Brian Chan
Movement of a waving sheet over a thin layer of viscous fluid in the lubrication limit
Model of sea snail locomotion using lubrication pressures in a thin layer of Newtonian fluid
INTRODUCTION
Gastropods employ a vastly different mode of locomotion than most other land animals. Using their ‘foot’, a flexible, extensiblesheet of muscle, the snailrides on a layer of its own mucus.The motion of the snail is directly coupled to stresses within this viscous fluid film.
When snails move over a transparent substrate, alternate light and dark bands can sometimes be seen moving along the sole the foot. These bands, termed waves and interwaves, move in the line of motion and, depending on the species of snail, can be moving front-to-back or back-to-front (Denny). Waves that move in the direction of the snail’smotion are classified as direct waves, whereas waves moving in the opposite motion are known as retrograde waves. Waves can also either be monotaxic or ditaxic; monotaxic waves span the entire foot, while ditaxic waves are of alternate phase on right and left sides of the foot.
M.W. Denny has made numerous observations and done several experiments on the adhesive motion of the banana slug, Ariolimax columbianus, which uses direct waves. It was found that most land-dwelling snails use direct waves for motion.
It has been observed that most aquatic snails observed,periwinkles (Littorina sp.), and several limpets mainly used retrograde waves. We hypothesize that these aquatic snails may partially rely on lubrication pressures to generate thrust forces.A large volume of lubricating fluid is required to sustain motion using sinusoidal wave motion; therefore it may only be feasible for aquatic species. This hypothesis is reinforced by the observation thatall of the land snail species studied,Ariolimax columbianus, Helix aspersa, Limax maximus, Achatinaachatina, only employed direct waves in locomotion.
Using lubrication theory, valid for analyzing viscous fluids within thin gaps, we modeled the possible motion of a moving snail.
We have built a mechanical snail that uses a mechanically actuated waving foot to move over a layer of glycerol, a viscous Newtonian fluid.
THEORY:
We will model the motion of the snail’s foot as a lubrication layer. In the lubrication limit,
-Pressure only varies in the x-direction
-x-velocities are assumed to be much greater than y-velocities
-inertial effects are negligible.
First we consider the lab reference frame.
The velocity of the ground is zero, while the snail moves to the left with -v_s. The wave of the snail’s moves to the right, the speed of the wave crests in relation to the snail is v_w, in relation to the lab frame it is v_w-v_s.
An analysis of this motion is greatly simplified when we follow the wave crests.Because the fluid is incompressible and the shape of the boundary is now fixed, the flow rate Q is constant for all points along X. In the reference frame of the wave, the top boundary moves with -v_wave and the ground moves with -v_w+v_s
In the lubrication approximation, the governing relation between pressure and velocity is:
Integration of the pressure-velocity equation (1) gives a parabolic velocity profile that follows the equation
where the constants A and B are solved using the boundary condition imposed by the no-slip condition on the snail foot and the ground.
B is easily found by setting (y)=0
Substituting back into the equation for B and solving for A
The flow rate, Q, is the integral of u dy and it is a constant.
The pressure is given by the integral of the pressure gradient.
where:
If h, x are made non-dimensional
h* = h/a
x* = x/L
If we assume a periodic model, there is no net pressure difference over a wavelength.This allows us to solve for Q.
The normalized pressure is shown in the following graph:
There is a high pressure zone immediately before the dip in the sine wave, where the wave trough is compressing the fluid, and a low pressure zone behind the dip, where the trough is pulling apart the fluid.The magnitudes of the pressure peak and dip increase as the average height approaches the amplitude of foot motion, since the snail must force fluid through a thinner gap.
Thrust comes from the pressure forces acting on the sloped sections of the snail’s foot, expressed as the integral:
The vertical force is the integral of pressure per unit of foot.
However, the pressure can have any arbitrary unknown constant due to the integration of dp/dx, and depends on the end effects of a finite length snail, and leakage out the sides when the width is finite length. This force (and thus the relationship between snail weight and gap height) was calculated using a computer code to simulate a finite length snail, and the experiments were run with side walls to minimize the leakage of fluid out the sides of the snail.
STEADY STATE SNAIL VELOCITY:
The steady state velocity of the snail arises when the shear forces from fluid viscosity balance the propulsive force from pressure.
rearranging terms, we can see that the ratio of snail velocity to the wave velocity is a function only of the average height-to-amplitude ratio of the foot.
EXPERIMENT: ROBOSNAIL
We have constructed a mechanical snail that uses the waving foot motion. The Robosnail is capable of speeds up to 0.6 cm/s on glycerol, a Newtonian fluid.
Design:
The Robosnail is powered by an external DC power source, capable of 1.5 , 3.0 and 4.5 volts. The motor is a DC Mabuchi connected to a variable-speed gear box. A pulley connects the gear box to a shallow helix made of brass, which passes through an array of aluminum plates perforated with slots. The sheets are directly glued onto a flexible foam sheet. When the helix is spun by the motor and gearbox, it causes the plates to translate up and down inside their tracks, in a moving sinusoidal wave pattern that is transferred directly to the foam sheet.
When placed on a viscous layer of glycerol, the robosnail moves in the opposite direction as the waves it generates, as was predicted by the lubrication model.
1. Find whether water snails lift the foot
2. Compare the energy required to move a certain distance using flapping sheet with water, or flat sheet on mucus.
References:
Yasukawa, Saito, Kenichi, Ito: Learned in stomach foot walking of the snail development of the membrane structural soft somatotype actuator which