Abstract
Incited by public fascination and engineering application, waterskipping of rigid stones and spheres has received considerable study. While these objects can be coaxed to ricochet, elastic spheres demonstrate superior waterskipping ability, but little is known about the effect of large material compliance on water impact physics. Here we show that upon water impact, very compliant spheres naturally assume a disklike geometry and dynamic orientation that are favourable for waterskipping. Experiments and numerical modelling reveal that the initial spherical shape evolves as elastic waves propagate through the material. We find that the skipping dynamics are governed by the wave propagation speed and by the ratio of material shear modulus to hydrodynamic pressure. With these insights, we explain why softer spheres skip more easily than stiffer ones. Our results advance understanding of fluidelastic body interaction during water impact, which could benefit inflatable craft modelling and, more playfully, design of elastic aquatic toys.
Introduction
Waterskipping has been studied for centuries with diverse motivations including: the ancient art of stone skipping^{1–4}; naval application^{5–10}; watersurface craft^{11,12}; and biological^{13,14} and biomimetic^{15} waterwalking. While water ricochet of rigid objects has been well studied, the physics underlying the water impact of highly deformable elastic solids remains poorly understood^{12,16,17}. Compliant bodies such as inflatable boats^{12} and elastic aquatic toys^{18,19} exhibit behaviour that is not readily explained within the traditional framework for rigid objects. For such elastic bodies, an understanding of the coupling between the material response and hydrodynamic loading is essential in unravelling the overall dynamics.
An object obliquely impacting a water surface with sufficient inertia will carve a cavity on the air–water interface^{20} and experience a pressuredriven hydrodynamic force dependent on object velocity, geometry and orientation^{4,8,21,22,23}. Waterskipping occurs when the upward vertical component of this force is large enough to lift the object off the water surface^{8}. Studies born from naval applications ranging from cannonball skipping tactics^{5,8} to the dambusting Wallis bomb^{10} have revealed an upper bound on the impact angle β_{o} (angle between the freesurface and object velocity vector) below which rigid spheres will skip on water^{7,8,10,21}. Diskshaped stones are more amenable to skipping, particularly if one orients the stone at just the right angle^{2}. Further research has revealed more details regarding the oblique water impact of these and other canonical rigid body geometries^{23,24,25,26}. The referenced ricochet events are dominated by inertia, with negligible contributions from viscous and surface tension forces^{4,20}. In this regime, the physics of waterskipping also generalize to the waterwalking ability of basilisk lizards^{13,14} and some birds^{14,27}, and to surface craft slamming^{11}.
In this work, we investigate the skipping of deformable elastic solid spheres on water. We observe that elastic spheres can skip for impact angles nearly three times larger than predicted for rigid spheres. Experiments and numerical modelling show that the spheres deform throughout impact in response to elastic waves propagating in the material. In some cases these elastic waves actually interact with the air–water interface to create nested cavities. We determine how the deformed geometry scales with material properties and initial impact kinematics. Using an analytical model to relate deformation to the hydrodynamic lift force, we identify the mechanisms by which elastic spheres skip so readily on water. Furthermore, we compute the normal and tangential restitution coefficients and find, surprisingly, that they display analogous behaviour to liquid droplets bouncing on inclined liquid films^{28}. Based on our findings about single impact events, we explain how elastic spheres are able to achieve multiple successive skips on water.
Results
Elastic sphere skipping phenomena
Prior research has reported an upper bound on the impact angle of below which rigid spheres (density ρ_{s}) will skip on water (density ρ_{w})^{7,8,10,21}. We have found that elastic spheres skip at much larger values of β_{o}, raising the question of how the elastic response enables this enhanced skipping behaviour. To investigate the mechanisms underlying elastic sphere skipping, we film the water impact of custommade elastomeric spheres with a highspeed camera viewing from the side. Rigid and elastic spheres having nearly the same radius R and density ρ_{s} are shown experimentally impacting the water in Fig. 1a,b. Each sphere strikes with approximately the same speed U_{o} and impact angle β_{o}, but the elastic sphere has a shear modulus G that is four orders of magnitude smaller than that of the rigid sphere material. Within a few milliseconds after impact, the elastic sphere deforms dramatically and rides along the front of a cavity on the air–water interface before lifting off the surface. By the time the elastic sphere is two diameters above the surface (t≈25 ms), the rigid sphere has plunged nearly the same distance below it. The elastic sphere evidently experiences a larger upward vertical force from the water.
The extreme sphere deformation is more evident in Fig. 1c, which shows that the watercontacting surface assumes the shape of a disk with a larger radius than that of the undeformed sphere. The disk is oriented at an attack angle α that, unlike for skipping stones^{4}, changes in time throughout the impact. Large amplitude oscillations excited by the impact can persist in the sphere after lifting off the surface (Fig. 1b and Supplementary Movie 1) or even while the sphere is still in contact with the water (Fig. 1d and Supplementary Movie 2). In the latter case, the sphere vibrations form a group of nested cavities, or a socalled matryoshka cavity^{29}, named after Russian nesting dolls (Fig. 1d). This phenomenon is in contrast to rigid sphere skipping, for which the cavity is asymmetric, but smooth (Fig. 1e).
Sphere deformation modes
To examine the sphere deformations more thoroughly, we implement a fully coupled numerical finiteelement model in Abaqus (ref. 30) (see Methods section). Figure 2a shows the results of a numerical simulation carried out with the same sphere material properties (R, G and ρ_{s}) and impact conditions (U_{o} and β_{o}) as the experimental test shown in Fig. 2b. The numerics reveal an elastic wave propagating around the circumference of the sphere in a counterclockwise direction. We classify this type of wave propagation, depicted in the line drawing of Fig. 2c, as vibration mode 1^{−}. In some cases this elastic wave impacts the air–water interface at time t_{w}, thus initiating a matryoshka cavity (as seen numerically and experimentally in Fig. 2a,b). While tempting to attribute these kinematics to rigid body rotation, we find that the elastic wave propagation time t_{w} is typically much smaller than the measured period of rigid rotation (see Methods and Supplementary Movie 2).
The Abaqus numerical model predicts two additional vibration modes, generally occurring with increasing impact speed and/or decreasing shear modulus. In mode 2 (Fig. 2d, Supplementary Movie 1), the sphere assumes an ellipsoidal shape with oscillating major and minor axes. In mode 1^{+} (Fig. 2e) an elastic wave again propagates around the circumference of the sphere, but in the clockwise direction. Finally, we observe that the attack angle α of the deformed watercontacting face evolves as a result of the elastic wave propagation (Fig. 2a,f and Supplementary Movie 2).
Skipenhancing mechanisms
Based on our experimental observations and numerical simulations (Figs 1 and 2), we hypothesize that the elastic response of the sphere enhances skipping through two mechanisms: (1) by taking on the shape of a flat disk with an increased wetted area; and (2) by acquiring a favourable attack angle.
To connect the suggested skipenhancing mechanisms to the vertical force acting on the sphere, we propose an analytical model of the coupled fluidstructure interaction. The sphere is idealized as an incompressible, neoHookean hyperelastic solid^{31} with shear modulus G, radius R and density ρ_{s}. During water impact, the sphere deforms into a diskshaped ellipsoid inclined at attack angle α(t) to the water surface (Fig. 3a). A set of equations can then be written for the motion of the centre of mass (COM) and sphere deformation in terms of general forces and tractions acting on the body (see Methods section).
To couple the sphere response to the fluid loading, we extend an existing hydrodynamic force model for circular diskshaped skipping stones^{4} by approximating the watercontacting face as a circular disk with radius λ_{eq}R, where λ_{eq} approximates the sphere deformation (see Fig. 3a and Methods section). The force is modelled as
where U_{B} and β_{B} describe the velocity of the watercontacting face and the wetted area S_{w} is proportional to (λ_{eq}R)^{2}. The equation for the vertical motion of the COM is then
where d_{2} is the vertical coordinate of the COM and g is gravity. We can now predict how the hypothesized mechanisms relate to skipping. First, a larger area S_{w} increases the magnitude of the hydrodynamic force term in equation 2. For a neoHookean material, S_{w} increases with decreasing shear modulus G for a given applied compressive stress^{31}. Second, a smaller attack angle increases cos α thereby increasing the vertical force component that lifts the sphere off the surface. We note that the governing equations for the sphere deformation predict a steadystate solution in which the attack angle evolves as in response to a circumferential wave propagating in mode 1^{−} (see Supplementary Note 1). While the sphere deformation during impact is not steadystate, we nonetheless expect α to be governed by the speed of elastic waves propagating in the sphere through the characteristic distance R such that . Therefore, we predict that a more compliant sphere (smaller G) will assume a smaller rate of change of α, thereby increasing the upward vertical force that enables skipping.
Numerical simulations verify the expected dependence of these two mechanisms on G. Measurements from the Abaqus results show that the rate of change of the attack angle scales as for mode 1^{−} deformations (Fig. 3b). Additionally, we find that the maximum value of λ_{eq} achieved during impact, which we call λ_{max}, increases with a decrease in the dimensionless term (Fig. 3c), which is the ratio of material stiffness to hydrodynamic pressure. Therefore, a smaller G yields a larger stretch and larger wetted area, as well as a smaller rate of change of α, as predicted.
To confirm that these mechanisms indeed enhance skipping, we perform experiments and simulations over a range of impact conditions and sphere properties and measure the minimum impact speed required to skip U_{min}, as a function of G (Fig. 4). Above a certain value of G (≈10^{3}−10^{4} kPa, depending on β_{o}), we recover the rigid skipping regime, in which U_{min} is independent of shear modulus but is very sensitive to β_{o}. For rigid spheres impacting above , where ρ*=ρ_{s}/ρ_{w}, prior research suggests spheres may broach (that is, become completely submerged before exiting), but not skip^{8} (Fig. 4f). For stiffness values below the rigid regime, the elasticity of the sphere becomes important and U_{min} decreases monotonically with decreasing shear modulus. Our analytical model accurately predicts the experimental and numerical results in this regime. The minimum speed is also much less sensitive to β_{o} in the elastic skipping regime and as a result we observe skipping at impact angles nearly three times larger than predicted for rigid spheres (Fig. 4 and Supplementary Fig. 1).
While our results show that reducing shear modulus has the predicted effect on wetted area and attack angle (Fig. 3b,c), it is unclear whether one or both of these mechanisms are responsible for the observed improved skipping performance. To isolate the mechanisms we consider the limiting case of small G, for which and thus cos α≈1 over typical impact timescales. The expected dependence of U_{min} on G in this limit can be rationalized by scaling analysis of equation 2 (see Methods and Supplementary Note 2), which gives
where t_{c} is the collision time (that is, time in which the sphere is in contact with the water). For threshold skipping cases, we expect the characteristic acceleration to be small compared with gravity and thus, to first order, . Furthermore, in the small G limit our numerical modelling shows that (Fig. 3c). Applying this dependence and solving for U_{min}, we find . Figure 4a shows that U_{min} approaches the G^{1/5} relation in the limit of small G, indicating that the only mechanism by which reducing shear modulus enhances skipping in this limit is through the increased wetted area. However, for larger G (>≈10 kPa), U_{min} deviates from the G^{1/5} relation as the coupling between shear modulus and α becomes important (Fig. 4a–d). As stiffness continues to increase, ultimately the amplitude of the deformations (affecting both S_{w} and α) become negligible and the sphere is effectively rigid (Fig. 4a,e). Consequently, we conclude that decreasing shear modulus below this rigid boundary causes an increase in the upward vertical force that promotes skipping through both of the hypothesized mechanisms, save for the limit of small G where decreasing shear modulus only affects lift by increasing S_{w}.
Skipping regimes
As the impact events devolve from clear skipping to water entry, we observe a transitional regime characterized by a matryoshka cavity, in which the sphere still skips (Fig. 1d; Supplementary Movie 2). The matryoshka phenomenon occurs when the total contact time of the sphere with water t_{c} is longer than the wave time t_{w} associated with mode 1^{−} elastic wave propagation, such that t_{c}/t_{w}>1. We define t_{w} as the time from impact until the circumferential elastic wave strikes the air–water interface (Fig. 2a,b). Experiments over a range of sphere properties and impact conditions reveal that t_{c}/t_{w} is governed by the impact angle β_{o} and the ratio (Fig. 5). The dependence on these terms can be rationalized by scaling analysis with U_{o} replacing U_{min} in equation 3. When t_{c}≈t_{w}, the characteristic sphere acceleration is much greater than g such that and we find (see Methods section). Furthermore, based on the propagation speed of mode 1^{−} elastic waves, we expect , which is confirmed experimentally (Supplementary Fig. 2). Combining the scaling for each time gives . We can now examine the evolution of the timescale ratio in the vicinity of the transitional regime in the limit of shallow (β_{o}→0) and steep impact angles, making use of the relationship between λ_{max} and shown in Fig. 3c. Here, we consider steep impact angles to be , where is the maximum impact angle at which we have observed elastic sphere skipping . In the shallow β_{o} limit, t_{c}/t_{w}≈1 occurs at values of , for which λ_{max}→1 (Fig. 3c); thus, we anticipate for shallow angles. For steep β_{o}, the transitional regime occurs for , for which (Fig. 3c), and we expect . These limiting relations capture the evolution of t_{c}/t_{w} observed experimentally (Fig. 5) and provide insight into the differences observed at different impact angles. We see that for steeper β_{o}, the sphere deformation has a larger effect on the collision time, which gets manifested as a higher sensitivity of t_{c}/t_{w} to .
Based on our findings regarding the transitional skipping regime, we hypothesize that the same dimensionless parameters (that is, β_{o} and ) govern all deformation modes and associated skipping behaviour. An empirical regime diagram indeed shows that these parameters classify all observed skip types (Fig. 6). Mode 1^{+} is promoted by shallow β_{o}, large U_{o} and/or small G. As stiffness becomes large relative to hydrodynamic pressure, the vibration type traverses the mode 2 and mode 1^{−} regimes. Our analytical model correctly predicts the boundary between the mode 1^{−} skipping (t_{c}/t_{w}<1) and mode 1^{−} transitional (t_{c}/t_{w}>1) regimes (marked by red line on Fig. 6). A small, radiusdependent overlap region exists between the transitional regime and water entry. Finally, we have observed that the vibration mode associated with the transitional and water entry regimes is exclusively mode 1^{−}.
Skipping sphere rebound
We quantify the rebound characteristics of skipping spheres by computing the normal and tangential coefficients of restitution, and , where U_{e} is the exit velocity and n and t refer to components normal and tangential to the freesurface, respectively. Here, we find some interesting similarities between skipping elastic spheres and liquid droplets bouncing on inclined liquid layers^{28}. Following the work on bouncing liquid droplets, we plot and as a function of (Fig. 7), which is equivalent to the normal Weber number with shear modulus G replacing the Laplace pressure σ/R_{d}, where R_{d} is the droplet radius and σ is surface tension of the liquid droplet. Interestingly, we find and , which are the same scaling relations found for liquid droplets bouncing on inclined liquid layers^{28}. The restitution coefficients deviate from these trends when t_{c}/t_{w}>1. In these cases, sphere vibrations interact with the water via the matryoshka cavity causing a significant decrease in bouncing efficiency. For both elastic skipping spheres and bouncing droplets, the restitution coefficients are always less than one as part of the initial translational kinetic energy goes into postimpact vibrations in the sphere or droplet^{28,32}.
We build on the bouncing droplet analogy to speculate on the lower bound of validity of the U_{min}∝G^{1/5} scaling relation in the limit G→0 (Fig. 4). When the relative magnitude of droplet surface tension becomes small for liquid droplets impacting liquid layers, bouncing does not occur and the droplet completely merges with the liquid layer^{28}. We conjecture about a similar limit for elastic spheres with G→0 and impacting with , where σ_{w} is the surface tension of water. In this limit, we expect the surface tension force from the water to act prominently on the sphere^{33} and to inhibit sphere reformation, thus preventing recovery of translational kinetic energy from deformational potential energy during impact. As a result, we hypothesize that sphere skipping would ultimately cease in this limit. We anticipate these dynamics would become relevant when G≈σ_{w}/R (see Supplementary Fig. 3 and Supplementary Note 3). To validate these predictions is beyond the scope of the present work.
Discussion
Perhaps the most mesmerizing manifestation of elastic sphere water impact is continual skipping across water (Supplementary Movie 3). To confirm that our physical description of a single skip generalizes to multiple skip events, we predict the placement of successive impacts on the regime diagram (Fig. 6). An experimental investigation using isolated water tanks validates the predictions and shows the sphere traversing through the vibration modes with each impact (Fig. 8). As to how repeated skipping is sustained over very long skipping trajectories (Supplementary Movie 3), we gain insight from the behaviour of the restitution coefficients (Fig. 7). First, ɛ_{t} is consistently larger than ɛ_{n}, which causes β_{o} to decrease and thus become more favourable with each skip. Second, the restitution coefficients actually become larger as decreases, until the sphere enters the mode 1^{−} skipping and transitional regimes. Therefore, one could say it becomes easier to skip with every skip.
While toy elastic balls may bestow upon the casual sportsman the ability to break the world stone skipping record (88 skips by K. Steiner, Guinness World Records), we believe the physics underlying the elastic sphere impact are common to the large deformation hydroelastic response of surfaceriding and skipping compliant bodies. Models of these structures, such as inflatable boats, typically ignore extreme elastic deformation even though it is known to affect drag, stability and slamming loads^{12}. The mechanisms of form and force augmentation, as well as the secondary vibrationinduced fluid interactions that we have revealed, can be exploited for functional advantage and incorporated into higherfidelity hydroelastic models.
Methods
Sphere fabrication and material properties
In order to control material properties, custom elastomeric spheres were fabricated from a high performance platinumcure silicone rubber called Dragon Skin produced by SmoothOn, Inc., which consists of two liquid constituent parts. Once the two constituents are mixed, the material sets without requiring heat treatment. The shear modulus was varied by adding a silicone thinner to the mixture before setting, which reduces the material shear modulus by decreasing the polymer crosslinking density. Sphere materials with three different shear moduli were fabricated by adding 0, 1/3 and 1/2 parts thinner by mass ratio. Before setting, the liquid mixture was placed in a vacuum chamber to remove any entrained air. For our experiments, spheres were fabricated by curing the liquid mixture in smooth, machined aluminium moulds to produce spheres with three different radii: 20.1±0.8 mm; 26.2±0.8 mm; and 48.8±0.9 mm. A rigid sphere with R=25.8±1 mm was fabricated from Nylon DuraForm PA using selective laser sintering. The uncertainty on each sphere radius represents the 95% confidence interval based on several independent measurements. A thin Lycra casing was loosely fitted around each sphere in order to prevent undesired particles from adhering to the surface and to reduce the friction between the sphere and the launching mechanism from which it was fired.
The silicone rubber was so compliant that traditional uniaxial ‘dogbone’ testing on our Instron machine was not feasible as the forces generated were too small to be reliably measured. To overcome this, we performed a test in which the spheres were compressed on the Instron to generate a quasistatic forcedisplacement curve. This test setup was then numerically modelled in Abaqus with the sphere material described by a neoHookean hyperelastic constitutive model, parameterized by the shear modulus G ^{31} . We then varied G to find the value that produced the best fit between the numerically simulated and experimentally measured forcedisplacement curves. The elastomeric spheres used in our experiments had shear moduli of 97.2, 28.5 and 12.3 kPa corresponding to 0, 1/3 and 1/2 parts thinner, respectively. According to the manufacturer of the rigid sphere (3D Systems—Quickparts Solutions), the elastic modulus of the selective laser sintering Nylon DuraForm PA material is 1.59 × 10^{6} kPa, which—assuming a Poisson’s ratio of 0.4—gives a shear modulus of G=5.66 × 10^{5} kPa.
Sphere skipping experiments and data processing
Spheres were launched at the water surface from a variableangle, pressuredriven cannon consisting of a pressure chamber for compressed air, a sliding cylindrical piston and interchangeable barrels. Sliding the cylindrical piston allowed air to flow from the pressure chamber into the barrel, thus forcing the sphere to accelerate out of the barrel and strike the water surface with impact speed U_{o} and angle β_{o}. Impact events were illuminated with diffuse white back lighting and filmed with either NAC GX3 or Photron SA3 highspeed cameras acquiring at 1,000–2,000 frames per second (fps).
The impact speed U_{o} and angle β_{o} were measured from images of the sphere before water impact using a crosscorrelation algorithm. The mean uncertainties on U_{o} and β_{o} are ±1.09 m s^{−1} and ±1.75°, respectively (computed at 95% confidence level). The same algorithm was used to measure the exit speed and angle of the sphere after lifting off the surface. Also, the rigid body rotation of the sphere was estimated after skipping by tracking reference markers on the exterior of the Lycra casing. It was found for mode 1^{−} skip types that the rotational kinetic energy was typically <8% of the translational kinetic energy after water exit. Furthermore, for these impacts the wave time t_{w} was typically <30% of the period of rigid body rotation measured after skipping. The collision time t_{c}, wave time t_{w} and vibration mode classification were all determined from manual inspection of the highspeed images. The minimum skipping speed U_{min} was found experimentally by performing successive experiments with identical conditions but with increasing speed until the sphere skipped.
Abaqus numerical model
Details of the Abaqus numerical model of the elastic sphere impact are contained in reference^{30}; clarifications relevant for the present work are summarized here. The finiteelement model uses the built in coupled euler–lagrange functionality of Abaqus/Explicit, which couples the contact region between the Lagrangian (sphere) and Eulerian (fluid) domains using a penalty method. Direct numerical simulation of the compressible Navier–Stokes equations is performed in the Eulerian domain. For the solid, conservation of momentum is solved with an incompressible neoHookean constitutive model describing the sphere. For all numerical model results presented herein, the mesh consists of eightnoded Eulerian hexahedral elements with spatial resolution of 3 mm. The sphere radius R, density ρ_{s} and shear modulus G, as well as the impact speed U_{o} and angle β_{o} were set to match experimental values. The threedimensional computational domain consists of a water tank (length=30 R, depth=6.5 R) with a symmetry plane coinciding with the plane of motion. For the numerical results presented in Fig. 4a, U_{min} is the average of the impact speeds for a skipping case and the nonskipping case with the nearest U_{o}. The numerical marker error bars reported in Fig. 4a represent ±1/2 of the speed difference between the two cases.
Analytical model of elastic sphere skipping
An approximate analytical approach to modelling the impact between a compliant elastomeric sphere and a fluid surface is outlined (for a complete derivation, see Supplementary Note 1). Here, we describe the sphere deformation and motion using a set of reduced, scalar generalized coordinates that are governed by a system of ordinary differential equations (ODEs). We begin by defining a fixed Cartesian coordinate system {e_{1}, e_{2}, e_{3}} (Fig. 3a). We assume the sphere moves only in the e_{1}−e_{2} plane and undergoes no rigid body rotation. The sphere deformation is first described by a rigid displacement d that describes the motion of the COM in terms of the generalized coordinates d_{1}, d_{2}. This is followed by a volume preserving stretch V that deforms the sphere into an ellipsoid. The coordinates of a material particle before deformation (x) and after deformation (y) are related by y=d+Vx. The velocity and acceleration fields follow as and . The stretch V and its time derivatives , can be written in terms of the principal stretches λ_{1}, λ_{2} and λ_{3}=1/λ_{1}λ_{2}, which are aligned with the bodyfixed {m_{1}, m_{2}, m_{3}} coordinates, respectively (see Fig. 3a and Supplementary Note 1). The {m_{1}, m_{2}, m_{3}} coordinate system is inclined at attack angle α relative to the freesurface. We introduce a virtual velocity field where the kinematic variables are associated with virtual rates of change , , , , about the current (deformed) state. The governing equations for the generalized coordinates d_{1}, d_{2}, λ_{1}, λ_{2} and α are obtained from the principle of virtual work (that is, weak form of the momentum conservation equation)^{31}
where σ is the Cauchy stress tensor, D is the stretch rate, b are body forces, t are traction forces and V and A denote integration over the volume and surface of the deformed solid, respectively. The term involving the external traction can be rewritten as
where F represents the resultant hydrodynamic force acting on the solid and the second term on the righthand side represents the virtual power associated with a force dipole tending to distort the elastomer. Expressing equations 4 and 5 in terms of the generalized coordinates (see Supplementary Note 1) and then setting each of , , , , to be nonzero in turn yields a set of coupled second order nonlinear ODEs in terms of general forces and tractions:
where F_{h} and F_{v} are the horizontal and vertical force components in {e_{1}, e_{2}, e_{3}} coordinates, respectively, and t_{i} and y_{i} are the components of the traction vector and position vector on the surface of the ellipsoid in {m_{1}, m_{2}, m_{3}} coordinates, respectively.
Next, we propose an approximate model for the hydrodynamic forcing on the sphere. It is beyond the scope of this work to derive an analytical expression for the dynamic pressure distribution over the wetted sphere surface. Rather, our goal is to generate a simplified model that captures the firstorder sphere motion and deformation during an oblique water impact. Thus, the hydrodynamic force F is computed using equation 1, which follows from work on skipping stones^{4} by considering the deformed sphere to be a circular disk. The disk radius is described by an equivalent principal stretch λ_{eq} computed by equating the area of the equivalent circular disk with the crosssectional area of the deformed sphere in the m_{1}m_{3} plane, π(λ_{eq}R)^{2}=π(λ_{1}R)(R/λ_{1}λ_{2}), which gives . Furthermore, S_{w}, U_{B} and β_{B} can be written in terms of the generalized coordinates describing the sphere deformation (see Supplementary Note 1). Without describing the pressure distribution, we cannot specify the centre of pressure and, thus, cannot define the y_{1} coordinate at which the traction vector acts in equation 8, which governs the attack angle α. To overcome this, we determine α from our Abaqus numerical model (Fig. 3b). With α prescribed, the remaining ODEs for d_{1}, d_{2}, λ_{1} and λ_{2} (equations 6, 7, 9 and 10) can be solved without further simplification. Inserting the hydrodynamic force yields equation 2 and the remaining ODEs are:
Equations 2 and 11, 12, 13 are solved using a fourthorder Runge–Kutta solver in Matlab, with α and prescribed based on the Abaqus numerical model results.
Scaling analysis
Using U_{min} as the characteristic speed, (λ_{max}R)^{2} as the characteristic wetted area and as the characteristic acceleration, equation 2 can be estimated as
For cases that barely skip, we have observed t_{c}≈O(10^{−1})s, which gives a characteristic acceleration of m s^{−2} (with R≈O(10^{−2})m). Therefore, in the case of the minimum impact speed, we expect gravity to be an order of magnitude larger than the sphere acceleration such that
In the limit of small G, simulations show that sin (α+β_{B}) cos α is of order 1 (see Supplementary Note 2) and (see Fig. 3c). Thus, equation 15 leads to
To determine how t_{c}/t_{w} evolves, we consider a scaling analysis of equation 14 when t_{c}≈t_{w}, which is typically O(10^{−2})s for our experiments. We find the characteristic acceleration m s^{−2} and thus can neglect gravity. Replacing U_{min} with U_{o} in equation 14 and solving for t_{c} gives . Simulations show that is order 1 (see Supplementary Note 2). Thus, the collision time scales as
Additional information
How to cite this article: Belden, J. et al. Elastic spheres can walk on water. Nat. Commun. 7:10551 doi: 10.1038/ncomms10551 (2016).
Change history
10 June 2016
A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.
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Acknowledgements
J.B., T.T.T. and R.C.H. acknowledge funding from the Office of Naval Research, Navy Undersea Research Program (grant N0001414WX00811), monitored by Miss Maria Medeiros. J.B. and M.A.J. acknowledge funding from the Naval Undersea Warfare Center InHouse Lab Independent Research program, monitored by Mr Neil Dubois. We thank J. Bird, N. Dubois, A. Hellum, J. Marston, B. Roden and S. Thoroddsen for reading and commenting on the manuscript and C. Mabey for his photography.
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J.B., T.T.T. and R.C.H. designed the research; J.B., R.C.H., T.T.T. and M.A.J performed the experiments; J.B., R.C.H. and T.T.T. analysed the data; M.A.J developed the Abaqus numerical model; A.F.B. and J.B. developed the analytical model; and J.B. wrote the original manuscript and all authors helped revise it.
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Correspondence to Jesse Belden.
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Supplementary information
Supplementary Information
Supplementary Figures 13, Supplementary Notes 13 and Supplementary References (PDF 257 kb)
Supplementary Movie 1
Elastic spheres deform dramatically upon water impact. An elastic sphere impacting the water surface assumes a favorably oriented disklike shape that promotes skipping. Large amplitude elastic oscillations initiated upon impact persist after the sphere lifts off the water surface. This video demonstrates vibration mode 2 in which the sphere assumes an ellipsoidal shape with oscillating major and minor axes. (MOV 25713 kb)
Supplementary Movie 2
Sphere vibrations induce a matryoshka cavity. In what we refer to as the transitional impact regime, elastic spheres carve nested disturbances on the surface forming a matryoshka cavity. When viewed from the side (left hand video frame), the disturbances appear to result from rigid body rotation of the sphere. However, when viewed in a timesynced top view (right hand video frame), the stitching on the sphere casing indicates minimal rigid rotation. Rather, the video reveals an elastic wave propagating around the circumference of the sphere as the source of the nested cavities. A matryoshka cavity forms when the collision time t_{c} is longer than the elastic wave propagation time t_{w}. (MOV 14155 kb)
Supplementary Movie 3
Elasticity allows spheres to walk on water. The liftenhancing dynamics of elastic spheres enable multiple skips over long trajectories. Modeling and laboratory observations suggest that the impact angle β_{o} decreases with each skip, which prolongs skipping despite U_{o} diminishing. This video qualitatively demonstrates this behaviour as an elastic sphere skips across a lake. (MOV 2680 kb)
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Belden, J., Hurd, R., Jandron, M. et al. Elastic spheres can walk on water. Nat Commun 7, 10551 (2016). https://doi.org/10.1038/ncomms10551
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