REVIEW 2 major objections 3 minor 90 references
Locomotion on a lubricating fluid with spatial viscosity variations
T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read A crawler on a lubricating film is always slowed when crossing a viscosity boundary, for both retrograde and direct travelling waves.
desk verdict Worth reviewing; the longitudinal-wave averaging formula (88) rests on an unstated amplitude-frequency scaling that the authors need to fix. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The carrying object is the general locomotion velocity formula (equation 23), which expresses the instantaneous speed $U_0$ as a ratio of integrals of the surface heights, surface velocities, and position-dependent viscosity; it generalizes the standard crawling formula to spatially varying viscosity and two moving boundaries. The second carrying element is the multiple-time-scale expansion: with $T=\omega t$ as the fast variable and $t$ as the slow variable, the interface position $x_*$ is expanded in powers of $1/\omega$, and the solvability condition $\langle\partial x_1/\partial T\rangle=0$ picks out the slow drift equation. The jump is encoded in the parameter $K(x_*)=m/[k(\lambda M/2+x_*m)]$ with $m=\mu_1-\mu_2$ and $M=\mu_1+\mu_2$, which controls the reduction factor $1-O(K^2\cos^2(kx_0/2))$ appearing in both equations (72) and (88).
What would settle it
Numerically integrate the full instantaneous velocity formula (23) for a purely longitudinal wave with a fixed amplitude $A_x$, crossing a sharp viscosity interface, for several frequencies $\omega$; if the measured average speed does not approach the uniform-viscosity value from below with a correction of the form $1-O(K^2\cos^2(kx_0/2))$, then equation (88) is falsified. A laboratory check would be to measure the average speed of a small mechanical crawler crossing a viscosity step and look for a speed increase rather than a decrease.
Extended reading notes
Core claim
The paper claims that crossing a sharp viscosity interface always reduces the average crawling speed, regardless of whether the surface wave is transverse or longitudinal. For a purely transverse wave the averaged velocity is $$\bar{U}$_0^{{(y)}}$=$U_0^{{(y)}}$\left[1-$4K^{2}$\$cos^{2}$(kx_0/2)\right]$$ and for a purely longitudinal wave it is $$\bar{U}$_0^{{(x)}}$=$U_0^{{(x)}}$\left[1-$2K^{2}$\$cos^{2}$(kx_0/2)\right],$$ both strictly smaller in magnitude than the uniform-viscosity speed. The reduction is largest when the front of the crawler penetrates the more viscous layer and the rear exits from it, and it grows as the viscosity contrast $|q|=|\mu_1-\mu_2|/(\mu_1+\mu_2)$ increases. For a linear viscosity gradient, the transverse-wave speed is reduced by the factor $1-\eta^2$, while the longitudinal-wave speed is unchanged at leading order. These conclusions come from the derived lubrication formula combined with multiple-time-scale averaging, in which the fast oscillation of the surface deformation feeds into the slow displacement of the body.
Load-bearing premise
The multi-scale derivation assumes that the longitudinal surface amplitude is tied to the wave frequency in a way the paper does not state, namely that $2\omega A_x$ is of order one as $\omega\to\infty$; if that ordering fails, the leading-order slow equation for the crawler's position does not follow and the correction term in equation (88) is not rigorously justified.
Editorial extensions
If this is right
- A retrograde crawler crossing any viscosity jump loses speed in the averaged sense, and the loss becomes more pronounced as the viscosity contrast between the two layers increases.
- A direct crawler also slows across a viscosity jump, despite having an instantaneous velocity that oscillates at first order in the amplitude, because the slow modulation of the oscillation produces a higher-order cumulative correction.
- On a linear viscosity gradient, the averaged speed of a transverse-wave crawler is reduced by a factor $1-\eta^2$ that does not depend on the sign of the gradient, so moving into either higher or lower viscosity costs speed.
- For a longitudinal-wave crawler on a gentle gradient, the leading-order averaged speed is exactly the uniform-viscosity value; corrections appear only at higher order.
- With uniform viscosity but a wavy substrate, the same velocity formula predicts that roughness increases the speed of a retrograde crawler and decreases the speed of a direct crawler.
Reading between the lines
- Inference: the same two-time-scale mechanism should produce a slow drift for any periodic gait, not just sinusoidal waves; a gait whose fast oscillations correlate with the local viscosity gradient will accumulate a net speed correction even if the phase-averaged instantaneous velocity vanishes.
- Inference: the appendix formula for y-dependent viscosity suggests a testable extension: a vertically stratified film may change the sign or size of the speed reduction, since the effective viscosity felt by the shear flow would vary through the gap.
- Inference: the roughness result implies one could engineer a substrate with periodic corrugations to selectively slow direct crawlers or speed up retrograde crawlers, offering a passive control mechanism for soft robots on thin films.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies two-dimensional crawling on a thin Newtonian film whose viscosity varies in the streamwise direction. It first derives a general lubrication-theory formula for the instantaneous crawling velocity, Eq. (23), and evaluates it for a sinusoidal surface composed of transverse and longitudinal travelling waves. For a sharp viscosity interface, a multiple-time-scale analysis produces closed-form averaged velocities, Eqs. (72) and (88), which both predict a speed reduction relative to uniform viscosity for retrograde and direct crawlers respectively. The paper then treats a linear viscosity gradient, finding a leading-order slowdown for a transverse wave but not for a longitudinal wave, and, in an appendix, applies the general formula to substrate topography, obtaining opposite corrections for the two wave modes.
Significance. If the central results hold, they give clean, falsifiable predictions: a viscosity interface reduces the average crawling speed for both retrograde and direct waves, with an explicit dependence on the viscosity contrast and on the crawler position. The general velocity formula (23) usefully extends the Chan et al. lubrication result to spatially varying viscosity and two moving boundaries, and the uniform-viscosity limits correctly reduce to known results, including the Taylor-sheet velocity for the longitudinal wave. The paper also provides numerical checks of the slow dynamics and, notably, the derivation contains no fitted parameters. The main caveat is a missing asymptotic ordering in the multiple-time-scale analysis of the longitudinal-wave case, which affects the formal justification of Eq. (88).
major comments (2)
- [III B 2, Eqs. (75)-(76)] The multiple-time-scale expansion (65)-(66) treats the right-hand side of Eq. (75) as O(1) in the 1/omega expansion, but this requires B = 2 omega A_x = O(1), i.e. A_x = O(omega^{-1}). With the stated assumption A_x = O(epsilon), the right-hand side is O(omega epsilon), and the leading-order balance partial x_0/partial T = 0 does not follow. The transverse-wave equation (64) has the same problem and requires A_y = O(omega^{-1/2}) for U_0^{(y)} to be O(1). These amplitude-frequency scalings are never stated, and the paper uses the same epsilon for both waves and for the estimate tau_dur/tau_osc = O(epsilon^{-2}), so the ordering is internally ambiguous. The derivation should state the scaling explicitly and verify consistency with the small-amplitude expansion.
- [III B 2, Eqs. (85)-(88)] Even if one grants A_x = O(omega^{-1}), Eq. (88) is not established as a uniformly valid average over a full interface crossing. In that scaling U_0^{(x)} = O(omega^{-1}), so the crossing time is O(omega), while Eq. (85) gives d x_1/dt = O(1); hence omega^{-1} x_1 = O(1) at the end of the crossing and the expansion x_* = x_0 + omega^{-1} x_1 + O(omega^{-2}) is not asymptotic on the interval used in the crossing. The authors should either introduce a further rescaled slow time tau = t/omega, or derive the averaged velocity by a method that does not require x_1 to remain small for the full crossing. The numerical check in Fig. 5(b), computed at omega = 2 pi with A_x = 0.01, does not resolve this asymptotic question.
minor comments (3)
- [III B 2, after Eq. (76)] The solvability condition is written as <partial x_1/partial t> = 0; it should be <partial x_1/partial T> = 0 at this order, or <partial x_2/partial T> = 0 at the next order.
- [Fig. 5(b) caption] The caption uses Delta x* for both the unaveraged deviation and the averaged deviation; the text defines Delta x_* in Eq. (89) and Delta x^* in Eq. (90), so distinct symbols would avoid confusion.
- [III B 2, Eq. (79)] The symbol x_1(t) is used both for the O(omega^{-1}) correction field and for the arbitrary function of t added after integrating in T; a separate notation, such as X_1(t), would improve clarity.
Circularity Check
No circularity: the speed-reduction results are derived from lubrication theory and multiple-time-scale averaging, not from fitted inputs or self-citations.
full rationale
The paper is self-contained. The general velocity formula (23) is obtained by integrating the lubrication equations (1)-(11) and imposing force balances (17)-(21); no parameter is fitted to the target speeds. The instantaneous crawler velocity across the viscosity jump, Eq. (54), follows by substituting the prescribed travelling-wave surface (38) and the step viscosity (44) into this formula. The averaged velocities (72) and (88) are then obtained by a systematic multiple-time-scale expansion of the slow-fast ODE (56)/(75), with the leading uniform-viscosity limits (59)-(60) checked against independent benchmarks (Chan et al. and the Taylor sheet). References to Walker et al. are methodological citations for the multiple-time-scale technique and do not carry the physical claim. The possible asymptotic-ordering weakness in Section IIIB2 (B=2ωAx must be treated as O(1) for the 1/ω balance, which is not stated) is a formal rigor concern about the expansion, not circularity: it does not identify the predicted speed with an input or with a fitted value. No fitted input is renamed as a prediction, and no load-bearing conclusion relies solely on self-citation.
Assumptions & free parameters
assumptions (7)
- domain assumption Lubrication approximation: vertical length scale Y is much smaller than horizontal scale X, so the Stokes equations reduce to equations (1) and (2).
- domain assumption The fluid is incompressible and Newtonian, with viscosity varying only with x (not y) in the main text.
- domain assumption Surface deformations are small (amplitudes of order epsilon, epsilon << 1), allowing asymptotic expansions.
- domain assumption Pressure at the two ends of the fluid region is equal (Delta p = 0), and the body is force-free.
- domain assumption The viscosity jump is modeled as a Heaviside step function, valid when the viscosity variation length scale is of the order of the film height.
- domain assumption The viscosity gradient is linear and the gradient strength is not necessarily small.
- domain assumption The crawler body length is one wavelength, and the interface position evolves as dx*/dt = -U0 in the crawler frame.
Cite this review
Pith. "Pith review of Locomotion on a lubricating fluid with spatial viscosity variations." pith.science (2026). https://pith.science/paper/B24QVTU5
@misc{pith2026241215656,
author = {Pith},
title = {Pith review of: Locomotion on a lubricating fluid with spatial viscosity variations},
year = {2026},
howpublished = {\url{https://pith.science/paper/B24QVTU5}},
note = {Machine review of arXiv:2412.15656}
}
read the original abstract
We studied locomotion of a crawler on a thin Newtonian fluid film whose viscosity varied spatially. We first derived a general locomotion velocity formula with fluid viscosity variations via the lubrication theory. For further analysis, the surface of the crawler was described by a combination of transverse and longitudinal travelling waves and we analysed the time-averaged locomotion behaviours under two scenarios: (i) a sharp viscosity interface and (ii) a linear viscosity gradient. Using the asymptotic expansions of small surface deformations and the method of multiple time-scale analysis, we derived an explicit form of the average velocity that captures nonlinear, accumulative interactions between the crawler and the spatially varying environment. (i) In the case of a viscosity interface, the time-averaged speed of the crawler is always slower than that in the uniform viscosity, for both the transverse and longitudinal wave cases. Notably, the speed reduction is most significant when the crawler's front enters a more viscous layer and the crawler's rear exits from the same layer. (ii) In the case of a viscosity gradient, the crawler's speed becomes slower for the transverse wave, while for the longitudinal wave, the corrections are of a higher order compared with the uniform viscosity case. As an application of the derived locomotion velocity formula, we also analysed the impacts of a substrate topography to the average speed. Our analysis illustrates the fundamental importance of interactions between a locomotor and its environment, and separating the time scale behind the locomotion.
Figures
Figures from the paper (5 more)
Reference graph
Works this paper leans on
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[1]
Remarks on the case of uniform viscosity Before proceeding to a detailed analysis of the nonlinear effects on the locomotion speed, here we discuss the case of uniform viscosity. Note that without a viscosity jump (i.e.,µ1 =µ2 =µ and thus m = 0 and M = 2µ, leading toq = 0), the nondimensional parameter K now vanishes as K(x; 0) = 0 [see equation (55)] and...
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[2]
Before moving to the next section, here we remark on the validity of the small amplitude expansion
Thus, when this condition is satisfied, the crawler locomotion becomes either direct or retrograde only by changing the relative phase of the waves with their amplitudes kept constant. Before moving to the next section, here we remark on the validity of the small amplitude expansion. For a pure transverse wave with Ay = 0.3, the relative error,ϵrel =|(Unu...
-
[3]
R. M. Alexander and G. Goldspink, Mechanics and energetics of animal locomotion (Chapman and Hall London, 1977)
1977
-
[4]
Case of transverse wave (retrograde crawler) We start with the transverse wave case with Ax = 0 and Ay , 0. The locomotion velocity [equation (54)] together with equation (56) gives dx∗ dt =−U(y) 0 ( 1− K(x∗) sin(kx∗) cos (kx∗− 2ωt)− 4K2(x∗) cos2 kx∗ 2 ! [1 + cos (kx∗− 2ωt)] ) , (62) where U(y) 0 =−3ωA2 y/(h2 0k)(< 0) is the velocity in the absence of the...
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[5]
−BK(x0) cos kx0 2 ! sin kx0− 2T′ 2 ! − * −BK(x0) cos kx0 2 ! sin kx0− 2T 2 !+ T=T′ # dT′, (80) or explicitly, E(t, T) =−BK(x0) cos kx0 2 !
Case of longitudinal wave (direct crawler) We now consider locomotion with a longitudinal surface wave with Ax , 0 and Ay = 0. As seen in equation (54), the leading-order oscillation term is of O(ϵ), being distinct from the transverse wave case, where the leading order is of O(ϵ2). The 13 0 1000 2000 3000 t 0.06 0.05 0.04 0.03 0.02 0.01 0.00 ∆x∗, ∆x0 (a) ...
2000
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[6]
Then, the instantaneous velocity, U0 [equa- tion (B4)], reduces to U0 =− 3ω h2 0k A2 y− 3ω h2 0k AyBy cosψ + O(ϵ4), (B8) where we setϕ = 0 without loss of generality
Case of transverse wave (retrograde crawler) We first consider the case of the transverse wave (i.e., Ax = 0 and Ay , 0). Then, the instantaneous velocity, U0 [equa- tion (B4)], reduces to U0 =− 3ω h2 0k A2 y− 3ω h2 0k AyBy cosψ + O(ϵ4), (B8) where we setϕ = 0 without loss of generality. We then write equation (B5) in the form dψ dt = a− b cosψ, (B9) wher...
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[7]
Case of longitudinal wave (direct crawler) Next, we use procedures similar to those in section B 1 to obtain an effective velocity under the conditions Ax , 0 and Ay = 0. The instantaneous velocity [equation (B4)] reads as U0 = kω 2 A2 x− ω h0 AxBy sinψ + O(ϵ4), (B14) and we write the time evolution ofψ [equation (B5)] in the form dψ dt = c− d sinψ, (B15)...
-
[8]
V ogel,Life in moving fluids: the physical biology of flow-revised and expanded second edition (Princeton university press, 1994)
S. V ogel,Life in moving fluids: the physical biology of flow-revised and expanded second edition (Princeton university press, 1994)
1994
Show all 90 references
-
[9]
S. J. Gray, Animal locomotion (Norton New York, 1968)
1968
-
[10]
M. W. Denny and J. M. Gosline, The physical properties of the pedal mucus of the terrestrial slug, Ariolimax columbianus, Journal of experimental Biology 88, 375 (1980)
1980
-
[11]
Brackenbury, Fast locomotion in caterpillars, Journal of insect physiology 45, 525 (1999)
J. Brackenbury, Fast locomotion in caterpillars, Journal of insect physiology 45, 525 (1999)
1999
-
[12]
K. J. Quillin, Kinematic scaling of locomotion by hydrostatic animals: ontogeny of peristaltic crawling by the earthworm lumbricus terrestris, Journal of Experimental Biology 202, 661 (1999)
1999
-
[13]
R. M. Alexander, Principles of animal locomotion (Princeton university press, 2003)
2003
-
[14]
Chapman, Of the movement of worms, Journal of Experimental Biology 27, 29 (1950)
G. Chapman, Of the movement of worms, Journal of Experimental Biology 27, 29 (1950)
1950
-
[15]
J. B. Keller and M. S. Falkovitz, Crawling of worms, Journal of theoretical biology 104, 417 (1983)
1983
-
[16]
Bolotnik, M
N. Bolotnik, M. Pivovarov, I. Zeidis, and K. Zimmermann, The undulatory motion of a chain of particles in a resistive medium, ZAMM- Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte Mathematik und Mechanik 91, 259 (2011)
2011
-
[17]
S. L. Miller, The classification, taxonomic distribution, and evolution of locomotor types among prosobranch gastropods, Journal of Molluscan Studies 41, 233 (1974)
1974
-
[18]
M. W. Denny, A quantitative model for the adhesive locomotion of the terrestrial slug, Ariolimax columbianus, Journal of experimental Biology 91, 195 (1981)
1981
-
[19]
DeSimone, F
A. DeSimone, F. Guarnieri, G. Noselli, and A. Tatone, Crawlers in viscous environments: linear vs non-linear rheology, International journal of non-linear mechanics 56, 142 (2013)
2013
-
[20]
Tanaka, K
Y . Tanaka, K. Ito, T. Nakagaki, and R. Kobayashi, Mechanics of peristaltic locomotion and role of anchoring, Journal of the Royal Society Interface 9, 222 (2012)
2012
-
[21]
Kuroda, I
S. Kuroda, I. Kunita, Y . Tanaka, A. Ishiguro, R. Kobayashi, and T. Nakagaki, Common mechanics of mode switching in locomotion of limbless and legged animals, Journal of the Royal Society interface 11, 20140205 (2014)
2014
-
[22]
B. Chan, N. Balmforth, and A. Hosoi, Building a better snail: Lubrication and adhesive locomotion, Physics of fluids 17 (2005)
2005
-
[23]
Z x2 x1 µF(x) (h2− h1)3 dx #,
addressed a similar problem and they concluded that longitudinal deformation has subleading order contribution in a usual lubrication setting. The models have been extended to include non-Newtonian properties of the lubricating fluids [24, 25], elasticity of the body of the cr...
2024 arXiv
-
[24]
Wilkening and A
J. Wilkening and A. Hosoi, Shape optimization of a sheet swimming over a thin liquid layer, Journal of Fluid Mechanics 601, 25 (2008)
2008
-
[25]
J. M. D. Olmsted, Notes on the locomotion of certain Bermudian mollusks, Journal of Experimental Zoology 24, 223 (1917)
1917
-
[26]
Jones and E
H. Jones and E. Trueman, Locomotion of the limpet, Patella vulgata L, Journal of Experimental Biology 52, 201 (1970)
1970
-
[27]
J. H. Lai, J. C. del Alamo, J. Rodr´ıguez-Rodr´ıguez, and J. C. Lasheras, The mechanics of the adhesive locomotion of terrestrial gastropods, Journal of Experimental Biology 213, 3920 (2010)
2010
-
[28]
Denny, The role of gastropod pedal mucus in locomotion, Nature 285 (1980)
M. Denny, The role of gastropod pedal mucus in locomotion, Nature 285 (1980)
1980
-
[29]
S. P. Holmes, A. Cherrill, and M. S. Davies, The surface characteristics of pedal mucus: a potential aid to the settlement of marine organisms?, Journal of the Marine Biological Association of the United Kingdom 82, 131 (2002)
2002
-
[30]
D. F. Katz, On the propulsion of micro-organisms near solid boundaries, Journal of Fluid Mechanics 64, 33 (1974)
1974
-
[31]
Lauga and A
E. Lauga and A. Hosoi, Tuning gastropod locomotion: Modeling the influence of mucus rheology on the cost of crawling, Physics of Fluids 18 (2006)
2006
-
[32]
S. S. Pegler and N. J. Balmforth, Locomotion over a viscoplastic film, Journal of Fluid Mechanics 727, 1 (2013)
2013
-
[33]
Argentina, J
M. Argentina, J. Skotheim, and L. Mahadevan, Settling and swimming of flexible fluid-lubricated foils, Physical review letters99, 224503 (2007)
2007
-
[34]
Balmforth, D
N. Balmforth, D. Coombs, and S. Pachmann, Microelastohydrodynamics of swimming organisms near solid boundaries in complex fluids, Quarterly journal of mechanics and applied mathematics 63, 267 (2010)
2010
-
[35]
P. W. Miller and J. Dunkel, Gait-optimized locomotion of wave-driven soft sheets, Soft matter 16, 3991 (2020)
2020
-
[36]
Iwamoto, D
M. Iwamoto, D. Ueyama, and R. Kobayashi, The advantage of mucus for adhesive locomotion in gastropods, Journal of theoretical biology 353, 133 (2014)
2014
-
[37]
S. Lee, J. W. Bush, A. Hosoi, and E. Lauga, Crawling beneath the free surface: Water snail locomotion, Physics of Fluids 20 (2008)
2008
-
[38]
Crowdy, S
D. Crowdy, S. Lee, O. Samson, E. Lauga, and A. Hosoi, A two-dimensional model of low-Reynolds number swimming beneath a free surface, Journal of Fluid Mechanics 681, 24 (2011)
2011
-
[39]
S. Joo, S. Jung, S. Lee, R. H. Cowie, and D. Takagi, Freshwater snail feeding: lubrication-based particle collection on the water surface, Journal of the Royal Society Interface 17, 20200139 (2020)
2020
-
[40]
Huang, H
Z. Huang, H. Yang, K. Xu, J. Wu, and J. Zhang, Collecting di fferently sized particles on water surface by maneuvering pedal waves on 25 the foot of the water snail Pomacea canaliculata, Soft Matter 18, 7850 (2022)
2022
-
[41]
K. M. Wheeler, G. C´arcamo-Oyarce, B. S. Turner, S. Dellos-Nolan, J. Y . Co, S. Lehoux, R. D. Cummings, D. J. Wozniak, and K. Ribbeck, Mucin glycans attenuate the virulence of Pseudomonas aeruginosa in infection, Nature microbiology 4, 2146 (2019)
2019
-
[42]
Liebchen, P
B. Liebchen, P. Monderkamp, B. Ten Hagen, and H. L ¨owen, Viscotaxis: Microswimmer navigation in viscosity gradients, Physical review letters 120, 208002 (2018)
2018
-
[43]
M. R. Stehnach, N. Waisbord, D. M. Walkama, and J. S. Guasto, Viscophobic turning dictates microalgae transport in viscosity gradients, Nature Physics 17, 926 (2021)
2021
-
[44]
Shoele and P
K. Shoele and P. S. Eastham, Effects of nonuniform viscosity on ciliary locomotion, Physical Review Fluids 3, 043101 (2018)
2018
-
[45]
Datt and G
C. Datt and G. J. Elfring, Active particles in viscosity gradients, Physical review letters 123, 158006 (2019)
2019
-
[46]
V . A. Shaik and G. J. Elfring, Hydrodynamics of active particles in viscosity gradients, Physical Review Fluids6, 103103 (2021)
2021
-
[47]
J. Gong, V . A. Shaik, and G. J. Elfring, Swimming efficiency in viscosity gradients, Journal of Fluid Mechanics 998, A2 (2024)
2024
-
[48]
P. S. Eastham and K. Shoele, Axisymmetric squirmers in Stokes fluid with nonuniform viscosity, Physical Review Fluids 5, 063102 (2020)
2020
-
[49]
J. Gong, V . A. Shaik, and G. J. Elfring, Active spheroids in viscosity gradients, Journal of Fluid Mechanics984, A26 (2024)
2024
-
[50]
G. I. Taylor, Analysis of the swimming of microscopic organisms, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 209, 447 (1951)
1951
-
[51]
Dandekar and A
R. Dandekar and A. M. Ardekani, Swimming sheet in a viscosity-stratified fluid, Journal of Fluid Mechanics 895, R2 (2020)
2020
-
[52]
Kamal and E
C. Kamal and E. Lauga, Resistive-force theory of slender bodies in viscosity gradients, Journal of Fluid Mechanics 963, A24 (2023)
2023
-
[53]
Esparza L ´opez and E
C. Esparza L ´opez and E. Lauga, Rate invariance and scallop theorem in viscosity gradients, Physical Review Fluids8, 063301 (2023)
2023
-
[54]
Z. Liu, Y . Chao, Z. Zheng, and L. Zhu, Spontaneous oscillation of an active filament under viscosity gradients, Soft Matter 20, 3845 (2024)
2024
-
[55]
Gonzalez-Gutierrez, S
J. Gonzalez-Gutierrez, S. Osorio-Ramirez, F. J. Solorio-Ordaz, and R. Zenit, Dynamics of a helical swimmer crossing an interface between two immiscible fluids, Physical Review Fluids 4, 083102 (2019)
2019
-
[56]
Esparza L ´opez, J
C. Esparza L ´opez, J. Gonzalez-Gutierrez, F. Solorio-Ordaz, E. Lauga, and R. Zenit, Dynamics of a helical swimmer crossing viscosity gradients, Physical Review Fluids 6, 083102 (2021)
2021
-
[57]
Coppola and V
S. Coppola and V . Kantsler, Green algae scatter off sharp viscosity gradients, Scientific reports 11, 399 (2021)
2021
-
[58]
C. Feng, J. J. Molina, M. S. Turner, and R. Yamamoto, Dynamics of microswimmers near a liquid–liquid interface with viscosity di ffer- ence, Physics of Fluids 35 (2023)
2023
-
[59]
J. Gong, V . A. Shaik, and G. J. Elfring, Active particles crossing sharp viscosity gradients, Scientific Reports13, 596 (2023)
2023
-
[60]
Ishimoto, H
K. Ishimoto, H. Gad ˆelha, E. A. Ga ffney, D. J. Smith, and J. Kirkman-Brown, Coarse-graining the fluid flow around a human sperm, Physical review letters 118, 124501 (2017)
2017
-
[61]
J. S. Guasto, K. A. Johnson, and J. P. Gollub, Oscillatory flows induced by microorganisms swimming in two dimensions, Physical review letters 105, 168102 (2010)
2010
-
[62]
B. M. Friedrich and F. J ¨ulicher, Chemotaxis of sperm cells, Proceedings of the National Academy of Sciences 104, 13256 (2007)
2007
-
[63]
Shiba, S
K. Shiba, S. A. Baba, T. Inoue, and M. Yoshida, Ca2+ bursts occur around a local minimal concentration of attractant and trigger sperm chemotactic response, Proceedings of the National Academy of Sciences 105, 19312 (2008)
2008
-
[64]
Drescher, R
K. Drescher, R. E. Goldstein, and I. Tuval, Fidelity of adaptive phototaxis, Proceedings of the National Academy of Sciences107, 11171 (2010)
2010
-
[65]
De Maleprade, F
H. De Maleprade, F. Moisy, T. Ishikawa, and R. E. Goldstein, Motility and phototaxis of Gonium, the simplest di fferentiated colonial alga, Physical Review E 101, 022416 (2020)
2020
-
[66]
B. J. Walker, K. Ishimoto, E. A. Gaffney, C. Moreau, and M. P. Dalwadi, Effects of rapid yawing on simple swimmer models and planar Jeffery’s orbits, Physical Review Fluids 7, 023101 (2022)
2022
-
[67]
E. A. Ga ffney, M. P. Dalwadi, C. Moreau, K. Ishimoto, and B. J. Walker, Canonical orbits for rapidly deforming planar microswimmers in shear flow, Physical Review Fluids7, L022101 (2022)
2022
-
[68]
M. P. Dalwadi, C. Moreau, E. A. Ga ffney, K. Ishimoto, and B. J. Walker, Generalised Jeffery’s equations for rapidly spinning particles. Part 1. Spheroids, Journal of Fluid Mechanics 979, A1 (2024)
2024
-
[69]
M. P. Dalwadi, C. Moreau, E. A. Ga ffney, B. J. Walker, and K. Ishimoto, Generalised Jeffery’s equations for rapidly spinning particles. Part 2. Helicoidal objects with chirality, Journal of Fluid Mechanics 979, A2 (2024)
2024
-
[70]
B. J. Walker, K. Ishimoto, C. Moreau, E. A. Ga ffney, and M. P. Dalwadi, Emergent rheotaxis of shape-changing swimmers in poiseuille flow, Journal of Fluid Mechanics 944, R2 (2022)
2022
-
[71]
B. J. Walker, K. Ishimoto, and E. A. Ga ffney, Systematic parameterizations of minimal models of microswimming, Physical Review Fluids 8, 034102 (2023)
2023
-
[72]
Reynolds, IV
O. Reynolds, IV . on the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil, Philosophical transactions of the Royal Society of London , 157 (1886)
-
[73]
Cameron, Principles of Lubrication (Longman Publishing Group, 1986)
A. Cameron, Principles of Lubrication (Longman Publishing Group, 1986)
1986
-
[74]
L. G. Leal, Advanced transport phenomena: fluid mechanics and convective transport processes , V ol. 7 (Cambridge university press, 2007)
2007
-
[75]
Blake, Infinite models for ciliary propulsion, Journal of Fluid Mechanics 49, 209 (1971)
J. Blake, Infinite models for ciliary propulsion, Journal of Fluid Mechanics 49, 209 (1971)
1971
-
[76]
DeSimone and A
A. DeSimone and A. Tatone, Crawling motility through the analysis of model locomotors: two case studies, The European Physical Journal E 35, 1 (2012)
2012
-
[77]
Lissmann, The mechanism of locomotion in gastropod molluscs: I
H. Lissmann, The mechanism of locomotion in gastropod molluscs: I. kinematics, Journal of Experimental Biology 21, 58 (1945)
1945
-
[78]
Lissmann, The mechanism of locomotion in gastropod molluscs: II
H. Lissmann, The mechanism of locomotion in gastropod molluscs: II. kinetics, Journal of Experimental Biology 22, 37 (1945)
1945
-
[79]
J. R. V ´elez-Cordero and E. Lauga, Waving transport and propulsion in a generalized Newtonian fluid, Journal of Non-Newtonian Fluid Mechanics 199, 37 (2013). 26
2013
-
[80]
D. J. Smith, E. Ga ffney, H. Gadˆelha, N. Kapur, and J. Kirkman-Brown, Bend propagation in the flagella of migrating human sperm, and its modulation by viscosity, Cell motility and the cytoskeleton 66, 220 (2009)
2009
-
[81]
Fang-Yen, M
C. Fang-Yen, M. Wyart, J. Xie, R. Kawai, T. Kodger, S. Chen, Q. Wen, and A. D. Samuel, Biomechanical analysis of gait adaptation in the nematode Caenorhabditis elegans, Proceedings of the National Academy of Sciences 107, 20323 (2010)
2010
-
[82]
Ishimoto, C
K. Ishimoto, C. Moreau, and J. Herault, Robust undulatory locomotion via neuromechanical adjustments in a dissipative medium, arXiv preprint arXiv:2405.01802 (2024)
2024 arXiv
-
[83]
Nosrati, A
R. Nosrati, A. Driouchi, C. M. Yip, and D. Sinton, Two-dimensional slither swimming of sperm within a micrometre of a surface, Nature communications 6, 1 (2015)
2015
-
[84]
Unnikrishnan, R
S. Unnikrishnan, R. L. Scott, E. Ogundele, M. A. Azad, K. Ishimoto, S. S. Suarez, and C. K. Tung, Hybrid motility mechanism of sperm at viscoelastic fluid-solid interface, Scientific Reports 14, 21841 (2024)
2024
-
[85]
O’Brien, The gliding motion of a bacterium: Flexibacter strain BH3, The ANZIAM Journal 23, 2 (1981)
R. O’Brien, The gliding motion of a bacterium: Flexibacter strain BH3, The ANZIAM Journal 23, 2 (1981)
1981
-
[86]
Siddiqui, R
A. Siddiqui, R. Burchard, and W. Schwarz, An undulating surface model for the motility of bacteria gliding on a layer of non-Newtonian slime, International journal of non-linear mechanics 36, 743 (2001)
2001
-
[87]
J. Du, J. P. Keener, R. D. Guy, and A. L. Fogelson, Low-Reynolds-number swimming in viscous two-phase fluids, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 85, 036304 (2012)
2012
-
[88]
Ishimoto, E
K. Ishimoto, E. A. Ga ffney, and D. J. Smith, Squirmer hydrodynamics near a periodic surface topography, Frontiers in cell and develop- mental biology 11, 1123446 (2023)
2023
-
[89]
Li, Slow motion of a sphere near a sinusoidal surface, Journal of Fluid Mechanics 975, A31 (2023)
G. Li, Slow motion of a sphere near a sinusoidal surface, Journal of Fluid Mechanics 975, A31 (2023)
2023
-
[90]
D. Nie, K. Zhang, and J. Lin, Enhanced speed of microswimmers adjacent to a rough surface, Physical Review E 110, 045101 (2024)
2024
Reviewed August 11, 2026 · model on record in the stance chip above.
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