## Abstract

The desert-dwelling sandfish (*Scincus scincus*) moves within dry sand, a material that displays solid and fluidlike behavior. High-speed x-ray imaging shows that below the surface, the lizard no longer uses limbs for propulsion but generates thrust to overcome drag by propagating an undulatory traveling wave down the body. Although viscous hydrodynamics can predict swimming speed in fluids such as water, an equivalent theory for granular drag is not available. To predict sandfish swimming speed, we developed an empirical model by measuring granular drag force on a small cylinder oriented at different angles relative to the displacement direction and summing these forces over the animal movement profile. The agreement between model and experiment implies that the noninertial swimming occurs in a frictional fluid.

The locomotion of organisms (*1*, *2*), whether by running, flying, swimming, or crawling, is the result of multiple-degree-of-freedom nervous and musculoskeletal systems interacting with an environment that often flows and deforms in response to movement. Nearly all experiments and models of terrestrial locomotion have been developed for running and walking on rigid, flat, no-slip frictional substrates for which the complication of substrate flow is not considered. In contrast, complexity in interaction with the environment in aquatic and aerial locomotion ( swimming and flying) is well recognized (*3*). Determining mechanisms for propulsion or lift in these media is always possible in principle because the rules of interaction with fluids are worked out: They require solving Navier-Stokes hydrodynamics in the presence of moving boundary conditions. A major challenge in biology is to understand the locomotion of organisms that walk, crawl, or burrow on or within terrestrial substrates like sand (*4*), soil (*5*), and muddy sediments (*6*) that display both solid and fluidlike behavior. In such materials, validated theories such as the Navier-Stokes equations for fluids do not exist, and visualization techniques [such as particle image velocimetry in fluids (*7*)] are nearly nonexistent. Understanding of the mechanics of subsurface movement has ecological importance and could reveal how the actions of small burrowing organisms can transform entire landscapes (*8*).

Animal burrowing and movement within granular media is relevant to desert organisms like scorpions, snakes, and lizards that move within sand to escape heat and predators and hunt for prey (*9*, *10*). Desert sand [which covers 6 to 10% of land surface (*11*)] is an example of a granular material, a collection of dissipative particles that interact through contact forces and in bulk can display solid and fluidlike features (*12*) when disturbed. A key parameter that controls the response of granular media to intrusion is the volume fraction φ, the ratio of material volume to total occupied volume. In dry granular media in natural environments, φ depends on the history of the sand (for example, perturbations by wind or animal burying and digging), and can vary between 0.57 and 0.64 (*13*). The response of granular media to intrusion depends on φ: Closely packed material at high φ must expand to flow, whereas loosely packed material at low φ consolidates (*14*). The effects of φ on drag are largely unexplored, although we have recently found that vertical penetration resistance doubles as φ increases by just 0.08 (*15*).

To investigate how rheological features of the material influence the locomotor mode and performance of an organism moving within sand, we used high-speed x-ray imaging to study a small (~10 cm) desert-dwelling lizard, the sandfish, that inhabits the Saharan desert of Africa and moves within granular media of different φ. The sandfish’s above-ground burial process has been described (*9*), and it is hypothesized that its counter-sunk lower jaw and smooth scales with low friction and low wear properties (*16*) aid swimming and digging. However, little is known about how the animal moves subsurface. Although it has been hypothesized that body motion plays an important role in thrust production (*9*, *17*) in sand-dwelling lizards, a recent study using nuclear magnetic resonance (NMR) to visualize subsurface motion proposed that the sandfish used its limbs in a paddling motion along with undulations on its body to generate thrust subsurface (*18*). However, the observational quality and number of trials were limited. We show through detailed biological experiment and development of a model of the locomotion that subsurface sandfish motion is not driven by limbs but by an undulatory gait without use of limbs. Thrust and drag result from frictional grain flow, which leads to features of the swimming that are intermediate to movement in fluids and on solid surfaces, and surprisingly are independent of φ.

To perform controlled studies of the burial and swimming of the sandfish (11 animals; mass = 16.2 *±* 4 g, body length *L* = 8.3 *±* 3.3 cm, defined as the length from snout tip to the base of the tail), we used a fluidized bed (fig. S1) (*19*) to prepare a 10-cm-deep container of 0.27 *±* 0.04 mm glass beads into naturally occurring loosely packed and closely packed volume fractions φ = 0.58 and φ = 0.62, respectively. The sandfish took 672 *±* 239 ms to complete its entire burial process, independent of volume fraction (Student’s *t* test, *P >* 0.05) (movie S1). Once below the surface, the dorsal x-ray video imaging (movies S2 and S3) revealed that the animal no longer used limbs for propulsion. Instead, the animal placed its limbs against its sides and executed an undulatory motion (Fig. 1, E and F) with large-amplitude axial oscillation, using its body to propel itself at speeds of up to *~*10 cm/s. We observed similar limbless undulatory motion in beads with mean diameters ranging from 0.1 mm to 3 mm. To quantify the lack of limb use, we measured the distance between pairs of limbs (fore and hind) (Fig. 1G). As the animal ran across the surface, this interlimb distance oscillated around 2.75 *±* 0.59 cm, indicating the sprawl of the limbs during the diagonal gait (*1*, *9*). As it began to bury, the interlimb distance decreased significantly (Student’s *t* test, *P <* 0.01) until it became a constant (1.72 *±* 0.53 cm) close to body width (1.23 *±* 0.51 cm) because limbs were placed close to the side of the body. Like in (*18*), side-view x-ray imaging of the sandfish subsurface showed that the animal undulates in a plane at *~* 22.2 *±* 3.7° with respect to the horizontal and buries consistently to a depth of 2.1 *±* 0.5 cm (surface to top of mid-point of the back), independent of volume fraction (Student’s *t* test, *P >* 0.05).

We characterized the undulatory kinematics by digitizing the midline from the dorsal x-ray images (Figs. 1F and 2A). The undulatory motion fit well (*r*^{2} *>* 0.85, *P <* 0.01) the form of a posteriorly traveling single-period sinusoidal wave (Fig. 2A) such that *y* = *A*sin 2π(*x* + *v*_{w}*t*)/λ, with *y* the displacement away from the midline of a straight animal, *A* the amplitude, λ the wavelength, *f* the wave frequency, and *v*_{w} = *f*λ*¸* the wave speed. From the fit, we deduced these parameters as well as the forward swimming speed of the center of mass *v _{x}* (which was not equal to the wave speed; see discussion below). During a swim, there was no change in either

*A*(

*P*= 0.12) or λ (

*P*= 0.66), and neither

*A*nor λ varied significantly for the different φ (

*P >*0.05 comparing each treatment by using a repeated measures one-way analysis of variance) (Fig. 2B). The large-amplitude undulations over the entire body are unlike the kinematics of other undulatory swimming organisms on the sandfish-sized scale [such as eels (

*20*)], which tend to increase in amplitude posteriorly. The undulations do resemble those of eel movement on land (

*20*) as well as small (

*<*1 mm) swimmers (for example, nematodes) in fluids at low Reynolds (

*Re*) number, in which the inertia of the material surrounding the organism plays no role (

*21*,

*22*).

The sandfish increased swimming speed by increasing wave frequency *f* (Fig. 2C) (Student’s *t* test, *P <* 0.01), and the slope of the *v*_{x} versus *f* relationship was independent of volume fraction (Student’s *t* test, *P >* 0.05). We found, however, that *v*_{x} is not simply the product of *f* and λ, but instead the slope is reduced by a constant factor η so that *v*_{x} = *η f* (Fig. 2C) with η = 0.54 *±* 0.12 and η = 0.49 ± 0.09 for loosely and closely packed material, respectively. Because *v*_{w} = λ*f*, this implies that the average forward speed is a fraction of the wave speed. Thus, the animal is not moving in a tube because successive undulations do not trace a continuous path; this can be seen in Fig. 1F. Instead, tracer particles placed in the bulk reveal that there is backward displacement of material as the animal moves forward. Slipping while progressing is common to undulatory swimmers in deformable media across length scales (for example, eels and spermatozoa in fluids) (*1*) and is characterized by the wave efficiency (η) (*21*), defined as the ratio between the forward speed of the animal (*v _{x}*) and the velocity of the wave traveling down its body (

*v*

_{w}): η =

*v*

_{x}

*/v*

_{w}. Because λ for the sandfish is independent of

*f*in our experiments, η is the slope of the

*v*

_{x}/λ versus

*f*curve and is shown in Fig. 2C. Typical wave efficiencies of organisms in moving in fluids at low

*Re*(such as nematodes) are 0.25 (

*23*), nearly a factor of two smaller than the sandfish, whereas organisms undulating (creeping) along the solid surface of an agar-air interface can have η close to unity [η

*~*0.8 − 0.9 (

*21*)] because deformations in the surface allow movement that is effectively in a tube. η for the sandfish in granular media is intermediate to that for fluids and solids and is independent of φ, even though materials with different φ have penetrations resistances that can vary by a factor of 2 (

*15*).

Prediction of η for undulatory movement in fluids and solids has been accomplished by use of models of interaction of the organism and the environment: In low *Re* Newtonian fluids, forces are purely viscous (Stoke’s law) such that force is proportional to instantaneous velocity (*22*, *23*), and in undulatory crawling on solids force is determined with static/dynamic friction coefficients (*22*). Although the equivalent force laws are not available in granular media, features of the sandfish motion allow us to hypothesize about the medium in which it swims: (i) It is well known that for slow enough disturbances, the drag in granular systems is dominated by friction between flowing grains, and thus this makes drag independent of velocity (*14*, *24*–*26*). (ii) At the observed body speeds and accelerations during undulatory motion, inertial forces due to accelerating material are small as compared with frictional drag forces (*ma*/*F ≈* 0.2, where *F ≈* 2 N is estimated for a small cylinder dragged in sand and *ma ≈* 0.4 N is found from the product of the acceleration of the sandfish body and a mass of sand surrounding it), and thus inertia can be neglected. (iii) Because grain interactions are dissipative, disturbances typically dampen in short times, and thus changes in the static configuration dominate the drag force.

Therefore, we hypothesize that the animal swims in a so-called “frictional fluid” (*27*), such that flowing grain-grain and grain-animal friction determine the thrust and drag forces. Such a frictional fluid superficially resembles low *Re* swimming in that there is no inertia, but the mechanism for drag is frictional, rather than viscous. Movement in a frictional fluid differs from noninertial crawling on a surface in which anisotropic friction [originating from, for example, deforming the surface and pushing off as does nematode on agar (*21*) or using overlapping belly scales or changes in weight distribution as does a snake (*28*)] generates net propulsive force without net flow of material.

To test this hypothesis, we developed a model inspired by similar local force, noninertial models [typically called Resistive Force Theory (RFT) (*23*)] such that the body of the organism is partitioned into elements, each of which generates thrust and experiences drag when moving through a media and whose forces are not influenced by force fields from other segments. If *F*_{N} and *F*_{L} are the normal and lateral forces (perpendicular and parallel to the element respectively) acting on the sides of the sandfish body, net forward force on the element can be written as *F*_{x} = *F*_{N} sinθ − *F*_{L} cosθ, where θ is the angle between the forward direction of the animal and the orientation of an element of the organism (Fig. 3A). The forces for each element are integrated over the length of the body (snout to tail tip) and for a full period of a traveling wave to obtain average swimming speed. Unlike in fluids, we have no validated theory to calculate the force on an intruder as a function of its angle relative to displacement or for varying φ. Therefore, we used a combination of measurements (*19*) of the granular thrust and drag forces on a stainless steel cylindrical rod [with grain-surface friction coefficient similar to the sandfish (*16*)], insights from Fig. 3B, and an empirical fitting function that incorporates drag on the sidewalls and end-faces of the cylinder (equation S8) so as to approximate *F*_{N} and *F*_{L} for elements of the sandfish body as:
*F*_{N} = 2*lr*(*C*_{S} sinβ_{0} + *C*_{F} sinψ)*F*_{L} = 2*lrC*_{F} cosψ (1)where tanβ_{0} = cotγ_{0} sin ψ, ψ is the angle of the segment with respect to its local velocity vector, *l* and *r* are the length and radius of the segment (assumed cylindrical), and the constants *C*_{S}, *C*_{F}, and γ_{0} characterize the material response to drag. Plots of *F*_{N} and *F*_{L} as a function of ψ are shown in Fig. 3, C and D. Equation 1 represents the data well (*R*^{2} = 0.97) (solid lines shown in Fig. 3, C and D), and the fit coefficients are given in table S1.

*F*_{L} is well approximated by the cosine function, similar to viscous fluids or frictional material, indicating that its source is the resolved component of friction between the grains and the body. The form of *F*_{N} is not well approximated by the corresponding sine component (Fig. 3C, dashed gray lines) but rather increases more rapidly for ψ < 30° and increases more slowly for ψ > 30°. We hypothesize that above this angle [close to the angle at which internal slip planes or shear bands form and move (*14*)], the force increases slowly because of the formation of a solid region of grains that moves along with the rod (*14*, *26*). Preliminary numerical simulation indicates that the size of the solid region grows slowly with ψ for ψ > 20°. The fitting function (Eq. 1) incorporates these effects in *F*_{N} and *F*_{L} through terms proportional to the angular component of *C*_{S} (which is determined primarily by grain-object frictional forces) and an extra term in *F*_{N} that increases rapidly and then saturates for ψ > 20°. In this term, the coefficient *C*_{S} is set by grain-grain frictional force and γ_{0}, which is related to the internal slip angle. *C*_{F} ≈ *C*_{S}*/*2, indicating that grain-grain friction is larger than grain-surface friction.

To determine the wave efficiency (and thus swimming speed for a given frequency), we balanced thrust and drag over a cycle (net force, *v _{x}*). Because

*v*

_{x}and

*v*

_{w}appear only as a ratio (from the kinematics in equation S2) in the argument of the integral in Equation S7, and do not appear in the

*F*

_{N}and

*F*

_{L}terms (fig. S2B), the force balance predicts η. Numerical integration of the model over a measured organism sinusoid from Fig. 2 with constant cylindrical cross-section predicts 0.4 ≤ η ≤ 0.65 independent of φ (Fig. 4A), which is in accord with experimental observation; the bounds on η are set by assuming the head drag is zero or that of a flat plate because measurements for the drag force on an object with the exact morphology of the shovel-shaped head are unavailable. The model predicts a maximum in forward progress per cycle when

*A*/λ ≈ 0.2, and we find that the measured kinematics fall close to this speed (Fig. 4B). The maximum is due to a competition between increased η (less slipping) from increased

*A*and decreased λ because for an inextensible animal λ decreases as

*A*increases. An analytic low-amplitude approximation (equations S9 to S14) demonstrates the relevant scaling of η.

It is remarkable that η does not change significantly for different φ, especially because drag forces between closely and loosely packed material differ by ≈ 80%. The model indicates that η is essentially constant because *F*_{N} and *F*_{L} scale by the same ratio of 2.1 between closely and loosely packed material (Fig. 3, C and D, insets), and it is this ratio that determines η (equation S14). The model implies that η is greater than in low-*Re* fluids because of the greater thrust to drag force ratio in the frictional granular media.

The agreement between experiment and model indicates that the assumptions of velocity independence, local interaction, and the use of the average *F*_{N} and *F*_{L} for all phases of the motion are good approximations at these slow swimming speeds. Although the assumption that all body and head segments encounter material with the same φ is probably not true (because the body encounters grains disturbed by the head and other body segments), because *F*_{N} and *F*_{L} scale in the same ratio in all φ and this ratio determines η, wave efficiency should be independent of position along the body. The model assumptions break down in the approximation that the instantaneous force (Fig. 3B) can be replaced by the average force. This is a good approximation except for *F*_{N} in closely packed material (Fig. 3B, red curves); enhancement of *F*_{N} because of peak stress significantly greater than the mean could explain why, contrary to expectation, the range of *f* (and *v*_{x}) that the animal accessed was greater in closely packed material and the maximal swimming speed the animal was able to access was ≈50% larger in closely packed material (Fig. 2C).

In conclusion, we used high-speed x-ray imaging to show that the sandfish lizard propels itself within granular media using a large-amplitude traveling-wave oscillation of its body without using its limbs and that the wave efficiency of this motion is independent of the volume fraction of the medium. Using a model that balances thrust and drag with measured granular drag laws, we can predict the wave efficiency and optimal kinematics. We demonstrated that the animal swims in a medium with properties intermediate to those of low *Re* viscous fluids and frictional solid surfaces and that are unique to dense granular beds, namely velocity-independent forces and enhanced normal-drag forces. Our results demonstrate that burrowing and swimming in complex media can have similar complexity to movement in air or water, and that organisms can exploit the solid and fluidlike properties of these media to move effectively within them.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/325/5938/314/DC1

Materials and Methods

Figs. S1 to S3

Table S1

References

Movies S1 to S3

## References and Notes

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- This work was supported by NSF Physics of Living Systems grant PHY-0749991 and the Burroughs Wellcome Fund. We thank R. Full and T. Papenfuss for helpful discussion. We thank S. Steinmetz and N. Gravish for experimental assistance and P. Umbanhowar for discussion of the drag model and careful reading of the manuscript.