

    Brennen, C.E. 1982 A review of added mass and fluid inertial forces . Tech. Rep. CR 82.010 Naval Civil Engineering Laboratory

    1982

  4. [12]

    , Coupier, G

    Bureau, L. , Coupier, G. & Salez, T. 2023 Lift at low R eynolds number . Eur. Phys. J. E 46 (11), 111

    2023

  5. [13]

    , Picardi, G

    Calisti, M. , Picardi, G. & Laschi, C. 2017 Fundamentals of soft robot locomotion . J. R. Soc. Interface. 14 (130), 20170101

    2017

  6. [14]

    2015 Apparatus and method for orthosonic lift by deflection

    Colasante, D.A. 2015 Apparatus and method for orthosonic lift by deflection. U.S. P atent US8967965B1

    2015

  7. [15]

    2016 Youtube videos

    Colasante, D. 2016 Youtube videos. Available at https://www.youtube.com/watch?v=kG6vXGidQbo (2016) and https://www.youtube.com/watch?v=ruDpMhlKy6M (2024). Accessed on April 9, 2025

    2016

  8. [16]

    , Rycroft, C.H

    Derr, N.J , Dombrowski, T. , Rycroft, C.H. & Klotsa, D. 2022 Reciprocal swimming at intermediate R eynolds number . J. Fluid Mech. 952 , A8

    2022

  9. [17]

    , Hierold, C

    Fedder, G.K. , Hierold, C. , Korvink, J.G. & Tabata, O. 2015 Resonant MEMS: fundamentals, implementation, and application\/ . John Wiley & Sons

    2015

  10. [18]

    & Leshansky, A

    Fouxon, I. & Leshansky, A. 2018 Fundamental solution of unsteady S tokes equations and force on an oscillating sphere near a wall . Phys. Rev. E 98 (6), 063108

    2018

  11. [19]

    , Rubinstein, B

    Fouxon, I. , Rubinstein, B. , Weinstein, O. & Leshansky, A. 2020 Fluid-mediated force on a particle due to an oscillating plate and its effect on deposition measurements by a quartz crystal microbalance . Phys. Rev. Lett. 125 (14), 144501

    2020

  12. [20]

    , Argentina, M

    Gazzola, M. , Argentina, M. & Mahadevan, L. 2014 Scaling macroscopic aquatic locomotion . Nat. Phys. 10 (10), 758--761

    2014

  13. [21]

    , Koike, Y

    Hashimoto, Y. , Koike, Y. & Ueha, S. 1996 Near-field acoustic levitation of planar specimens using flexural vibration . J. Acoust. Soc. Am. 100 (4), 2057--2061

    1996

  14. [22]

    , Liu, R

    Hong, Q. , Liu, R. , Yang, H. & Zhai, X. 2009 Wall climbing robot enabled by a novel and robust vibration suction technology. In 2009 IEEE International Conference on Automation and Logistics\/ , pp. 331--336 . IEEE

    2009

  15. [23]

    2006 Hydrodynamic lubrication\/

    Hori, Y. 2006 Hydrodynamic lubrication\/ . Springer

    2006

  16. [24]

    1966 The unsteady laminar flow between two parallel discs with arbitrarily varying gap width

    Ishizawa, S. 1966 The unsteady laminar flow between two parallel discs with arbitrarily varying gap width . Bulletin of JSME 9 (35), 533--550

    1966

  17. [25]

    , Ramanarayanan, S

    Jia, C. , Ramanarayanan, S. , Sanchez, A.L. & Tolley, M.T. 2023 Controlling the motion of gas-lubricated adhesive disks using multiple vibration sources . Front. Robot. AI. 10 , 1231976

    2023

  18. [26]

    & Wilson, S.D.R

    Jones, A.F. & Wilson, S.D.R. 1975 On the failure of lubrication theory in squeezing flows . J. Tribol

    1975

  19. [27]

    , Weishaupt, K

    Koch, T. , Weishaupt, K. , Gl\"aser, D. & others 2021 DuMux 3 an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling . Computers & Mathematics with Applications 81 , 423--443

    2021

  20. [28]

    & Hori, Y

    Kuroda, S. & Hori, Y. 1976 A study of fluid inertia effects in a squeeze film (in japanese) . Journal of Japan Society of Lubrication Engineers 21 (11), 740--747

    1976

  21. [29]

    1968 Fluid inertia effects in squeeze films

    Kuzma, D.C. 1968 Fluid inertia effects in squeeze films . Applied Scientific Research 18 , 15--20

    1968

  22. [30]

    & Lifshitz, E.M

    Landau, L.D. & Lifshitz, E.M. 1986 Course of Theoretical Physics vol. 7: Theory of Elasticity\/ . 3rd edn. Pergamon

    1986

  23. [31]

    1962 Isothermal squeeze films

    Langlois, W.E. 1962 Isothermal squeeze films . Quarterly of Applied Mathematics 20 (2), 131--150

    1962

  24. [32]

    2007 Floppy swimming: Viscous locomotion of actuated elastica

    Lauga, E. 2007 Floppy swimming: Viscous locomotion of actuated elastica . Phys. Rev. E 75 (4), 041916

    2007

  25. [33]

    , Tung, R

    Lee, J.W. , Tung, R. , Raman, A. , Sumali, H. & Sullivan, J.P. 2009 Squeeze-film damping of flexible microcantilevers at low ambient pressures: theory and experiment . J. Micromech. Microeng. 19 (10), 105029

    2009

  26. [34]

    Li, X. , Li, N. , Tao, G. , Liu, H. & Kagawa, T. 2015 Experimental comparison of B ernoulli gripper and vortex gripper . Int. J. Precis. Eng. Man. 16 , 2081--2090

    2015

  27. [35]

    , Zhao, Z

    Liu, Y. , Zhao, Z. & Chen, W. 2023 Theoretical investigation of the levitation force generated by underwater squeeze action . Jpn. J. Appl. Phys. 62 (3), 034001

    2023

  28. [36]

    , Mani, M

    Mandre, S. , Mani, M. & Brenner, M.P. 2009 Precursors to splashing of liquid droplets on a solid surface . Phys. Rev. Lett. 102 (13), 134502

    2009

  29. [37]

    , Chivilikhin, S

    Melikhov, I. , Chivilikhin, S. , Amosov, A. & Jeanson, R. 2016 Viscoacoustic model for near-field ultrasonic levitation . Phys. Rev. E 94 (5), 053103

    2016

  30. [38]

    & Bucher, I

    Minikes, A. & Bucher, I. 2003 Coupled dynamics of a squeeze-film levitated mass and a vibrating piezoelectric disc: numerical analysis and experimental study . J. Sound Vib. 263 (2), 241--268

    2003

  31. [39]

    1965 A review of squeeze films

    Moore, D.F. 1965 A review of squeeze films . Wear 8 (4), 245--263

    1965

  32. [40]

    Naghdi, P. M. 1973 The theory of shells and plates . In Linear theories of elasticity and thermoelasticity: linear and nonlinear theories of rods, plates, and shells\/ , pp. 425--640 . Springer

    1973

  33. [41]

    1980 Historical review of WIG vehicles

    Ollila, R.G. 1980 Historical review of WIG vehicles . J. Hydronaut. 14 (3), 65--76

    1980

  34. [42]

    & Pratap, R

    Pandey, A.K. & Pratap, R. 2007 Effect of flexural modes on squeeze film damping in MEMS cantilever resonators . J. Micromech. Microeng. 17 (12), 2475

    2007

  35. [43]

    , Cuttle, C

    Peng, G.G. , Cuttle, C. , MacMinn, C.W. & Pihler-Puzovi \'c , D. 2023 Axisymmetric gas--liquid displacement flow under a confined elastic slab . Phys. Rev. Fluids 8 (9), 094005

    2023

  36. [44]

    , Koch, T

    Poulain, S. , Koch, T. , Mahadevan, L. & Carlson, A. 2025 Hovering of an actively driven fluid-lubricated foil, arXiv:arXiv: 2501.17080

    2025

  37. [45]

    & Roychowdhury, A

    Pratap, R. & Roychowdhury, A. 2014 Vibratory MEMS and Squeeze Film Effects\/ , pp. 319--338 . New Delhi: Springer India

    2014

  38. [46]

    1977 Life at low R eynolds number

    Purcell, E.M. 1977 Life at low R eynolds number . Am. J. Phys. 45 (1), 3--11

    1977

  39. [47]

    2024 Fluid-elastic interactions near contact at low R eynolds number

    Rallabandi, B. 2024 Fluid-elastic interactions near contact at low R eynolds number . Annu. Rev. Fluid Mech. 56 , 491--519

    2024

  40. [48]

    2024 On the emergence of attractive load-bearing forces in vibration-induced squeeze-film gas lubrication

    Ramanarayanan, S. 2024 On the emergence of attractive load-bearing forces in vibration-induced squeeze-film gas lubrication . PhD thesis, University of California, San Diego

    2024

  41. [49]

    , Coenen, W

    Ramanarayanan, S. , Coenen, W. & S \'a nchez, A.L. 2022 Viscoacoustic squeeze-film force on a rigid disk undergoing small axial oscillations . J. Fluid Mech. 933

    2022

  42. [50]

    & S \'a nchez, A.L

    Ramanarayanan, S. & S \'a nchez, A.L. 2022 On the enhanced attractive load capacity of resonant flexural squeeze-film levitators . AIP Advances 12 (10)

    2022

  43. [51]

    & S \'a nchez, A.L

    Ramanarayanan, S. & S \'a nchez, A.L. 2024 The role of fluid--structure coupling in the generation of an attractive squeeze-film force . J. Fluid Mech. 1001 , A52

    2024

  44. [52]

    1991 On the aerodynamics of animal flight in ground effect

    Rayner, J.M.V. 1991 On the aerodynamics of animal flight in ground effect . Proc. Roy. Soc. B 334 (1269), 119--128

    1991

  45. [53]

    , Argentina, M

    Rojas, N.O. , Argentina, M. , Cerda, E. & Tirapegui, E. 2010 Inertial lubrication theory . Phys. Rev. Lett. 104 (18), 187801

    2010

  46. [54]

    , Feng, K

    Shi, M. , Feng, K. , Hu, J. , Zhu, J. & Cui, H. 2019 Near-field acoustic levitation and applications to bearings: a critical review . Int. J. Extreme Manuf. 1 (3), 032002

    2019

  47. [55]

    , Cacucciolo, V

    Shintake, J. , Cacucciolo, V. , Floreano, D. & Shea, H. 2018 Soft robotic grippers . Adv. Mater. 30 (29), 1707035

    2018

  48. [56]

    & Mahadevan, L

    Skotheim, J.M. & Mahadevan, L. 2005 Soft lubrication: The elastohydrodynamics of nonconforming and conforming contacts . Phys. Fluids 17 (9)

    2005

  49. [57]

    , Terada, D

    Takasaki, M. , Terada, D. , Kato, Y. , Ishino, Y. & Mizuno, T. 2010 Non-contact ultrasonic support of minute objects . Phys. Procedia 3 (1), 1059--1065

    2010

  50. [58]

    1967 Film notes for low- R eynolds-number flows

    Taylor, G.I. 1967 Film notes for low- R eynolds-number flows. National Committee for Fluid Mechanics Films. Available at https://web.mit.edu/hml/ncfmf.html. Accessed on June 11, 2025

    1967

  51. [59]

    & Saffman, P.G

    Taylor, S.G. & Saffman, P.G. 1957 Effects of compressibility at low R eynolds number . Journal of the Aeronautical Sciences 24 (8), 553--562

    1957

  52. [60]

    & Bourgin, P

    Tichy, J.A. & Bourgin, P. 1985 The effect of inertia in lubrication flow including entrance and initial conditions . J. Appl. Mech

    1985

  53. [61]

    & Winer, W.O

    Tichy, J.A. & Winer, W.O. 1970 Inertial considerations in parallel circular squeeze film bearings . J. Lubric. Tech

    1970

  54. [62]

    & Woinowsky-Krieger, S

    Timoshenko, S. & Woinowsky-Krieger, S. 1959 Theory of plates and shells\/ , 2nd edn. McGraw-hill New York

    1959

  55. [63]

    & Persson, B.N.J

    Tiwari, A. & Persson, B.N.J. 2019 Physics of suction cups . Soft matter 15 (46), 9482--9499

    2019

  56. [64]

    & Bentwich, M

    Tuck, E.O. & Bentwich, M. 1983 Sliding sheets: lubrication with comparable viscous and inertia forces . J. Fluid Mech. 135 , 51--69

    1983

  57. [65]

    , Hashimoto, Y

    Ueha, S. , Hashimoto, Y. & Koike, Y. 2000 Non-contact transportation using near-field acoustic levitation . Ultrasonics 38 (1-8), 26--32

    2000

  58. [66]

    , Lambert, P

    Vandaele, V. , Lambert, P. & Delchambre, A. 2005 Non-contact handling in microassembly: Acoustical levitation . Precis. Eng. 29 (4), 491--505

    2005

  59. [67]

    2004 Compact models for squeezed-film dampers with inertial and rarefied gas effects

    Veijola, T. 2004 Compact models for squeezed-film dampers with inertial and rarefied gas effects . J. Micromech. Microeng. 14 (7), 1109

    2004

  60. [68]

    , Bendall, S

    Waltham, C. , Bendall, S. & Kotlicki, A. 2003 Bernoulli levitation . Am. J. Phys. 71 (2), 176--179

    2003

  61. [69]

    , Liu, J

    Wei, Z. , Liu, J. , Zheng, X. , Sun, Y. & Wei, R. 2021 Influence of squeeze film damping on quality factor in tapping mode atomic force microscope . J. Sound Vib. 491 , 115720

    2021

  62. [70]

    , Adibnazari, I

    Weston-Dawkes, W.P. , Adibnazari, I. , Hu, Y.-W. , Everman, M. , Gravish, N. & Tolley, M.T. 2021 Gas-lubricated vibration-based adhesion for robotics . Adv. Intell. Syst. 3 (7), 2100001

    2021

  63. [71]

    2018 Soft robotics

    Whitesides, G.M. 2018 Soft robotics . Angew. Chem. Int. Ed. 57 (16), 4258--4273

    2018

  64. [72]

    , Fenton Friesen, R

    Wiertlewski, M. , Fenton Friesen, R. & Colgate, J.E. 2016 Partial squeeze film levitation modulates fingertip friction . Proc. Natl. Acad. Sci. U.S.A. 113 (33), 9210--9215

    2016

  65. [73]

    & Goldstein, R.E

    Wiggins, C.H. & Goldstein, R.E. 1998 Flexive and propulsive dynamics of elastica at low R eynolds number . Phys. Rev. Lett. 80 (17), 3879

    1998

  66. [74]

    , Riveline, D

    Wiggins, C.H. , Riveline, D. , Ott, A. & Goldstein, R.E. 1998 Trapping and wiggling: elastohydrodynamics of driven microfilaments . Biophys. J. 74 (2), 1043--1060

    1998

  67. [75]

    Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known . J. Physiol. 127 (3), 553--563

    1955

  68. [76]

    , Lauga, E

    Yu, T.S. , Lauga, E. & Hosoi, A.E. 2006 Experimental investigations of elastic tail propulsion at low reynolds number . Phys. Fluids 18 (9)

    2006

  69. [77]

    , Bertin, V

    Zhang, Z. , Bertin, V. , Essink, M.H. , Zhang, H. , Fares, N. , Shen, Z. , Bickel, T. , Salez, T. & Maali, A. 2023 Unsteady drag force on an immersed sphere oscillating near a wall . J. Fluid Mech. 977 , A21

    2023

  70. [78]

    , Liu, R

    Zhu, T. , Liu, R. , Wang, X. D. & Wang, K. 2006 Principle and application of vibrating suction method. In 2006 IEEE International Conference on Robotics and Biomimetics\/ , pp. 491--495 . IEEE

    2006

  71. [79]

    & Cimbala, J

    Çengel, Y. & Cimbala, J. 2013 Fluid Mechanics\/ . McGraw Hill

    2013

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.