Crawling Beneath the Free Surface: Water Snail Locomotion ͒ Sungyon Lee,1 John W

PHYSICS OF FLUIDS 20, 082106 ͑2008͒

Crawling beneath the free surface: Water snail locomotion ͒ Sungyon Lee,1 John W. M. Bush,2 A. E. Hosoi,1 and Eric Lauga3,a 1Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 2Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 3Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, Cambridge, La Jolla, California 92093-0411, USA ͑Received 14 December 2007; accepted 10 June 2008; published online 26 August 2008͒ Land snails move via adhesive locomotion. Through muscular contraction and expansion of their foot, they transmit waves of shear stress through a thin layer of mucus onto a solid substrate. Since a free surface cannot support shear stress, adhesive locomotion is not a viable propulsion mechanism for water snails that travel inverted beneath the free surface. Nevertheless, the motion of the freshwater snail, Sorbeoconcha physidae, is reminiscent of that of its terrestrial counterparts, being generated by the undulation of the snail foot that is separated from the free surface by a thin layer of mucus. Here, a lubrication model is used to describe the mucus flow in the limit of small-amplitude interfacial deformations. By assuming the shape of the snail foot to be a traveling sine wave and the mucus to be Newtonian, an evolution equation for the interface shape is obtained and the resulting propulsive force on the snail is calculated. This propulsive force is found to be nonzero for moderate values of the capillary number but vanishes in the limits of high and low capillary number. Physically, this force arises because the snail’s foot deforms the free surface, thereby generating curvature pressures and lubrication flows inside the mucus layer that couple to the topography of the foot. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2960720͔

I. INTRODUCTION all of which use waves of contraction that propagate in the direction of their motion ͑“direct waves”͒. Waves traveling in Engineers often look to nature’s wide variety of locomo- the opposite direction ͑“retrograde waves”͒ were examined tion strategies to inspire new inventions and robotic by Jones and Trueman.11 A vital insight was later provided devices.1–4 More generally, scientists across all disciplines by Denny,12 who turned the focus of study from the snail are interested in understanding the physical mechanisms be- foot to the properties of the pedal mucus. Pedal mucus has a hind different styles of biolocomotion. The mechanism of finite yield stress that allows it to act as an adhesive under terrestrial snail locomotion has been investigated and eluci- small strains and to flow like a viscous liquid beyond its dated over the past couple of decades; conversely, the pro- yield point. Thus, the snail is able to create regions of flow in pulsion of water snails that crawl beneath a free surface has the mucus by locally shearing it, while the rest of the mucus ͑ yet to be considered. The purpose of this paper is to propose is effectively glued to the solid substrate; these regions or ͒ a propulsive mechanism for water snails. shear waves propagate along the length of the foot, enabling the snail to move. Denny used the nonlinear nature of the Gastropod locomotion has been of scientific interest for mucus to rationalize the locomotion of Ariolimax columbi- more than a century.5–7 Three distinct modes of locomotion anus, a terrestrial slug.13 Recently, it was found that mucus have been examined: ciliary motion, pedal waves, and swim- with shear-thinning properties results in energetically favor- ming. Ciliary locomotion, characterized by the beating of 14 able locomotion, which is well supported by the experi- large arrays of cilia on the animal’s foot, is usually distin- mental studies of the mucus properties. A detailed investiga- guished from pedal waves by indirect means, such as lack of tion of the rheology of mucus was conducted by Ewoldt visible muscle undulation, uniform adherence of the foot to et al.,15 who also tested different synthetic slimes. The ver- 8 the substrate, and uniform gliding of the snail body. This satility of snail locomotion has also inspired Chan et al.2 to particular type of locomotion is mostly employed by various build a robotic snail, the first mechanical device to utilize the marine and freshwater snails. nonlinear properties of this synthetic mucus. A significant effort has gone towards understanding Terrestrial snails that employ adhesive locomotion are pedal wave locomotion by terrestrial snails. Lissman9,10 was only a fraction of the species in the class of gastropods. a pioneer in constructing a mechanistic model of the snail Traditionally, gastropods have been divided into four sub- foot undergoing such locomotion. In Ref. 9, he studied three classes: prosobranchia, opisthobranchia, gymnomorpha, and species of terrestrial snails ͑Helix, Haliotis, and Pomatias͒, pulmonata, the latter of which consists primarily of terrestrial snails. Many species have not been thoroughly investigated ͒ a Author to whom correspondence should be addressed. Electronic mail: but exhibit interesting locomotive behavior. For instance, 16 [email protected]. opisthobranchs, which have reduced or absent shells, swim

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FIG. 1. ͑Color online͒ Snail ͑Sorbeoconcha physidae͒ crawling smoothly FIG. 2. A trail of mucus behind the snail crawling upside down beneath the underneath the water surface while the surface deforms. Note the surface free surface. Photo courtesy of David Hu and Brian Chan ͑MIT͒. deﬂection associated with the undulatory waves propagating from nose to tail along its foot. Photo courtesy of David Hu and Brian Chan ͑MIT͒.

generation of propulsive forces is discussed in Sec. IV, together with the main conclusions and a summary of the sim- plifying assumptions used in our analysis. or burrow.17,18 A still more puzzling mode of locomotion is observed in certain species of water snails. 19 In 1910, Brocher remarked on water snails that can II. OBSERVATIONS swim inverted beneath the water surface; since then, other qualitative descriptions have been reported. Milne and Several common freshwater snails, Sorbeoconcha phys- Milne20 observed the foot of a pond snail “pulsing with slow idae, were collected from Fresh Pond, MA. About 1 cm in waves of movement from aft to fore along its length,” sug- length, this particular snail can crawl beneath the water sur- gesting that direct waves are employed for propulsion. The face at speeds as high as 0.2 cm/s, comparable to its speed presence of a trail of snail mucus was also reported. on solid substrates, and perform a 180° turn in 3 s. It is Goldacre21 measured the surface tension of this thin trailing rendered neutrally buoyant by trapping air in its shell. film to be approximately 10 dyn/cm; he also remarked that The undulation of the snail foot causes surface deforma- the creature was “grasping the film” as evidenced by the film tions with a characteristic wavelength of 1 mm and ampli- being pushed sideways as the snail advanced. Deliagina and tude of 0.2–0.3 mm ͑see Fig. 1͒. This deformation appears to Orlovsky22 made similar observations while studying feeding travel in the opposite direction of the snail motion, suggest- patterns of Planorbis corneus. This particular freshwater ing the generation of retrograde waves, contrary to the ob- snail crawls at about 15 mm/s, a speed comparable to that servations of Milne and Milne.20 Another notable feature of on land, while the cilia apparent on the organism’s sole “beat water snail propulsion is the presence of a trail of mucus ͑see intensely.” Cilia-aided crawling beneath a free surface was Fig. 2͒. For land snails, this mucus layer is typically observed on marine snails as early as 1919. Copeland23 con- 10–20 ␮m in thickness;15 as with land snails its rheological cluded that the locomotion of Alectrion trivittata, which characteristics may also play a significant role in underwater crawls upside down on the surface, relied solely on the cili- locomotion. Since these water snails are also able to crawl on ary action. He conducted a similar study on Polinices dupli- solid substrates, one might venture that their mucus proper- cata and Polinices heros, both of which were observed to use ties do not differ too greatly from those of land snails. both cilia and muscle contraction for locomotion on hard surfaces.24 Only ciliary motion was employed by the young Polinices heros when crawling inverted beneath the surface. III. MODEL It is therefore clear that freshwater and some marine A. Assumptions snails have the striking ability to move beneath a surface that is unable to sustain shear stresses. In this paper, we attempt a The crawling of water snails beneath the free surface has first quantitative rationalization of these observations. We four distinct physical features: a free surface with finite sur- use a simplified model based on the lubrication approxima- face tension ␴, a layer of ͑presumably͒ non-Newtonian mu- tion to show that a free-moving organism located underneath cus, coupled deformations of the foot and the surface, and a a free surface can move using traveling-wave-like deforma- matching of the flow inside the mucus to that around the tions of its foot. We first present our observations of the snail. To isolate the critical influence of the first feature, we propulsion of the freshwater snail Sorbeoconcha physidae in consider in this paper a simplified model system character- Sec. II. We introduce our model based on the lubrication ized by a Newtonian mucus layer and small deformations of approximation in Sec. III and present solutions for small- the foot and the interface, with hopes of providing physical amplitude motion of the foot. The physical picture for the insight into the propulsion mechanism.

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B. General equations Choosing a characteristic velocity Uϳ1cm/s, a mucus thickness Hϳ20 ␮m, and a ͑post-yield͒ mucus viscosity mucus ␯ϳ10−2 m2 /s ͑Ref. 15͒ suggests a Reynolds number of the Vw − Vs flow within the mucus layer to be ReϵUH/␯ϳ10−5. Thus, we neglect inertia and start with incompressible Stokes equa- ˆ tions: h2 ˆ yˆ h1 v =0, ͑1a͒ snail foot · ٌ Vs xˆ zˆ ⌸ =0, ͑1b͒ lab frame · ٌ where v is the velocity field inside the mucus and ⌸ is the (a) stress tensor. Normal and tangential stress boundary conditions at the surface may be expressed as mucus 2πλˆ n · ⌸ · n = ␴␬, ͑2a͒ hˆ 2 hˆ t · ⌸ · n =0, ͑2b͒ 1 snail foot where n and t denote, respectively, unit vectors normal Vw -outward͒ and tangent to the free surface, and ␬=−ٌ·n de͑ notes the curvature of the free surface. We limit our attention wave frame to the two-dimensional case, for which the interface shape is (b) given by yˆ =hˆ͑xˆ͒͑see Fig. 3͒ and ␬ may be expressed as ˆ /͑ ˆ 2͒3/2 FIG. 3. ͑Color online͒ A close-up view of the mucus and the snail foot hxˆxˆ 1+h , where the “hat” notation denotes dimensional xˆ ˆ variables. The mucus is assumed to be a Newtonian fluid; undergoing a simple sinusoidal deformation of wavelength 2␲␭. The pre- ˆ thus, the stress tensor is given by scribed shape of the snail foot is denoted as h1; the resultant shape of the ˆ free surface, h2, is to be solved for. The known constant speed of the wave, ˆ ˆ Vw, is set relative to the snail that is translating with an unknown speed, Vs. ⌸ =−pÎ +2␮e, ͑3͒ In laboratory frame ͑a͒, the wave is moving in the negative xˆ direction with ˆ ˆ ˆ Vw −Vs while the snail is moving in the positive xˆ direction with Vs.Inthe frame moving with the wave ͑b͒, the snail body appears to move in the .where eϵ 1 ٌ͕͑v͒+ٌ͑v͒T͖ is the rate-of-strain tensor 2 positive xˆ direction with Vˆ . In the frame moving with the snail, we assume that the w gastropod foot undergoes periodic deformations in the form ˆ of a traveling wave moving at a speed Vw. Our approach is as equations of motion. The governing equations and boundary follows: given the shape of the foot, we solve for the shape conditions ͓Eqs. ͑1a͒, ͑1b͒, ͑2a͒, ͑2b͒, and ͑3͔͒ are nondimen- of the liquid-air interface together with the velocity field in sionalized using the following set of characteristic scales: the mucus layer, and then calculate the resulting propulsive ␭ˆ ͑ ͒ force on the snail. xˆ = x, 4a

C. Lubrication analysis yˆ = Hˆ y, ͑4b͒ To eliminate temporal variations, we consider the frame ͑ ͒ ˆ ͑ ͒ ͑ ͒ ˆ uˆ,vˆ = Vw u,av , 4c of reference moving at the wave speed Vw relative to the snail ͑Fig. 3͒. We define Vˆ and the ͑unknown͒ snail speed w ˆ ˆ ͑ ͒ ˆ Vs = VwVs, 4d Vs to be positive when the snail moves in the positive xˆ direction while the wave travels in the opposite direction. ␮␭ˆ ˆ ͑ ͒ Vw The thickness of the mucus tens of microns is observed pˆ = p. ͑4e͒ to be small relative to a typical wavelength of foot deforma- Hˆ 2 tion ͑millimeters͒. Thus, we apply the lubrication ap- proximation25–28 and reduce the governing equations based Based on standard lubrication theory, the equations of motion are reduced to the following: on Hˆ /␭ˆ ϵaӶ1, where Hˆ is the characteristic thickness of the vץ uץ ␭ˆ ␲ mucus film and is the wavelength divided by 2 . More 0= + , ͑5a͒ yץ xץ specifically, terms of order a or higher are discarded in the

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2u In order to obtain the motion of the snail, it is necessaryץ pץ 0=− + , ͑5b͒ -y2 to consider the forces acting on the organism. Since its moץ xץ tion occurs at low Reynolds numbers, the snail is force-free; hence, the forces from the ͑internal͒ mucus ﬂow and those :p from the external ﬂow around the body must sum to zeroץ 0= , ͑5c͒ yץ

while the boundary conditions may be expressed as ͑ ͒ Fint + Fext =0. 10 ͑ ͒ u =1 aty = h1, 6a ˆ ˆ More specifically, Fext is equal to −Fdragex, where Fdrag is the ͑ ͒ v =0 aty = h1, 6b magnitude of the drag force from the external flow; Fint is the traction caused by the flow in the mucus on the foot of the snail and can be expressed as the integral of ⌸·n , where n u f fץ 0= at y = h , ͑6c͒ is outward normal to the foot of the snail. In the lubrication 2 ץ y limit, Eq. ͑10͒ reduces to

a3 ͑ ͒ h2,xx =−p at y = h2, 6d ␲ Ca ␮Vˆ 2n dh du wˆ ͩ w ͪ͵ ͫͯp 1 + ͯ ͬdx = Fˆ , ͑11͒ a dx dy drag where x and y are the horizontal and vertical coordinates of 0 y=h1 the system while z points out of the page, and u and v are the velocity components in the x and y directions, respectively. ͑ ͒ which is a scalar equation representing force balance in the x The shape of the snail foot is described by h1 x while ͑ ͒ direction. Here n is the number of waves generated by the h2 x denotes the unknown free surface shape. Note that ϵ␮ ˆ /␴ foot and wˆ is the width of the foot in the z direction. Physi- Ca Vw is not a capillary number in the traditional sense ͑ ͒ ˆ cally, the left-hand side of Eq. 11 is the propulsive force since Vw is not necessarily the characteristic speed of the that arises from the internal flow of the mucus and balances flow in the lubrication layer. In order to allow surface tension the drag from the external flow. Note that Eqs. ͑10͒ and ͑11͒ effects to remain relevant in the current problem, the curva- implicitly neglect the overlap regions between the internal ture term in Eq. ͑6d͒ is retained despite being multiplied by mucus flow and the external flow around the organism; we a3; this is a standard practice in thin film problems with will derive in Sec. III H the asymptotic limit in which this is surface tension ͑see, e.g., Ref. 29͒. a valid assumption. For convenience, we define a modified capillary number, ϵ␮ ˆ / 3␴ / 3 Ca Vw a =Ca a , so that the normal stress condition D. Solution for small-amplitude motion becomes In order to solve the model problem, we consider the 1 following limit for foot deformations. If ⌬Hˆ denotes the typi- h =−p at y = h . ͑7͒ 2,xx 2 ␧ϵ⌬ ˆ /␭ˆ Ca cal amplitude of the foot deformation, we define H and assume it to be small. Note that ␧ is a parameter inde- Then by integrating Eq. ͑5b͒ twice with respect to y and pendent of the geometrical aspect ratio a, as can be seen by ␧ applying necessary boundary conditions, we obtain an ex- considering the case where =0; in this limit, the dimension- ␧ pression for the velocity field in the mucus layer, less parameter is zero when the foot surface is flat, while a remains finite. We choose the foot shape as 1 1 1 u͑x,y͒ = h ͩh y − y2 + h2 − h h ͪ +1. ͑8͒ 2,xxx 2 1 2 1 Ca 2 2 ␧ ͑ ͒ h1 = sin x, 12 The resulting volume flux through the layer,

1 1 1 and solve for the associated layer proﬁle, h2, order by order Q = h − h + h ͫ ͑h − h ͒2ͩ h + h ͪ 2 1 2,xxx 2 1 1 2 as Ca 3 2 1 ͑ ͒ͩ ͪͬ ͑ ͒ + h1 h2 − h1 h1 − h2 , 9 ␧ ͑1͒ ␧2 ͑2͒ ͑␧3͒ ͑ ͒ 2 h2 =1+ h2 + h2 + O . 13

is constant since the mucus thickness does not vary with time in this moving reference frame. The resulting expression for the ﬂux is given by

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͑1͒ 2 ͑2͒ Q ͑␧h + ␧ h ͒ 1 ͑ ͒ ͑ ͒ ͑ ͒ 1 ͑ ͒ =1+ 2,xxx 2,xxx ͭ ͓1+␧͑h 1 − sin x͒ + ␧2h 2 ͔2ͫ1+␧ͩh 1 + sin xͪ + ␧2h 2 ͬͮ ␧͑ ͑1͒ ͒ ␧2 ͑2͒ 2 2 2 2 1+ h2 − sin x + h2 Ca 3 2 ͑1͒ 2 ͑2͒ ͑␧h + ␧ h ͒ ͑ ͒ ͑ ͒ ͑ ͒ 1 ͑ ͒ − 2,xxx 2,xxx ͭ␧ sin x͓1+␧͑h 1 − sin x͒ + ␧2h 2 ͔ͫ1+␧ͩh 1 − sin xͪ + ␧2h 2 ͬͮ, 2 2 2 2 Ca 2 ͑14͒

where Q=Q͑0͒ +␧Q͑1͒ +␧2Q͑2͒ +O͑␧3͒. Collecting terms of propulsive force is generated under strictly periodic bound- the same order, the leading order O͑1͒ simply states that ary conditions. Q͑0͒ =1, and then at order O͑␧͒ we obtain Instead, the three boundary conditions should be selected by the physical constraints on snail locomotion. Since the ͑ ͒ 1 ͑ ͒ Q͑1͒ = h 1 + h 1 − sin x. ͑15͒ moving gastropod is force- and torque-free, the first two con- 2 2,xxx 3Ca ditions should naturally be ͑ ͒ This third order linear ordinary differential equation has an ͚ Fy =0, 18a analytic solution given by x ͚ ␶ =0, ͑18b͒ ͑1͒ ͑1͒ ͩ ͪ h2 = Q + A1 exp − 1/3 C where F refers to forces in the y direction while ␶ is a torque ͱ y ͩ x ͪ ͩ 3x ͪ in the z direction. In all generality, the sum of all forces and + A exp / cos / torques acting on the snail must vanish at low Reynolds 2 2C1 3 2C1 3 numbers; the forces and torques from the thin film of mucus x ͱ3x C cos x + sin x must therefore balance the forces and torques generated by + A expͩ ͪsinͩ ͪ + , 3 2C1/3 2C1/3 C2 +1 the external flow around the body and those arising from gravity. For this analysis, we assume the snail does not ro- ͑ ͒ 16 tate, and that its shape is sufficiently symmetric that the ex- ϵ / ternal viscous torques and y force vanish. We also assume where C 1 3Ca, and A1, A2, and A3 are unknown constants. For convenience, Q͑1͒ is set to zero by arbitrarily setting the organism to be neutrally buoyant and homogeneous, so Qϵ1. Note that Q corresponds to the rate of mucus produc- the forces and torques due to gravity are zero. As a result, ͑ ͒ ͑ ͒ y z tion by the snail. Eqs. 18a and 18b only require the force and torque arising from the thin film to vanish. E. Boundary conditions The final boundary condition arises from consideration of the matching between the internal and the external flow The real challenge lies in identifying the three indepen- around the organism. By symmetry, we expect the swimming ϳ␧2 dent boundary conditions required to solve for A1, A2, and speed of the snail to be of order Vs . Since we are in the A3. As a logical starting point, we proceed by applying peri- Stokes regime, pressure differences across the moving gas- odic boundary conditions over each wavelength: tropod should scale linearly with the free stream velocity and ͑ ␲ ͒ ͑ ␲͑ ͒͒ ͑ ͒ h2,x 2 j = h2,x 2 j +1 , 17a

͑ ␲ ͒ ͑ ␲͑ ͒͒ ͑ ͒ h2,xx 2 j = h2,xx 2 j +1 , 17b patmatm ≡ 0 ͑ ␲ ͒ ͑ ␲͑ ͒͒ ͑ ͒ h2,xxx 2 j = h2,xxx 2 j +1 , 17c mucus where j ranges from 0 to n−1. In this limit, A1, A2, and A3 vanish, and these boundary conditions yield no motion of the snail, regardless of the value of Ca. Conducting a force bal- [1] ance on the mucus layer over one wavelength ͑as shown in p(2(2πjπ j) p(2(2π(j + 1))1)) Fig. 4͒ offers a simple explanation for this result: the peri- snail odic boundary conditions ensure that the pressure forces acting on the side control surfaces of the mucus layer precisely cancel. Since the top control surface of the mucus is exposed FIG. 4. ͑Color online͒ Free body diagram of a perfectly periodic mucus layer over one wavelength between nodes j and j+1. Pressures at these to ambient air pressure, there can be no net force that acts on ͑ ␲ ͒ ͑ ␲͑ ͒͒ ͑ ␲ ͒ nodes, p 2 j and p 2 j+1 , as well as the heights, h2 2 j and the bottom control surface. Thus, no equal and opposite force h ͑2␲͑j+1͒͒, are equal by the periodic boundary conditions. Above the ͓͑ ͔ ͒ 2 acts on the snail foot 1 in Fig. 4 , suggesting that no net mucus layer is open to atmosphere with patm set to zero.

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0.2

0.1

0 n /

p −0.1 n =30 o r prop p

F n =10 −0.2 n =5

−0.3

−0.4

10−3 10−2 10−1 100 101 102 CaCa (a)

10−5 1.6

10−5

— 10−10 — p p p o o r r

pro p

p pro –1 — F — F F n=5=5 n==55 −15 10 n=10=10 n==1010 3 −10 n=15=15 10 n==1515 n=20=20 n==2020 10−20 n=25=25 n==2525 n=30=30 n==3030 10−14 10−12 10−10 10−8 10−6 10−4 10−2 105 1010 CaCa CCaa (b) (c)

͑ ͒͑ ͒ FIG. 5. Color online a Dimensionless propulsive force Fprop, normalized by the number of wavelengths, n, as a function of the modiﬁed capillary number, ϵ␮ ˆ / 3␴ ͑ ͒ ͑ ͒ Ca Vw a , where the values of n range from 5 to 30 in increments of 5. In b and c , the absolute value of the dimensionless force Fprop is plotted on a logarithmic scale to show the power-law decay in the limits of Ca→0 and Ca→ϱ, respectively. The propulsive force exhibits a Ca−1 decay for large Ca, while it decays as Ca3 for small Ca.

2 occur at order ␧ as well. Thus, the pressure difference be- in particular, is a nontrivial function of Ca. Hence, unlike the tween the front and back of the snail is zero at O͑␧͒, the strictly periodic boundary condition case, the expression for order of our formulation, and the third boundary condition ͑1͒ h2 now contains a nonperiodic function that gives rise to a becomes nonzero propulsive force. p͑0͒ = p͑2n␲͒. ͑19͒ F. Crawling speed These three boundary conditions allow one to obtain a ͑1͒ To balance the thin-film propulsive force, it is necessary complete expression for h2 and subsequently solve for the ͑␧2͒ ˆ dimensionless propulsive force Fprop at O : to evaluate the external drag Fdrag caused by the motion of -the snail. For simplicity, the snail is modeled as approxi ץ 2n␲ dh1 u F = ͵ ͫͯp + ͯ ͬdx mately spherical, with radius Rˆ . Although approximate, yץ prop dx 0 y=h1 this model yields an order of magnitude estimate for the 2n␲ speed of our model snail. By nondimensionalizing Fˆ by ͵ ͓ ͑1͒ ͑1͒ ͑ ͑1͒ ͒ drag =3C − h2,xx cos x + h2,xxx h2 − sin x ␮ ˆ ͑ ͒ 0 Vwwˆ , the right-hand side of Eq. 11 becomes Fdrag Ϸ ␲ /␮* ˆ ͑2͒ ͔ ͑ ͒ 6 fVs , where Vs is the snail speed scaled by Vw and + h2,xxx dx. 20 ␮ ϵ␮/␮ * water, the viscosity ratio of mucus to water. A cor- ͑2͒ Note that the integral of h2,xxx vanishes by the matching pres- rection factor f accounts for the aspherical shape of the snail ͑ ͒ sure boundary condition, which is equivalent to h2,xx 0 as well as the influence of the free surface on the drag coef- ͑ ␲͒ ͑ ͒ =h2,xx 2n . Referring back to the constants in Eq. 16 , A1, ficient; for the present analysis, f will be treated as known

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90 pressure 3 80 p o r

prop p 70

F pressure f 60 (p) 0 50 40 30 –3

components o 20 shear CaCa 10 h2

10−2 10−1 100 101 CaCa h1 FIG. 6. Absolute magnitudes of components of dimensionless propulsive force due to pressure ͑solid line͒ and due to shear ͑dashed line͒ as a function 26 28 30 32 34 36 of Ca for n=10 ͑note that the shear force is negative͒. Hence, the total x propulsive force which is the sum of these two forces is nonzero only when (a) there is a difference between the two. 0 viscous for a given crawler. By combining this scaling with the O͑␧2͒ shear term in the force on the foot generated by the mucus layer, the following expression for Vs is obtained: –3 CaCa ␧2␮* Ϸ ͑ ͒ ͑ ͒ Vs Fprop Ca,n , 21 af h2 where Fprop, the total propulsive force function, is plotted in ͑ ͒ Fig. 5 a for different values of n. The exact formula for Fprop is not reproduced in this paper because it is long and not informative for the purpose of this analysis but is straightfor- h1 ward to calculate with symbolic packages. Note that the width of the snail foot, wˆ , is taken to be on the same order as 26 28 30 32 34 36 Rˆ ; thus, they drop out of Eq. ͑21͒. x (b) G. Results FIG. 7. ͑Color online͒ Dimensionless pressure ͑a, dashed line͒ and shear Figure 5 shows that the propulsive force vanishes in the stress ͑b, dashed line͒ within the mucus over two wavelengths for n=10. ͑ ͒ ͑ ͒ limits of both large and small surface tension. In the limit of The single dotted line in both a and b is the shape of the foot, h1, while the solid lines describe the shape of the interface, h2, for different values of infinitely large surface tensions ͑Ca→0͒, the interface be- Ca. Black arrows indicate the direction of increasing Ca. tween the air and mucus is undeformable and so is analogous to a flat surface that cannot sustain shear stress. A snail would simply slip on such a surface. The detailed behavior faster mode of locomotion for water snails. This points to a for small values of Ca is shown in Fig. 5͑b͒. The dimension- possible biological advantage of direct over retrograde ϳ3 waves. less force follows the power-law decay Fprop Ca for decreasing Ca for all values of n. Figure 6 quantifies the components of propulsive force due to pressure and shear. It is important to note that the With zero surface tension ͑Ca→ϱ͒, a pressure differ- ͑ ͒ ence across the interface cannot be sustained and so cannot force due to shear dashed line is negative. In the low Ca limit, these two components precisely cancel, leading to no drive the flow within the mucus; hence no propulsive force can be generated in this limit either. In this case, the force motion. When Ca is high, they both vanish. For intermediate ͉ ͉ϳ−1 values of Ca, a difference in the magnitudes of these forces follows for all n the power-law decay Fprop Ca as shown in Fig. 5͑c͒. results in a net propulsive force. Note the existence of a finite Note that the propulsive force goes from positive to value of Ca for which the propulsive force reaches zero, negative at a moderate value of Ca that, in the case of which is a surprising result of our model. n=10, is around 0.3. Physically, this implies that the snail Going back to the dynamic boundary condition Eq. ͑7͒, switches from retrograde waves to direct waves at this criti- one can calculate the pressure and shear stress distribution to O͑␧͒ inside the mucus layer. These are plotted in Fig. 7 for cal Ca. In addition, Fig. 5͑a͒ shows that the propulsive force ͑ ͒ n=10 along with the shape of the interface, h 1 . In the large exhibits two distinct maxima for retrograde and direct waves, 2 Ca limit, the interface shape exactly conforms to the shape of at values of Ca corresponding to 0.15 and 0.8, respectively. Since the maximum propulsive force for the direct waves is the foot h1; in the small Ca limit, the interface becomes flat. higher than that for the retrograde, the direct waves may be a At intermediate Ca, there exists an asymmetry in the inter-

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3 patmatm ≡ 0 pressure (p) 0 mucus

–3 CaCa p(0)(0) p(2(2nπ) h2 [1] snail

0123456 FIG. 9. ͑Color online͒ Free body diagram of an asymmetric mucus layer ͑ ͒ x across the foot of the snail. For simplicity, n=1 in this diagram. Pressures at the ends, p͑0͒ and p͑2n␲͒, are equal by the boundary condition; however, (a) they act over two different mucus thicknesses, resulting in a net pressure force. 3

pressure formed, it forces a lubrication ﬂow in the mucus layer above (p) 0 and leads to the deformation of the free surface. The resulting topography of the free surface, constrained by surface tension, is then exploited by the organism to generate a pro- –3 CaCa pulsive force.

h2 H. Matching internal and external flows In our analysis, we have neglected the fluid forces on the organism, Fmatch, arising from the intermediate matching re- h1 gion between the internal ͑mucus͒ and the external flows. 57 58 59 60 61 62 Since the propulsive force of the snail mostly arises from the x asymmetric shape of the free surface at the head and tail of (b) the foot, this requires further comment. Physically, since we are calculating the internal and ex- ͑ ͒ ͑ ͒ FIG. 8. Color online Dimensionless pressure dashed line and the inter- ternal flows separately, both need to be considered. In order face shape ͑solid line͒ in the front ͑a͒ and end ͑b͒ of the snail for n=10. The dotted line is the shape of the foot, and black arrows are in the direction of to first estimate the magnitude of Fmatch due to the external 30 decreasing surface tension. flow, we refer to the work by Berdan and Leal who studied the motion of a sphere near a deformable fluid-fluid interface. As an extension of previous work in which the interface ͑ ͒ 31,32 face shape, associated with the exponential term in Eq. 16 , is assumed to be flat, the work in Ref. 30 considers the that gives rise to the nonzero propulsive force. The interface limit of small interfacial deformation and its effects on the shape, pressure, and shear stresses are plotted for the first translating body. Unlike our current analysis, the velocity of ͑ wavelength and the last corresponding to the front and end the sphere is not governed by the shape of the free surface of the snail͒ for n=10 in Fig. 8. As shown in this figure, the but is fixed as Uˆ . The small parameter in this paper, ⑀ˆ, re- mucus thickness at the ends deviate substantially from ex- ⑀ˆ ␮ ˆ /␴ trapolated periodic values, creating an asymmetry between duces to a capillary number, = U , when gravitational the head and the tail. Thus, although the pressures at the ends effects are not included. Berdan and Leal showed that in the of the crawler are equal at O͑␧͒, there exists a net O͑␧2͒ case of a sphere moving parallel to the free surface, the de- pressure force acting on the side control surfaces, owing to formation of the interface only has a vertical force contribu- ͑⑀ˆ͒ the an O͑␧͒ difference in thickness of the mucus layer ͑see tion at O . In the current analysis, Fint and Fext, the forces ͑ ͒ ͑␧2͒ Fig. 9͒. To balance this net force, there has to be a force considered in Eq. 10 , are of O . Therefore, in order to acting on the bottom surface of the mucus layer; thus, there neglect Fmatch consistently, the following condition has to be exists an equal and opposite force acting on the foot of the satisfied: snail, corresponding to the propulsive force. ⑀ˆ2 Ӷ␧2, ͑22͒ As suggested by Fig. 5, surface tension is the essential ⑀ ␧ ingredient in this mode of locomotion, and the propulsive which requires one to consider how ˆ and are defined. ˆ ˆ ␧ϳ␮ ˆ /␴ force vanishes in both limits of asymptotically small ͑large Since U is Vs in our problem, we have ˆ Vs . Recalling Ca͒ and large ͑small Ca͒ surface tension. The snail would from Sec. III C, the capillary number Ca is defined in terms ˆ therefore have to tune the way it deforms its foot to exploit of the wave velocity Vw. Because it has been shown that ˆ / ˆ ␧2 ⑀ the property of the fluid-air interface. As the foot is de- Vs =Vs Vw scales as , ˆ can be expressed as

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␮Vˆ ␮Vˆ tween the free surface and the snail foot distinguishes water ⑀ˆ ϵ s ϳ ␧2 w ϵ ␧2Ca. ͑23͒ ␴ ␴ snail locomotion from that of their terrestrial counterparts. For the latter, the solid substrate on which the snail crawls is When one replaces Ca with a3Ca and rearranges the terms, fixed; hence, the shape of the snail foot alone determines the ͑ ͒ pressure and shear stresses generated within the mucus layer. the criterion to neglect Fmatch in Eq. 22 reduces to For water snails, however, the interface is deformed due to ␧2a6Ca2 Ӷ 1. ͑24͒ the flow created in the mucus by the foot undulation; the interface, in turn, affects the dynamics within the mucus Since a and ␧ are both small parameters asymptotically layer, creating pressure and shear stresses that act on the approaching zero in the lubrication analysis in the limit of foot. This nonlinear coupling between the foot geometry, sur- small deformation amplitude, ␧2a6 Ӷ1, Eq. ͑24͒ represents a face tension, and dynamics within the mucus layer makes the weak constraint on the validity of neglecting forces from the water snail locomotion a less straightforward mode of loco- intermediate region. motion. The second matching force to consider is that induced by A direct analogy exists between the thin film comprising the internal flow. Since there is, in general, a height difference between the mucus at the front and the back of the the mucus layer of water snails and those arising in coating snail, the fluid surface will be distorted at either end to match flows; for example, those used in photolithographic pro- with the flat surface far away. Our work will therefore be cesses to fabricate various electronic components. This class valid in the limit where the capillary forces resulting from of fluid problem has been well studied both experimentally33–35 and theoretically,33,34,36–40 an example of these distortions can be neglected, corresponding to an 41 asymptotic limit which we now characterize. which includes spin coating. Kalliadasis et al. used lubri- The two relevant length scales to consider for matching cation theory to show that in the limit of small Ca, the inter- the distorted fluid interface to the flat free surface in the facial features become less steep, an effect also captured by our model. Mazouchi and Homsy42 demonstrated that in the far field are the capillary length, ᐉ ϳͱ␴/␳g ͑␳ is the fluid c case of large capillary numbers, the shape of the free surface density͒, and the width of the snail, wˆ . The fluid interface nearly follows the topography. In the context of water snail ᐉˆ ᐉˆ ϳ ͑ᐉ ͒ will be distorted over a length where min c ,wˆ . The locomotion, we saw in Sec. III D that the free surface con- typical curvature pressure arising from surface distortion will forms to the shape of the foot in the same limit. be on the order of ϳ␴␦hˆ /ᐉˆ 2, acting on typical height differ- Our study is only the first step towards a quantitative ence ␦hˆ between the free surface near the snail and the far- understanding of gastropod crawling beneath free surfaces. It field height of the free surface, and therefore contributes to a is significant in that we have demonstrated the plausibility of force on the snail ͑per unit width͒ on the order of locomotion with the minimal ingredients: Newtonian fluids ϳ␴͑␦hˆ͒2 /ᐉˆ 2. Since ␦hˆ ϳ␧Hˆ ϳ␧a␭ˆ , the capillary force is on and small-amplitude deformations. Nevertheless, outstanding the order of ϳ␴␧2a2␭ˆ 2 /ᐉˆ 2. This force has to issues remain. In the case of adhesive snail locomotion on land, the non-Newtonian properties of snail mucus, such as a be compared with that arising from the external flow, given 12 finite yield stress and finite elasticity, play an essential role. by p Rˆ ͑again, per unit width͒ where p is the typical ext ext Non-Newtonian mucus is likewise expected to have a sig- magnitude of the pressure outside the organism as it is crawl- nificant effect for water snails. Furthermore, more systematic ϳ␮ ˆ / ˆ ˆ ϳ␮ ˆ ϳ␧2␮ ˆ ing. Since pext Vs R we have pextR Vs Vw. observational studies are needed to identify which water The matching condition becomes, therefore, ␴␧2a2␭ˆ 2 /ᐉˆ 2 snail species exhibit which modes of “inverted crawling.” As Ӷ␧2␮ ˆ 23,24 22 Vw, which is equivalent to reported by Copeland and Deliagina and Orlovsky, some species of water snails rely entirely on cilia for propul- ˆ 2 ˆ 2 2 R /ᐉ Ӷ aCan , ͑25͒ sion beneath the free surface. If such ciliary motion results in no free surface deformation, the physical mechanism exam- where we have used the estimate Rˆ ϳn␭ˆ . The second match- ined in this paper is of little relevance, and a closer look at ing condition, Eq. ͑25͒, requires that the number of wave- the cilia-induced flow is suggested. Alternatively, the non- lengths along the snail’s foot, n, be sufficiently large. Newtonian properties of the mucus may prove to be significant in this case. Categorizing different species according to IV. DISCUSSION their propulsion mechanism of crawling ͑i.e., cilia versus ͒ In this paper, we have presented a simplified model of muscle contraction and the constitutive properties of their water snail locomotion. The physical picture that emerges is mucus would provide a more complete physical picture of the following: the undulation of the snail foot causes normal this intriguing form of locomotion. stresses that deform the interface and drive a lubrication flow. The resulting stress distribution couples to the topogra- ACKNOWLEDGMENTS phy of the snail foot, leading to a propulsive force. This force vanishes in the limit of Ca→0, where the interface is flat, We thank Brian Chan and David Hu for the pictures. We and of Ca→ϱ, where the topographies of the interface and also thank George M. Homsy for helpful discussions. This the snail foot precisely match. A finite propulsive force is work was supported in part by NSF ͑Grant No. CTS- obtained for intermediate values of Ca. This interplay be- 0624830͒.

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