Abstract
A body in motion tends to stay in motion but is often slowed by friction. Here we investigate the friction experienced by centimetersized bodies sliding on water. We show that their motion is dominated by skin friction due to the boundary layer that forms in the fluid beneath the body. We develop a simple model that considers the boundary layer as quasisteady, and is able to capture the experimental behaviour for a range of body sizes, masses, shapes and fluid viscosities. Furthermore, we demonstrate that friction can be reduced by modification of the body’s shape or bottom topography. Our results are significant for understanding natural and artificial bodies moving at the airwater interface, and can inform the design of aerialaquatic microrobots for environmental exploration and monitoring.
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Introduction
Friction is the force that opposes the relative motion of solid bodies and fluids and can present itself in both dry and fluid forms. For the case of fluid friction, a body moving with speed U with respect to the surrounding fluid generally may experience three primary types of drag: Stokes drag, form drag and skin friction. The relative importance of Stokes drag and form drag depends on the Reynolds number Re = UL/ν, where L is the body size and ν is the fluid kinematic viscosity, and which represents the ratio between the fluid inertia and viscous stresses. Stokes drag tends to dominate for small Re and is proportional to U, while form drag is significant for larger Re and is proportional to U^{2}. Another component of hydrodynamic resistance is the skin friction due to the viscous boundary layer that forms in the vicinity of the body surface. In the case of unseparated laminar flow on a flat plate (\(Re\lesssim {10}^{6}\)), skin friction is proportional to U^{3/2} ^{1}. Quantifying these types of friction is of critical importance for the design of boats, robots, and projectiles as well as for understanding the motion of living organisms in water and air. In this report, we demonstrate that the dominant friction for centimetric, hydrophobic bodies sliding on the surface of water is skin friction and provide a simple model which rationalizes our observations. Our findings are particularly relevant for the study of natural^{2} and artificial bodies selfpropelling at the waterair interface. Natural systems include water striders, which move on the water surface by generating vortices^{3} with their superhydrophobic legs^{4}, fisher spiders^{5}, meniscusclimbing insects^{6}, and insects that selfpropel by generating surfacetension gradients^{7}. Artificial systems include magnetic^{8} and magnetocapillary^{9} swimmers, droplets^{10,11} and boats^{12,13} propelled by surfacetension gradients, droplets propelled by surface waves^{14,15,16}, which can find application in micromanipulation and microtransport and also can constitute the building blocks for artificial activematter systems at the fluid interface. Moreover, our findings are relevant for the design of robots^{3,17,18,19,20,21,22} and microrobots^{23,24} at the waterair interface for environmental monitoring and exploration.
Results
In our experiments sliding bodies, henceforth “sliders” (Fig. 1(b)) were supported at the waterair interface by virtue of the equilibrium between their weight, surface tension, and hydrostatic forces resulting in a maximum interface deformation d. The maximum deformation of the free surface over all trials is estimated to be approximately 12% of the slider’s diameter. A magnet was embedded in the sliders, which were accelerated along a straight line on the surface of a water bath through an inhomogeneous magnetic field generated by a coil (Fig. 1(a)). The coil was then turned off and the slider decelerated due to fluid friction (Fig. 1(c)).
The deceleration phase was recorded and the slider’s speed as a function of time was measured. Our setup allowed for continuous and repeatable runs of the experiment, which we show in the Supplementary Movie and describe in the Supplementary Materials. Extended details of the experimental setup can be found in the Methods section. The slider speed at the beginning of the deceleration phase was always in the range U_{0} = 6.1–16.0 cm/s with uncertainty 0.1 cm/s, below the phase speed of capillary waves \({U}_{c}\simeq 23\) cm/s, at which the slider with constant speed would generate a capillary wake^{25}. Visible capillary waves were generated during the initial acceleration period^{26,27}, however no additional waves were observed during the deceleration phase over which our measurements take place.
In the deceleration phase, the slider obeys the simple ordinary differential equation (o.d.e.) m dU/dt = −F_{D}, where U is the slider speed and F_{D} is a drag force. We proceeded by testing the solutions U(t) of the o.d.e. as obtained when three common fluid drag scalings are considered: Stokes drag F_{D} ∝ U, form drag F_{D} ∝ U^{2} and skin friction F_{D} ∝ U^{3/2}. The results are presented in Fig. 2 and show evidently that the skin frictiontype scaling completely captures the observed deceleration behavior. This is consistent with the orderofmagnitude estimate presented in the discussion section.
Skin friction is due to the viscous stress developed in the liquid boundary layer underneath the slider (Fig. 1(d)). We now show that in our experiments the boundary layer can be considered as quasisteady. In our experiments, 40 < Re_{R} < 3 · 10^{3}, where Re_{R} = U_{0}R/ν is the Reynolds number, which suggests laminar flow. The typical timescale t_{BL} over which the boundary layer develops can be derived by balancing fluid inertia and viscous stress
where μ = ρν is the liquid dynamic viscosity. A scaling analysis of Eq. 1 gives ρU_{0}/t_{BL} ~ μU_{0}/δ^{2}, where \(\delta \sim R/\sqrt{R{e}_{R}}\) is the boundary layer thickness from laminar boundary layer theory^{1}. This yields t_{BL} ~ R/U_{0}^{28}, which can be interpreted as an advection timescale. In the acceleration phase, the slider accelerates over a time t_{acc} ~ 1 s, larger than t_{BL} ~ 0.1 s. We thus conclude that the boundary layer is already fully developed at the start of the deceleration phase. The characteristic timescale of slider deceleration can be derived from the balance between the slider’s inertia and viscous stress
where A is the contact area between the slider and the water surface and \({\tau }_{w}\sim \mu \frac{\partial U}{\partial y}\) is the shear stress at the fluid interface. A scaling analysis of Eq. 2 gives mU_{0}/t_{d} ~ ρνU_{0}R^{2}/δ, where we considered the initial slider’s speed U_{0} as typical speed for both slider and fluid, A ~ R^{2} for a circular slider and t_{d} is a characteristic timescale. This balance ultimately allows us to define a characteristic timescale of deceleration
In all of our experiments \({t}_{d}\gg {t}_{BL}\), which allows us to treat the boundary layer as quasisteady. In other words, in the deceleration phase the boundary layer continuously adjusted to the slider speed over a timescale that is much smaller than the slider deceleration timescale. We validated this hypothesis by varying a number of system parameters, including slider radius R = 4–12 mm, mass m = 54–1337 mg and liquid viscosity ν = 0.01–0.04 cm^{2}/s. Collapse of the data using the characteristic time t_{d} and speed U_{0} is presented in Fig. 3 and it is effective for all slider radii, masses and fluid viscosities considered. The o.d.e. describing the system is thus
where β is the friction coefficient. We calculate β by using the classical Blasius solution derived for a steady boundary layer in a laminar flow beneath a semiinfinite and onedimensional flat plate^{1}. Under these assumptions, the drag force per unit width on one side of a plate of length L is \({f}_{D}=0.664\,\rho {U}^{2}L/\sqrt{R{e}_{L}}\). We integrate this force over the slider surface in contact with the bath and obtain \(\beta =\alpha \rho \,\sqrt{\nu }{R}^{\mathrm{3/2}}\), thus the friction force on the slider can be written as
with α = 1.64. In our experiments, the friction force is in the range F_{D} = 0.025–8.2 dynes (0.25–82 μN). The solution to Eq. 4 can thus be obtained by integration and written as
This solution is plotted on Fig. 3. The agreement with experimental data is satisfactory despite the assumptions in the theory and agrees up to a factor of order 1 on the friction coefficient β. In fact, by fitting each data set with Eq. 6 and with α as a free parameter, we obtain α = 1.3–2.1, close to the theoretical value α = 1.64. Values of α for specific circular sliders are reported in the Supplementary Table.
We then explored the possibility of modifying the friction properties by changing the slider shape and its orientation with respect to its direction of motion. Elliptical sliders were slid with their major axis parallel or perpendicular to the direction of motion. The slider contact surface was chosen to equal the contact surface of a circular slider of radius R = 4.00 ± 0.05 mm. Comparison between elliptical, circular sliders and theoretical prediction is presented on Fig. 4(a). For elliptical sliders, the deceleration time can be computed again from the balance between the slider’s inertia and viscous stress (now with A ~ ab) which yields \({t}_{d}=m/\rho \sqrt{\nu {U}_{0}a}b\), where a is the semiaxis parallel to the sliding direction and b is the semiaxis perpendicular to the sliding direction. Alternatively, one can obtain the theoretical force expression by integrating the drag force per unit width over the elliptical domain. The collapse of the experimental data is excellent and the agreement with the theoretical prediction is satisfactory (Fig. 4(a)). Fitting of the parameter yields α = 1.7–1.8 in all experimental configurations (ellipses and circular), again close to the expected theoretical value α = 1.64. We note that the timescale of deceleration t_{d} of the elliptical slider sliding with its major axis parallel to the direction of motion is larger, showing that the friction depends explicitly on the shape and the orientation of the slider. The elliptical slider with major axis along the direction of motion shows the longest deceleration time, and thus a decreased friction.
We then investigated the possibility of modifying skin friction by changing the topography of the slider bottom surface in contact with the fluid interface. Recent experiments on centimetric boats with superhydrophobic, grilled bottoms, have shown that the drag on these boats can be reduced with respect to boats with flat bottoms^{29}. We further explored this concept by performing experiments on circular sliders with grooved bottoms, in which the ridges were either parallel or perpendicular to the sliding direction (Supplementary Figure). The ridges lifted the slider above the interface and thus air could flow within the macroscopic grooves. Results are presented in Fig. 4(b). We found that ridges parallel to the direction of motion significantly reduce the skin friction, with the friction being lower as the spacing between ridges is increased. Surprisingly, sliders with ridges perpendicular to the direction of motion behave very similarly to sliders with flat bottom. This clearly demonstrates that the contact area is not the only parameter that matters, but the orientation of the body also has a dramatic effect on the friction of the slider.
Discussion
Our experiments demonstrate that centimetric bodies sliding at an airwater interface are decelerated by skin friction, which is dominant with respect to other forms of hydrodynamic drag. We can also understand this result by estimating the relative importance of Stokes drag and form drag with respect to skin friction. The Reynolds number in the liquid was always 40 < Re_{R} < 3 · 10^{3}, \(R{e}_{R}\gg 1\). Moreover, the depth of the bath h = 3.7 cm was chosen much larger than the maximum expected thickness \({\delta }_{max}\simeq 5\sqrt{2{\nu }_{m}R/{U}_{min}}\simeq 1.6\) cm of the boundary layer that would develop underneath the slider^{1}, where ν_{m} = 0.04 cm^{2}/s is the maximum kinematic viscosity explored and U_{min} = 1.0 cm/s is the minimum slider speed considered. We thus operated in the deep water regime avoiding the possibility of a Stokes (lubrication) regime in shallow water. The friction force is thus contributed by air drag F_{air} and skin friction F_{skin}. Skin friction in the regime of laminar flow scales as \({F}_{skin}\sim \rho \sqrt{\nu }\,{R}^{\mathrm{3/2}}{U}^{\mathrm{3/2}}\) ^{1}. Air drag scales as \({F}_{air}\sim {\rho }_{a}{u}^{2}R{h}_{s}\), where ρ_{a} = 1.2 · 10^{−3} g/cm^{3} is the density of air and h_{s} ~ 0.1 cm is the typical slider thickness. In our experiments, F_{air}/F_{skin} ~ 10^{−3}, thus air drag is negligible with respect to skin friction.
Finally, we discuss the condition under which we expect our quasisteady theory to remain applicable. In our experiments, the typical deceleration time \({t}_{d}=m/\rho \sqrt{U\nu {R}^{3}}\) is much larger than the time required for the boundary layer to set up t_{BL} = R/U, where U is the instantaneous body speed. From this, we define a “sliding number” as the ratio t_{BL}/t_{d}, which yields
We remark that the sliding number may also be interpreted as the ratio between the fluid inertia I_{BL} ~ ρR^{2}δU and the body inertia I_{b} ~ mU. A small sliding number is the condition for the freesliding behaviour to be adequately described by our quasisteady boundary layer model. In all of our experiments, S_{l} < 0.3 and the experimental behaviour is well captured by a simple model that considers the laminar boundary layer underneath the slider as quasisteady. The model appears to apply to different slider shapes, and we have shown that shapes elongated in the direction of motion yield lower overall friction. The free surface deformation due to the slider’s weight appears not to play a significant role in the process.
Skin friction can be further controlled by modifying the topography of the slider bottom, specifically if the bottom has grooved recessions parallel to the direction of motion. As the contact surface in topographic sliders is lower than in the full contact sliders, one might expect friction to be reduced for all topographic sliders. However, we find that friction reduction depends on the ridge orientation. This presumably results from the shear stress in a developing boundary layer being largest at the leading edge of any solid surface. Therefore, sliders with ridges perpendicular to the direction of motion have a larger projected ridge width. Furthermore, the boundary layer must restart in some manner at each successive ridge, again leading to additional drag as compared to the sliders with grooves parallel to the direction of motion.
Our work opens the door to further experimental and theoretical investigation of the friction experienced by small bodies moving on the surface of water as a function of their shape and bottom topography, with motivations of either drag reduction or propulsive efficiency through enhanced traction. Moreover, the experimental setup we have developed can be adapted for studying the drag on partially submerged centimetric bodies or to investigate the effect of an unsteady boundary layer^{28,30}, which is expected to become significant at sliding number \(Sl\gtrsim 1\).
Methods
Experimental setup
Sliders were manufactured using a Stereolithography (SLA) 3D printer (Formlabs Form 2), with cured resin density ρ_{s} = 1.15 g/cm^{3}. We produced circular sliders with radii in the range 4–12 mm with uncertainty 0.05 mm, elliptical sliders with semiaxes 5.00 ± 0.05 mm and 3.20 ± 0.05 mm, and sliders with radius R = 4.00 ± 0.05 mm with grooved bottoms. Masses ranged from m = 54 ± 2 mg to 1337 ± 2 mg. The slider mass was increased by adding brass shims and the overall slider thickness never exceeded 1.5 mm. Sliders were coated with a twopart commercial superhydrophobic spray coating (Ultraever Dry)^{31} that allowed the slider to rest stably at the interface and carry large loads without sinking^{32,33,34}. Each slider was embedded with a small permanent magnet of magnetic dipole moment \(\overrightarrow{p}\) (Fig. 1(a,d)), which allowed us to guide the slider on a liquid surface by means of an external inhomogeneous magnetic field \(\overrightarrow{B}\) through the force \({\overrightarrow{F}}_{B}=(\overrightarrow{p}\cdot \nabla )\overrightarrow{B}\). The magnetic dipole intensity was chosen in order to maximize the slider speed achievable in our experimental setup, while minimizing the influence of ambient magnetic field gradients. We used magnets with magnetic dipole intensity p = 0.074 A m^{2} for sliders of radius R = 10 mm and 12 mm and magnets with magnetic dipole intensity p = 0.019 A m^{2} for all other sliders. Sliders slid on the surface of a water bath in a rectangular glass tank with size 10.2 × 10.2 × 30.5 cm (Fig. 1(a)). The tank was surrounded by a 500turn coil with inner radius 10.1 cm and outer radius 11.4 cm (Pasco, EM6723), and its position adjusted such that the coil axis was aligned at the center of the airwater interface. The center of the coil was at a distance 11.5 cm from one of the tank’s short side wall. A small electromagnet was placed behind the same short side walls on the coil axis, and it served to retrieve the slider and fix its initial position. Before each measurement session, the glass tank was cleaned with ethanol and doublerinsed with deionized water. The glass tank was then filled with deionized water or deionized waterglycerol mixture up to a depth of h = 3.7 ± 0.2 cm. The uncertainty on the fluid viscosity was 0.0002 cm^{2}/s. The slider was then deposited on the bath’s surface and the tank covered with a transparent lid to avoid ambient contamination. A small opening was left in order to avoid accumulation of vapor on the lid, which would impede the tracking of the slider. The slider deceleration phase was recorded from the top with a CCD camera (Allied Vision Mako U130B) at 62.5 fps with resolution 1280 × 1024. Electromagnet, coil and camera were controlled by an Arduino and a custom relaybased circuit. In order to avoid magnet reorientation after switching off the coil, the whole setup was oriented along the Earth’s magnetic field. At least 39 trajectories were recorded per measurement session, in which all the parameters were kept constant. This allowed us to test the repeatability of our setup and measurement and define the variation between runs. The slider center of mass was tracked with a custom objecttracking algorithm implemented in MATLAB. The minimum slider speed considered was U_{min} = 1 cm/s, one order of magnitude larger than the maximum drift speed we observed as due to the influence of ambient magnetic field gradients, \({U}_{d}\lesssim 0.1\) cm/s.
Uncertainty on the parameter α
Each data set yielded the mean of at least 39 runs, which was fitted with the function \(U(t)=\mathrm{1/}{(\mathrm{1/}\sqrt{{U}_{0}}+ct)}^{2}\), where c is the fitting parameter. The parameter α was then computed as \(\alpha =2mc/\rho \sqrt{\nu }\,{R}^{\mathrm{3/2}}\) for circular sliders, and using an analogous formula for elliptical sliders. The uncertainty on α was calculated as the square root of the variance formula for independent variables
where σ_{c} is the value that yields 95% confidence bounds from the fitted value of c, σ_{m} = 2 mg, σ_{ρ} = 0.002 g/cm^{3}, σ_{ν} = 0.0002 cm^{2}/s and σ_{R} = 0.005 cm.
References
Kundu, P. K., Cohen, I. M. & Dowling, D. R. Fluid Mechanics (Elsevier, 2015).
Bush, J. W. & Hu, D. L. Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339–369 (2006).
Hu, D. L., Chan, B. & Bush, J. W. The hydrodynamics of water strider locomotion. Nat. 424, 663 (2003).
Gao, X. & Jiang, L. Biophysics: waterrepellent legs of water striders. Nat. 432, 36 (2004).
Suter, R., Rosenberg, O., Loeb, S., Wildman, H. & Long, J. Locomotion on the water surface: propulsive mechanisms of the fisher spider. J. Exp. Biol. 200, 2523–2538 (1997).
Hu, D. L. & Bush, J. W. Meniscusclimbing insects. Nat. 437, 733 (2005).
Andersen, N. M. The Semiaquatic Bugs (Hemiptera, Gerromorpha) (Scandinavian Science Press, 1982).
Snezhko, A., Belkin, M., Aranson, I. & Kwok, W.K. Selfassembled magnetic surface swimmers. Phys. Rev. Lett. 102, 118103 (2009).
Grosjean, G. et al. Remote control of selfassembled microswimmers. Sci. Rep. 5, 16035 (2015).
Izri, Z., Van Der Linden, M. N., Michelin, S. & Dauchot, O. Selfpropulsion of pure water droplets by spontaneous marangonistressdriven motion. Phys. Rev. Lett. 113, 248302 (2014).
Bormashenko, E. et al. Selfpropulsion of liquid marbles: Leidenfrostlike levitation driven by marangoni flow. J. Phys. Chem. C 119, 9910–9915 (2015).
Nakata, S., Doi, Y. & Kitahata, H. Synchronized sailing of two camphor boats in polygonal chambers. J. Phys. Chem. B 109, 1798–1802 (2005).
Karasawa, Y., Oshima, S., Nomoto, T., Toyota, T. & Fujinami, M. Simultaneous measurement of surface tension and its gradient around moving camphor boat on water surface. Chem. Lett. 43, 1002–1004 (2014).
Pucci, G., Fort, E., Ben Amar, M. & Couder, Y. Mutual adaptation of a Faraday instability pattern with its flexible boundaries in floating fluid drops. Phys. Rev. Lett. 106, 024503 (2011).
Pucci, G. Faraday instability in floating drops out of equilibrium: Motion and selfpropulsion from wave radiation stress. Int. J. Nonlinear Mech. 75, 107–114 (2015).
Ebata, H. & Sano, M. Swimming droplets driven by a surface wave. Sci. Rep. 5, 8546 (2015).
Song, Y. S. & Sitti, M. Surfacetensiondriven biologically inspired water strider robots: Theory and experiments. IEEE Trans. Robot. 23, 578–589 (2007).
Wu, L., Lian, Z., Yang, G. & Ceccarelli, M. Water dancer IIa: a nontethered telecontrollable water strider robot. Int. J. Adv. Robot. Syst. 8, 39 (2011).
Zhang, X. et al. Bioinspired aquatic microrobot capable of walking on water surface like a water strider. ACS Appl. Mater. Interfaces 3, 2630–2636 (2011).
Yuan, J. & Cho, S. K. Bioinspired micro/mini propulsion at airwater interface: A review. J. Mech. Sci. Technol. 26, 3761–3768 (2012).
Yan, J. et al. A miniature surface tensiondriven robot using spatially elliptical moving legs to mimic a water strider’s locomotion. Bioinspiration Biomim. 10, 046016 (2015).
Koh, J.S. et al. Jumping on water: Surface tension–dominated jumping of water striders and robotic insects. Sci. 349, 517–521 (2015).
Chen, Y. et al. A biologically inspired, flappingwing, hybrid aerialaquatic microrobot. Sci. Robotics 2, eaao5619 (2017).
Chen, Y., Doshi, N., Goldberg, B., Wang, H. & Wood, R. J. Controllable water surface to underwater transition through electrowetting in a hybrid terrestrialaquatic microrobot. Nat. Commun. 9, 2495 (2018).
Le Merrer, M., Clanet, C., Quéré, D., Raphaël, É. & Chevy, F. Wave drag on floating bodies. Proc. Nat. Acad. Sci. 108, 15064–15068 (2011).
Bühler, O. Impulsive fluid forcing and water strider locomotion. J. Fluid Mech. 573, 211–236 (2007).
Closa, F., Chepelianskii, A. & Raphael, E. Capillarygravity waves generated by a sudden object motion. Phys. Fluids 22, 052107 (2010).
Hall, M. G. The boundary layer over an impulsively started flat plate. Proc. R. Soc. Lond. A 310, 401–414 (1969).
Jiang, C., Xin, S. & Wu, C. Drag reduction of a miniature boat with superhydrophobic grille bottom. AIP Adv. 1, 032148 (2011).
Rayleigh, L. LXXXII. On the motion of solid bodies through viscous liquid. Phil. Mag. 21, 697–711 (1911).
Wang, L. et al. A study of the mechanical and chemical durability of ultraever dry superhydrophobic coating on low carbon steel surface. Colloids Surf. A 497, 16–27 (2016).
Jin, H. et al. Superhydrophobic and superoleophobic nanocellulose aerogel membranes as bioinspired cargo carriers on water and oil. Langmuir 27, 1930–1934 (2011).
Yong, J. et al. A bioinspired planar superhydrophobic microboat. J. Micromechanics Microengineering 24, 035006 (2014).
Suyambulingam, G. R. T. et al. Excellent floating and load bearing properties of superhydrophobic ZnO/copper stearate nanocoating. Chem. Eng. J. 320, 468–477 (2017).
Acknowledgements
The authors would like to thank the Brown OVPR Seed Award for partial support of this work.
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All the authors contributed to the design of the experimental setup, which was built by G.P. and I.H. G.P. performed the experiments. G.P. and D.M.H. analysed the data and wrote the manuscript.
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Pucci, G., Ho, I. & Harris, D.M. Friction on water sliders. Sci Rep 9, 4095 (2019). https://doi.org/10.1038/s4159801940797y
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DOI: https://doi.org/10.1038/s4159801940797y
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