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Brian Chan
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INTRODUCTION
Snails and slugs have a mode of locomotion very different from
most other animals. While the majority of creatures use legs to
step across solid surfaces, snails and slugs slide steadily
over surfaces, gliding on a thin layer of mucus. The mechanism
of their propulsion is not immediately obvious. As a snail
travels over a transparent surface, one notices a traveling
pattern of waves along the underside of its muscular foot
(from Japanese page)
We will model the motion of the snail’s foot as a lubrication
layer. As the wave moves along the bottom of the snail’s foot,
the pressure distribution should generate a thrust force on the
sloped areas of the snail’s foot.
THEORY:
first we consider the reference frame of snail:
The x- velocity of the top boundary is 0 while the ground moves
with v_snail. The traveling wave moves with v_wave (note that
v_wave is with respect to the snail and not the ground.)
Now consider the reference frame of the traveling wave
In this reference frame, q, the volume flow rate, is a constant
value. The top boundary moves with -v_wave and the ground moves
with -v_wave+v_snail.
The average of top and bottom velocities is
v
v av = -v wave + snail
2
If we assume a periodic model, there is no net pressure
difference over a wavelength.

dp
0 dx dx  0
use the lubrication equations to find p(x):
split the model to two parts
1) The first part consists of a flow that has no dp/dx where
the height is h_0. The volume flow rate is:
Q0  vav h0
At x locations where the height of the snail foot is greater
than h_0, there is a negative pressure gradient caused by a
backflow imposed by volume flow conservation. At x where h is
less than h_0, there is a positive pressure gradient because
more fluid needs to flow through a smaller gap.
Q0  vav h 
dp0 h3
dx 12
x h 

a3  sin(2 )  0 
x h  dp

a

Q0  vav a  sin(2 )  0   0 

a  dx
12

3
Solving for dp/dx:
x
sin(2 )
dp0 12 vav


3
dx
a2 
x h0 
 sin(2 )  

a

2) The second part consists of an unknown constant volume flow
rate Q1 to balance the pressure so that there is no net
pressure increase over one wavelength.
Q1 
h3 dp1
=constant
12 dx
dp1 Q112 
1

3
3
dx
a 
x h0 
 sin(2 )  

a

The total pressure is given by the integral
x
x
dp
dp
P( x)   1 dx   2 dx
dx
dx
0
0
 

x
x 

sin(2 )
12  vav  

dx  Q1
P( x) 
3
2

0  
a
avav
x h  
   sin(2 )  0  

a 
  

 

 
1

 dx 
3
0  
 
h
x
0 
  sin(2  )  a   
  

x
The net pressure change over one wavelength is zero:
P (0)  P ( )


dp
dp
0 dx1 dx  0 dx2 dx  0
 


 
x
 

 
sin(2 )

12  vav  
Q1 
1


 dx   0
P ( ) 
dx 
3
3
a 2  0  
avav 0  
x h0  
x h0   
   sin(2 )   
  sin(2 )    

a 

a   

  
Q1
we solve for the constant
using the above relation:
avav


x


sin(2 )


dx

3

x h  
0 
  sin(2 )  0  

a 
Q1

 
avav




1

 dx
3
0  
x h0  
  sin(2  )  a  
 

which is only a function of h_0/a
P ( )
The normalized pressure
is shown in the following graph:
  vav  
 2 
 a 

there is a high pressure zone immediately before the dip in the
sine wave, where the wave motion is compressing the fluid, and
a low pressure zone behind the dip, where the wave motion is
pulling apart the fluid.
The horizontal force (thrust) comes from the pressure forces
acting on the sloped sections of the snail’s foot, expressed as
the integral:

Fx   P
0
dy
dx
dx
 

x
 x* 

sin(2

)
12  vav  1  

dx  Q1
Fx 
3


0 
a
avav
x h  
0 
   sin(2 )  0  

a 
  
or
v   h 
Fx  av f 0  0 
a
a
where






1

 dx *  dy * dx *
3
0  
  dx * 
x h0  

  sin(2 )   

a 


x*
 


x
 x* 

sin(2

)
x* 
1
h 

dx  Q1 
f 0  o   12    
3
0


avav 0  
a
x h  
0 
   sin(2 )  0  
  sin(2

a 

  
Fx
The normalized x force is
on one
  vav  


 a 
is plotted as a function of h_0/a:




1
 dx *  dy * dx *
3
  dx * 
x h0  
)   


a 

wavelength of snail foot
likewise the vertical force is the integral of pressure per
unit of foot. Both forces increase as the mean height
approaches the amplitude of the wave.
STEADY STATE SNAIL VELOCITY:
the steady state velocity of the snail arises when the shear
force from fluid viscosity equals the propulsive force from
pressure.
the shear force contributions can once again be split into two
parts:
1)shear force resulting from v_av on both top and bottom
boundaries, which occurs when there is a nonzero pressure
gradient.

Fs1    dx
0
from lubrication (couette flow):
du h dp
 

dy 2 dx




x



sin(2 )
1 
6  vav  
1

  Q1 
 dx *
Fs1 
3
3





a 0 
avav 
x h 
x h  
  sin(2 )  0  
  sin(2 )  0   

a 

a   

 
or Fs1 
 vav 
a
 h0 
f1  
 a 


x


sin(2 )
 h0 



  Q1
where f1    6 
3
 
a
x h   avav
0
  sin(2 )  0  

a 
 
1




1

 dx *
3

x h  
  sin(2 )  0   

a   

2)shear force from equal opposing velocities on the top and
bottom boundaries +- delta_v = v_snail/2: The volume flow and
pressure is zero for this component.
du vsnail
 

dy
h
1
1
Fs 2   vsnail   dx *
h
0




v 
1
dx *
Fs 2  snail  
h
a




0
0
  sin(2 x*)  a  


1
or Fs 2 
 vsnail 
a
h 
f2  0 
a
The pressure force balances the sum of the shear forces from
both contributions:
Fx  Fs1  Fs 2
 vav 
a
 h  v   h  v   h 
f 0  0   av f1  0   snail f 2  0 
a
a
a
a
a
h 
h 
h 
vav f 0  0   vav f1  0   vsnail f 2  0 
a
a
a
vsnail
2
solving for v_snail/v_wave gives
vsnail
f 0  f1

vwave  f 0  f1 

 1
 2 
which is only a function of h_0/a:
since vav  vwave 
As the graph shows, the snail velocity approaches the wave velocity as the mean height
approaches the amplitude of the wave. This correlation applies to the two-dimensional model,
corrections must be made to account for edge effects. Real snails (and robosnail) do not have a
uniform wave on their underside- only the central section shows the wave when a moving snail
is seen from below.
EXPERIMENT: ROBOSNAIL
The mechanical snail is actuated by a DC motor, powered by two 1.5V AA batteries. The
underside is made of latex rubber, to which are attached 25 plastic plates which give it a
traveling wave motion. When the foot is geared to produce a traveling wave at 0.5 inch/s, the
snail travels at 0.14 inch/s
When the wave speed is increased to 1.5 inch/s, the snail moves at 0.23 inch/s
The ratio of v_snail to v_wave is 0.28 and 0.15 respectively. The height smaller than 1.5 times
the amplitude (from visual inspection) so the snail velocity should have been at least half that
of the wave velocity according to the two dimensional theory, however we must account for
viscous drag on the extra area of foot that was not used for driving the snail.
(to be continued)
References:
Yasukawa, Saito, Kenichi, Ito: Learned in stomach foot walking of the snail development of the
membrane structural soft somatotype actuator which
http://www.mec.m.dendai.ac.jp/mechatronics_e/introduction/kougai/h8/h8kata.htm
(there may have been some faulty translation)