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 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.
(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.
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.
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.
Article metrics loading...
Full text loading...