  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 non-zero for moderate
values of Capillary number but vanishes in the limits of high and low Capillary
numbers. 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.
