^{1}, John W. M. Bush

^{2}, A. E. Hosoi

^{1}and Eric Lauga

^{3,a)}

### Abstract

Land snails move via adhesivelocomotion. 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, adhesivelocomotion 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.

We thank Brian Chan and David Hu for the pictures. We also thank George M. Homsy for helpful discussions. This work was supported in part by NSF (Grant No. CTS-0624830).

I. INTRODUCTION

II. OBSERVATIONS

III. MODEL

A. Assumptions

B. General equations

C. Lubrication analysis

D. Solution for small-amplitude motion

E. Boundary conditions

F. Crawling speed

G. Results

H. Matching internal and external flows

IV. DISCUSSION

### Key Topics

- Land transportation
- 30.0
- Free surface
- 29.0
- Boundary value problems
- 17.0
- Surface tension
- 13.0
- Biological movement
- 12.0

## Figures

Snail (*Sorbeoconcha physidae*) crawling smoothly underneath the water surface while the surface deforms. Note the surface deflection associated with the undulatory waves propagating from nose to tail along its foot. Photo courtesy of David Hu and Brian Chan (MIT).

Snail (*Sorbeoconcha physidae*) crawling smoothly underneath the water surface while the surface deforms. Note the surface deflection associated with the undulatory waves propagating from nose to tail along its foot. Photo courtesy of David Hu and Brian Chan (MIT).

A trail of mucus behind the snail crawling upside down beneath the free surface. Photo courtesy of David Hu and Brian Chan (MIT).

A trail of mucus behind the snail crawling upside down beneath the free surface. Photo courtesy of David Hu and Brian Chan (MIT).

A close-up view of the mucus and the snail foot undergoing a simple sinusoidal deformation of wavelength . The prescribed shape of the snail foot is denoted as ; the resultant shape of the free surface, , is to be solved for. The known constant speed of the wave, , is set relative to the snail that is translating with an unknown speed, . In laboratory frame (a), the wave is moving in the negative direction with while the snail is moving in the positive direction with . In the frame moving with the wave (b), the snail body appears to move in the positive direction with .

A close-up view of the mucus and the snail foot undergoing a simple sinusoidal deformation of wavelength . The prescribed shape of the snail foot is denoted as ; the resultant shape of the free surface, , is to be solved for. The known constant speed of the wave, , is set relative to the snail that is translating with an unknown speed, . In laboratory frame (a), the wave is moving in the negative direction with while the snail is moving in the positive direction with . In the frame moving with the wave (b), the snail body appears to move in the positive direction with .

Free body diagram of a perfectly periodic mucus layer over one wavelength between nodes and . Pressures at these nodes, and , as well as the heights, and , are equal by the periodic boundary conditions. Above the mucus layer is open to atmosphere with set to zero.

Free body diagram of a perfectly periodic mucus layer over one wavelength between nodes and . Pressures at these nodes, and , as well as the heights, and , are equal by the periodic boundary conditions. Above the mucus layer is open to atmosphere with set to zero.

(a) Dimensionless propulsive force , normalized by the number of wavelengths, , as a function of the modified capillary number, , where the values of range from 5 to 30 in increments of 5. In (b) and (c), the absolute value of the dimensionless force is plotted on a logarithmic scale to show the power-law decay in the limits of and , respectively. The propulsive force exhibits a decay for large , while it decays as for small .

(a) Dimensionless propulsive force , normalized by the number of wavelengths, , as a function of the modified capillary number, , where the values of range from 5 to 30 in increments of 5. In (b) and (c), the absolute value of the dimensionless force is plotted on a logarithmic scale to show the power-law decay in the limits of and , respectively. The propulsive force exhibits a decay for large , while it decays as for small .

Absolute magnitudes of components of dimensionless propulsive force due to pressure (solid line) and due to shear (dashed line) as a function of for (note that the shear force is negative). Hence, the total propulsive force which is the sum of these two forces is nonzero only when there is a difference between the two.

Absolute magnitudes of components of dimensionless propulsive force due to pressure (solid line) and due to shear (dashed line) as a function of for (note that the shear force is negative). Hence, the total propulsive force which is the sum of these two forces is nonzero only when there is a difference between the two.

Dimensionless pressure (a, dashed line) and shear stress (b, dashed line) within the mucus over two wavelengths for . The single dotted line in both (a) and (b) is the shape of the foot, , while the solid lines describe the shape of the interface, , for different values of . Black arrows indicate the direction of increasing .

Dimensionless pressure (a, dashed line) and shear stress (b, dashed line) within the mucus over two wavelengths for . The single dotted line in both (a) and (b) is the shape of the foot, , while the solid lines describe the shape of the interface, , for different values of . Black arrows indicate the direction of increasing .

Dimensionless pressure (dashed line) and the interface shape (solid line) in the front (a) and end (b) of the snail for . The dotted line is the shape of the foot, and black arrows are in the direction of decreasing surface tension.

Dimensionless pressure (dashed line) and the interface shape (solid line) in the front (a) and end (b) of the snail for . The dotted line is the shape of the foot, and black arrows are in the direction of decreasing surface tension.

Free body diagram of an asymmetric mucus layer across the foot of the snail. (For simplicity, in this diagram.) Pressures at the ends, and , are equal by the boundary condition; however, they act over two different mucus thicknesses, resulting in a net pressure force.

Free body diagram of an asymmetric mucus layer across the foot of the snail. (For simplicity, in this diagram.) Pressures at the ends, and , are equal by the boundary condition; however, they act over two different mucus thicknesses, resulting in a net pressure force.

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