Optimizing Locomotion - American Physical Society

Transcription

Optimizing Low Reynolds
Number Locomotion
Anette (Peko) Hosoi
Hatsopoulos Microfluids Laboratory, MIT
What’s in This Talk?
1. Optimal stroke patterns for 3-link swimmers
2. Building a better snail
Swimming
Crawling
2
optimal three-link
Model of the swimmer.Our modelswimmer.
of the three-link swimmer inclu
Modelthe
of the
swimmer.model ofofthethe
threeof the surrounding flow,
geometry
andOur
kinematics
swi
Tiny Swimmers
of the dynamics
surrounding
flow,
the geometry
kine
equations governing
of the
system.
Each linkand
in the
sw
the dynamics
of the system.
rigid slender bodyequations
of lengthgoverning
2l and radius
b with characterisitic
asp
rigid slender
body
of lengthto2lbeand
radius
Figure 1). The Reynolds
number
is assumed
small,
Reb=with
ρU l
Figure
1).swimmer,
The Reynolds
number
is assumed
b
characteristic speed
of the
ρ is the
fluid density
and µtothe
characteristic
speed
of the effects,
swimmer,
is the
fluid
effects are thus neglected
relative
to viscous
andρthe
hydrody
effects
are thus neglected relative to viscous effect
governed by Stokes
equations
governed by Stokes equations
∇ · u = 0 , −∇p + µ∇2 u = 0 ,
subject to the boundary conditions
∇ · u = 0 , −∇p +
u = Us on S , u → 0 at ∞
u = Us on S , u →
where u and p are the velocity and the pressure fields in the flu
is the local velocity
at the
surface,
of the
swimmer.
Thepressur
hydro
where
u and
p areS,the
velocity
and the
is the local velocity at the surface, S, of the swi
3
3
3
First two images from http://www.astrographics.com/ - Dennis Kunkel
http://www.btinternet.com/~stephen.durr/volvoxtwo.html - Stephen Durr
heorem
optimal three-link
Modelthe
of the
geometry
andOur
kinematics
swi
Tiny
eed non-reciprocal
motion to Swimmers
swim at low Reynolds
of the dynamics
surrounding
flow,
the geometry
kine
equations governing
of the
system.
Each linkand
in the
sw
•
motion can
tion.
Figure
1).swimmer,
The Reynolds
number
is assumed
tothe
bE
characteristic
speed
of the
ρleaves
is the
fluid density
Tonyand
S. µYu,
Low
Re: reciprocal
motion
Ane
characteristic
speed
of the effects,
swimmer,
is the
fluid
effectsgyrating
are thus neglected
relative
to viscous
andρLauga,
the
hydrody
you
in place.
(Peko) E. Ho
effects
governed by Stokes
equations
2
∇ · u = 0 , −∇p + µ∇Introduction
u=0,
callop Theorem
Life at
Swimming
the dynamics
of the
system.
of lengthgoverning
2l and radius
asp
Around
th
Scallop
rigid slender
body
of lengthto2lbeand
radius
with
number
is assumed
small,
Reb=Theo
ρU l
∇ · u = 0 , −∇p +
Background
Scallop Theorem
Wiggins
u = Us on S , u → 0 at
∞ & Goldste
u =Experiments
Us on S , u →
Summary
at the
surface,
of the
swimmer.
Thepressur
hydro
where
u and
p areS,the
velocity
and
the
3
3
The 3-link swimmer
4
heorem
optimal three-link
Modelthe
of the
geometry
andOur
kinematics
swi
Tiny
eed non-reciprocal
motion to Swimmers
swim at low Reynolds
of the dynamics
surrounding
flow,
the geometry
kine
equations governing
of the
system.
Each linkand
in the
sw
•
motion can
tion.
Figure
1).swimmer,
The Reynolds
number
is assumed
tothe
bE
characteristic
speed
of the
ρleaves
is the
fluid density
Tonyand
S. µYu,
Low
Re: reciprocal
motion
Ane
characteristic
speed
of the effects,
swimmer,
is the
fluid
effectsgyrating
are thus neglected
relative
to viscous
andρLauga,
the
hydrody
you
in place.
(Peko) E. Ho
effects
governed by Stokes
equations
2
∇ · u = 0 , −∇p + µ∇Introduction
u=0,
callop Theorem
Life at
Swimming
the dynamics
of the
system.
of lengthgoverning
2l and radius
asp
Around
th
Scallop
rigid slender
body
of lengthto2lbeand
radius
with
number
is assumed
small,
Reb=Theo
ρU l
∇ · u = 0 , −∇p +
Background
Scallop Theorem
Wiggins
u = Us on S , u → 0 at
∞ & Goldste
u =Experiments
Us on S , u →
Summary
at the
surface,
of the
swimmer.
Thepressur
hydro
where
u and
p areS,the
velocity
and
the
3
3
The 3-link swimmer
4
rem
Scallop Theorem
orem
Background
Wiggins
& Goldstein
Scallop Theorem
3-link Swimmer
Experiments
Wiggins & Goldstein
Experiments
Summary
Summary
estigators
• Purcell (1977): proposed design
• Becker, Koehler and Stone (2003): optimized
• L.E.
vestigators
Becker,
S.A. Koehler,
and
H.A. Stone
geometry
(arm length/body
length and
stroke
••
••
•
angle)
L.E.
Becker,
S.A.A.E.
Koehler,
D.S.W.
Tam and
Hosoiand H.A. Stone
D.S.W.
Hosoi
B.
ChanTam
and and
A.E.A.E.
Hosoi
B. Chan and A.E. Hosoi
5
rem
Scallop Theorem
orem
Background
Wiggins
& Goldstein
Scallop Theorem
3-link Swimmer
Experiments
Wiggins & Goldstein
Experiments
Summary
Summary
estigators
• L.E.
vestigators
Becker,
S.A. Koehler,
and
H.A. Stone
geometry
(arm length/body
length and
stroke
••
••
•
angle)
L.E.
Becker,
S.A.A.E.
Koehler,
D.S.W.
Tam and
Hosoiand H.A. Stone
D.S.W.
Hosoi
B.
ChanTam
and and
A.E.A.E.
Hosoi
B. Chan and A.E. Hosoi
5
rem
Scallop Theorem
orem
Background
Wiggins
& Goldstein
Scallop Theorem
3-link Swimmer
Experiments
Wiggins & Goldstein
Experiments
Summary
Summary
estigators
• L.E.
vestigators
Becker,
S.A. Koehler,
and
H.A. Stone
geometry
(arm length/body
length and
stroke
••
••
•
angle)
L.E.
Becker,
S.A.A.E.
Koehler,
D.S.W.
Tam and
Hosoiand H.A. Stone
D.S.W.
Hosoi
B.
ChanTam
and and
A.E.A.E.
Hosoi
Can we do
B. Chan
and A.E. Hosoi
better?
5
Optimizing Kinematics
6
Fixed geometry
6
Fixed geometry
6
Fixed geometry
6
2
1
0
-1
-2
-2
-1
0
1
2
1
FIG. 2: Stroke sequences of three-link swimmers in the (Ω1 , Ω
?
square. Small swimmer diagrams correspond to successive config
efficiency, (−) optimal velocity and (−) the optimal ‘Purcell0s
Fixed geometry
-1
the stroke. The swimmer moves to the left when the trajectory
to the right otherwise.
6
-2
2
1
0
-1
-2
-2
-1
0
1
2
1
FIG. 2: Stroke sequences of three-link swimmers in the (Ω1 , Ω
?
square. Small swimmer diagrams correspond to successive config
efficiency, (−) optimal velocity and (−) the optimal ‘Purcell0s
Fixed geometry
-1
the stroke. The swimmer moves to the left when the trajectory
6
Kanso and Marsden (2005) - 3-link fish
Berman and Wang (2006) - insect flight
-2
1represented by a single closed curve in this space. A stroke patte
moves at a time appears as a square and will be referred to as the ‘
Model Swimmer
0original pattern proposed by Purcell (see Figure 2); it is also the
in the study by Becker et al [1].
-1 The hydrodynamic forces and torques, Fi = (F x , F y , τi ), on eac
i
i
2b
prescribe
j=1
1
As expected from the linearitykinematics
of Stokes equations (1), the force ve
2li
F(s)
U(s)
equations (3) and (4) integrated over each link
-2
3
-2
-1
0
1! 2 2
"
Fi =
(F · x, F · y, R × F)ds =
Aji V
Fi =
(Fix , Fiy , τi )
FIG. 2: Stroke sequences of three-link swimmers in the (Ω1 , Ω2 )-phase plane for: (−) opt
Λ(s)
form and are linear functions of the velocity. The coefficients of the
efficiency, (−) optimal velocity and (−) the optimal ‘Purcell0stroke’ which corresponds to
analytically for i = j and numerically using Gauss quadrature for
square. Small swimmer
diagrams correspond to successive configurations of the swimmer du
Ω2
In the low Reynolds number regime, the swimmer is force- and
-1
the stroke. The swimmerΛmoves
to slender
the leftbody
when
theinteracts
trajectory
followed
counterclockwise
i
the
only
withisthe
surrounding
flow and the
s=0
y
Ω1
R (s)
x
2li
vanish, thus
s = 2lhydrodynamical forces and torques
Solve
linear system
-2
0
1
3
3-2# "
3 -1 $
Vi = (ẋi , ẏi , Θ̇i )
"
"
force
Ẋj = 0 . opti
Fi =
Aji average
where V is the swimming speed averaged over one stroke
and Φ is the
balance
i=1
j=1
i=1
power
associated with the stroke.
resistive force
theory
- Lowest order:mechanical
Constraint:
links
are
attached
Equations
(5, 6 and
8) form
of
nine
first
order
FIG.
2: Stroke
sequences
ofa system
three-link
swimmers
in differen
the (Ω
Optimization
procedure.Without
lost
of
generality,
the
stroke
can
be
parametrized
NextO.order:
can and
incorporate
effects
integrated using a fourth order Runge-Kutt
10] J.-Avron,
Kenneth,
D. Oaknin,
New of
journal unknowns,
of physics which
7, 234is(2005).
efficiency, (−) optimal velocity and (−) the optimal ‘Purce
slenderness and
betweenΩlinks
twointeractions
periodic functions
Ω2 of period τform
. These
periodic functions
canofbe
1 andnon-dimensional
using two
theSwimming
characteristic
half-length
anre
a
velocities
11] E. Purcell, American journal of physics 45 (1), 3 (1977).
square. Small swimmer diagrams correspond to successive co
sented as a Fourier series. For
regular
functions,
Fourier
coeffici
length,
lref ,and
and differentiable
the period of the
stroke
as athe
reference
time,
τ.
and efficiencies
12] R. Cox, Journal of fluid mechanics 44 (4), 791 (1970).
the stroke. The
swimmeris moves
left when
the traject
decay rapidly and thus our optimization
procedure
based to
onthe
finding
the optimal
fir
13] D. Bertsekas, Nonlinear programming, 3rd Edition (Athena Scientific, Belmont, MA, 2005). 5
7 In
to the
right
otherwise.
coefficients of the Fourier series.
addition
to the stroke pattern, the geometry of
Optimized Stroke Patterns
•
•
Two cost functions
- Efficiency
- [useful work]/[energy dissipated]
- Unique parametrization that
-
optimizes efficiency for a given
curve
Speed
Symmetry axes
8
•
•
2
Two cost functions
- Efficiency
-
1
curve
0
Speed
Symmetry axes
-1
Distance
Efficiency
Arm Ratio Slenderness
Purcell
0.483
0.0077
0.809
Optimal
0.623
0.013
0.933
-2
-2
-1
0
1
2
As slender
as possible
FIG. 2: Stroke sequences of three-link swimmers in the (Ω1 , Ω2 )-phase pla
8
•
•
2
Two cost functions
- Efficiency
-
1
curve
0
Speed
Symmetry axes
-1
Distance
Efficiency
Purcell
0.483
0.0077
0.809
Optimal
0.623
0.013
0.933
-2
-2
-1
0
1
2
As slender
as possible
8
•
•
2
Two cost functions
- Efficiency
-
1
curve
0
Speed
Symmetry axes
-1
Distance
Efficiency
Purcell
0.483
0.0077
0.809
Optimal
0.623
0.013
0.933
-2
-2
-1
0
1
2
As slender
as possible
8
•
•
2
Two cost functions
- Efficiency
-
1
curve
0
Speed
Symmetry axes
-1
Distance
Efficiency
Purcell
0.483
0.0077
0.809
Optimal
0.623
0.013
0.933
-2
-2
-1
0
1
2
As slender
as possible
8
•
•
2
Two cost functions
- Efficiency
-
1
curve
0
Speed
Symmetry axes
-1
Distance
Efficiency
Purcell
0.483
0.0077
0.809
Optimal
0.623
0.013
0.933
-2
-2
-1
0
1
2
As slender
as possible
8
3-Link Race
D = distance
W = Total “work”
(viscous dissipation)
Thin line = path of
center of mass
Distance
Efficiency
Optimize geometry
Purcell
0.483
0.0077
0.809
Optimize geometry
AND kinematics
Optimal
0.623
0.013
0.933
9
As slender
as possible
Multiple Links
• Large N snake
• Analytic solution by Lighthill (in
Mathematical Biofluiddynamics)
‣ 41 degree angle
Multiple Links
• Large N snake
‣ 41 degree angle
small drag
coefficient
large drag
coefficient
Multiple Links
• Large N snake
‣ 41 degree angle
small drag
coefficient
large drag
coefficient
Effect of Slenderness
0.0125
0.0100
Efficiency E
from Becker et al.
0.0075
0.0050
0.0025
0.0000
Fully optimized stroke
’Purcell’ stroke
0
5
10
15
Slenderness log 1/κ
11
Effect of Slenderness
0.0125
0.0100
Efficiency E
from Becker et al.
0.0075
0.0050
Biological
systems
0.0025
0.0000
0
5
Fully optimized stroke
’Purcell’ stroke
10
15
Slenderness log 1/κ
11
Gastropod Locomotion
Locomotion is directly
coupled to stresses in
the thin fluid film
12
the thin fluid film
12
Retrograde vs
direct waves
F.Vles, C. R. Acad. Sci.,
Paris 145, 276 (1907)
the thin fluid film
12
What is Required for
Locomotion?
13
Locomotion?
13
Locomotion?
H
F
2
2
2
13
1
Locomotion?
H
F
2
2
2
1
Fshear stress
13
Locomotion?
H
F
2
2
2
1
Fshear stress
Couette flow in small gap
F = τ1 A = µ(τ1 )V1 A/H
= [V
τ12A(N= −
µ(τ
VcmF =
1)1 )V
+ 1VA/H
1 ] /N
!
"
Vcm =
[V
(N
−
1)
+
V
]
/N
11
F H2
1
Vcm =
−
!
"
AN
F H µ(τ11 ) µ(τ12 )
Vcm =
−
AN µ(τ1 ) µ(τ2 )
Each pad carries equal mass
13
Locomotion?
H
F
2
2
2
1
Fshear stress
F = τ1 A = µ(τ1 )V1 A/H
= [V
τ12A(N= −
µ(τ
VcmF =
1)1 )V
+ 1VA/H
] /N
F = τ1!
A = µ(τ1 )V11 A/H"
Vcm =
[V2 (N −
FH
1 1) + V11] /N
Vcm V=cm = [V2!(N − 1)
− + V1 ] /N
"
AN
F H µ(τ
11 ) µ(τ12 ) "
!
Vcm = F H
−
1
1
Vcm =AN µ(τ1 ) −µ(τ2 )
AN µ(τ1 ) µ(τ2 )
13
Locomotion?
H
F
2
2
2
1
Fshear stress
F = τ1 A = µ(τ1 )V1 A/H
= [V
τ12A(N= −
µ(τ
VcmF =
1)1 )V
+ 1VA/H
] /N
F = τ1!
A = µ(τ1 )V11 A/H"
Vcm =
[V2 (N −
FH
1 1) + V11] /N
Vcm V=cm = [V2!(N − 1)
− + V1 ] /N
"
AN
F H µ(τ
11 ) µ(τ12 ) "
!
Vcm = F H
−
1
1
Vcm =AN µ(τ1 ) −µ(τ2 )
AN µ(τ1 ) µ(τ2 )
Nonlinear characteristics first
measured by:
M. Denny, J. Exp. Biol. 91, 195 (1981)
M. Denny, Nature 285, 160 (1980)
13
RoboSnail II
Laponite
Carbopol
14
stress
“Tune” Material Properties
?
strain rate
15
stress
?
stress
strain rate
Shear Thickening
Newtonian
Shear Thinning
strain rate
15
stress
•
?
Perturb rheology of Newtonian
fluid (snail will crawl on any
non-Newtonian fluid)
stress
strain rate
Shear Thickening
Newtonian
Shear Thinning
•
strain rate
15
‣
shear thickening
‣
shear thinning
Which material properties are
“favorable”?
which
as σ terms
for simplicity,
the only
component is u
xy , as
f α − α +σβ,
all we
of denote
the other
on theand
right
handrelevant
side ofvelocity
Eq. (40)
steadyitstate
crawling if
then
d to show that
is positive
4βbecome
> 1 or equivalently, if 2I3 > I1 I2 . This
Substituting
this result back into E
∂σ
∂p
=
,
∂x
∂y
order solution for y0 and Eq (23) le
%
1
∂py /2h, leading to
g
=
3g
1
0 1= 0.
w(u)du dx
(41)
∂y
w≤
$
% be integrated once to obtain
% the stress
# boundary condition,
Using
these#equations
can
1
1
0
R1
0
w2 (x) 2w(x) −
w(u)du dx +
stress
−
#
w
$
w(u)dudp dx
(42)
$ Perturb
Finally,
# we find %the first order correc
•
We make the modeling assumption thatfluid
the mucus
as a slightly
(snailbehaves
will crawl
on anynon-Newtonian
2w(x) −
2w≥
#
1
?
R1
0
0σ(x, y)
w
%
w(u)du dx +
=
(y − h) + σ̄(x)·
dx
'
rheology of Newtonian
1
#
2w(x) −
w(u)du dx
(43)
R 1 shear stress, σ, given by
between 0shear rate, γ̇ (= ∂u/∂y),
fluid)
0
2w≥ and
w non-Newtonian
0
"
#
|σ|
σ
1−%
·
γ̇ =
µ
σ
(44)
∗
strain rate
Consequently, for
a given shear stres
‣
stress
shearisthickening
" > 0, that
when the mucus is shea
Shear Thickening
β > 0, which means
that the sign
of ∆Q/Q is the same as the sign of ", for
Newtonian
shear
thinning
to obtain the smallest flow rate, we need to pick
a shear-thinning
‣ " < 0, i.e.
the largest energy cost associated with locomotion, this criteria trumps the
Shear Thinning
Which
properties
are
a shear thinning fluid will decrease the overall
costmaterial
of locomotion
for the
“favorable”?
At this order we have to enforce th
strain rate
! λ
g2 dx = 0
•
15
0
herefore given by ∂y
=
µ dx
(y − y ) − %
µσ∗
dx
σ̄(x)
(y − y ) ,
y≥y .
g2S
hg
∗ 3
∗3
∗
−
y
)
−
y
] − Vs ,
(h − 2y ) − % II. [(h
y = h) + c =
2µ
3µσ∗ ASYMPTOTIC SOLUTION
(
Rheology Cost Function
(13)
Brief Article
e moving with the gastropod is given by
the boundary condition u(x, y = 0) = −c − Vs , and the fact that the velocity profile must be continuou
!
"
2
h
gh
(11) can be integrated
to give %g 2 S ∗4
The Author
∗
[y + (h − y ∗ )4 − 4y ∗3 h].
(14)
− y − Vs h −
=
2µ 3
12µσ∗
g2S
g
∗ 3
∗3
∗
∗
March
28,
2006
[(y
−
y
)
+
y
]
−
c
−
V
,
y
≤
y
(
y(y
−
2y
)
+
%
u(x,
y)
=
s
Mechanical
work done
in∗ done
crawling
(=
rate of toviscous
ssipation (equal to
the rate2µof mechanical
work
by the
gastropod
crawl)
3µσ
dissipation)
g
g2S
Brief Article
∗
u(x, y) =
y(y − 2y ) − %
[(y − y ∗ )3 − y ∗3 ] − c − Vs , y ≥ y ∗
(
# λ # h 2µ
3µσ∗
hλ 2
17 hλThe2 Author
E=
!σ |σ|" + ...
!σ " − #
E = dp
σ γ̇ dy dx.
(15)
|σ̄(x)|
and
≡
S(x)
in
order
to
simplify
the
notation.
The mucus velocity at y
e have defined dx0 ≡ g(x)
4µ
96
µσ
∗
0
σ̄(x)
March 28, 2006
"
!
to the moving gastropod is therefore given by 2
3
•
79 h
#185 !σ " − !|σ|"!σ |σ|"
!σ |σ|" 1 +
+ ...
Qs = #
g 2 S 8532σ∗ ∗ 3 !σ∗3
hg
432 µσ∗ ∗
|σ|"
[(h − y ) − y ] − Vs ,
(h − 2y ) − %
us (x) = u(x, y = h) + c =
Chemical cost associated
with mucus
2µ
3µσ∗ production ( ~ flux in
•
hλ 2
17 hλ 2
frame
moving
with
snail)
E
=
!σ |σ|" + ...
!σ
"
−
#
mucus flow rate in the frame moving with the gastropod is given
4µ by
96 µσ
Q=
#
0
h
∗
gh2
us (x)dy =
2µ
!
"
"
!
22 S
3 − !|σ|"!σ |σ|"
%g
h
h
79
185
∗4
∗ 4 !σ " ∗3
∗
−s V=s h# −
−y Q
!σ[y
+ ...
|σ|" +1(h
+ #− y ) − 4y h].
3
43212µσ
µσ∗ ∗
8532σ∗
!σ |σ|"
the average rate of energy dissipation (equal to the rate of mechanical work done by the gastropod to cr
by
# λ# h
E=
σ γ̇ dy dx.
0
0
16
=
µ dx
(y − y ) − %
µσ∗
dx
σ̄(x)
(y − y ) ,
y≥y .
g2S
hg
∗ 3
∗3
∗
−
y
)
−
y
] − Vs ,
(h − 2y ) − % II. [(h
y = h) + c =
2µ
3µσ∗ ASYMPTOTIC SOLUTION
(
(13)
Brief Article
the boundary condition u(x, y = 0) = −c − Vs , and the fact that the velocity profile must be continuou
!
"
2
h
gh
to give %g 2 S ∗4
The Author
∗
[y + (h − y ∗ )4 − 4y ∗3 h].
(14)
− y − Vs h −
=
2µ 3
12µσ∗
g2S
g
∗ 3
∗3
∗
∗
March
28,
2006
[(y
−
y
)
+
y
]
−
c
−
V
,
y
≤
y
(
y(y
−
2y
)
+
%
u(x,
y)
=
s
Mechanical
work done
in∗ done
crawling
(=
rate of toviscous
ssipation (equal to
work
by the
gastropod
crawl)
3µσ
dissipation)
g
g2S
Brief Article
∗
u(x, y) =
y(y − 2y ) − %
[(y − y ∗ )3 − y ∗3 ] −>c 0− Vs , y ≥ y ∗
(
# λ # h 2µ
3µσ∗
hλ 2
17 hλThe2 Author
E=
!σ |σ|" + ...
!σ " − #
E = dp
σ γ̇ dy dx.
(15)
|σ̄(x)|
and
≡
S(x)
in
order
to
simplify
the
notation.
e have defined dx0 ≡ g(x)
4µ
96
µσ
∗
0
σ̄(x)
March 28, 2006
"
!
3
•
79 h
#185 !σ " − !|σ|"!σ |σ|"
!σ |σ|" 1 +
+ ...
Qs = #
g 2 S 8532σ∗ ∗ 3 !σ∗3
hg
432 µσ∗ ∗
|σ|"
[(h − y ) − y ] − Vs ,
(h − 2y ) − %
us (x) = u(x, y = h) + c =
Chemical cost associated
with mucus
2µ
•
hλ 2
17 hλ 2
frame
moving
with
snail)
E
=
!σ |σ|" + ...
!σ
"
−
#
4µ by
96 µσ
Q=
#
0
h
∗
gh2
us (x)dy =
2µ
!
"
"
!
22 S
3 − !|σ|"!σ |σ|"
%g
h
h
79
185
∗4
∗ 4 !σ " ∗3
∗
−s V=s h# −
−y Q
!σ[y
+ ...
|σ|" +1(h
+ #− y ) − 4y h].
3
43212µσ
µσ∗ ∗
8532σ∗
!σ |σ|"
the average rate of energy dissipation (equal to the rate of mechanical work done by the gastropod to cr
by
# λ# h
E=
σ γ̇ dy dx.
0
0
16
=
(y − y ) − %
(y − y ) ,
y≥y .
(
Substituting
µσ∗ this
dx result
σ̄(x) back into Eq. (17e) (recall Qs is
solution
for y0 and Eq (23) leads to g0 y1 = 5(!σ̄|σ̄
g 2order
S
hg
∗ 3
∗3
∗
− y ) − y ] −SOLUTION
Vs ,
(13)
(h − 2y ) − % II. [(h
y = h) + c =
g1∗ =ASYMPTOTIC
3g0 y1 /2h, leading
to
2µ
3µσ
µ dx
Brief Article
1
the boundary condition u(x, y = 0) = −c − Vs , and the fact that the velocity profile must be 5continuou
!
"
(!
g1 =
2
2
%g
S
h
gh
to give
The Author
144 hσ∗
[y ∗4 + (h − y ∗ )4 − 4y ∗3 h].
(14)
− y ∗ − Vs h −
=
2µ 3
12µσ∗
g2S
g
∗first
3
∗3
∗ the mechanica
∗
March
28,
2006
Finally,
we
find
the
order
correction
to
[(y
−
y
)
+
y
]
−
c
−
V
,
y
≤
y
(
y(y
−
2y
)
+
%
u(x,
y)
=
s
Mechanical
work done
in∗ done
crawling
(=
rate of toviscous
ssipation (equal to
work
by the
gastropod
crawl)
3µσ
dissipation)
g
g2S
Brief Article
17 hλ(
∗
u(x, y) =
y(y − 2y ) − %
[(y − y ∗ )3 − y ∗3 ] −>c 0− Vs , y ≥ y ∗
E1 = −
# λ # h 2µ
3µσ∗
96 µσ∗
hλ 2
17 hλThe2 Author
E=
!σ |σ|" + ...
!σ " − #
E = dp
σ γ̇ dy dx.
(15)
|σ̄(x)|
≡
g(x)
and
≡
S(x)
in
order
to
simplify
the
notation.
e have defined dx
4µ
96
µσ
∗
0
0
σ̄(x)
Consequently, for a!given shear
applied by" the gastr
Marchstress
28, 2006
3
•
79 h
#185 !σ " − !|σ|"!σ |σ|"
" >Qs0,= that
isshear
when
the
mucus
is shear-thickening.
!σ |σ|"thickening
+ ...
#
1+
2
g S 8532σ∗ ∗ 3 !σ∗3
hg
432 µσ∗ ∗
|σ|"
[(h − y ) − y ] − Vs ,
(h − 2y ) − %
associated
with mucus
2µ
3µσ∗ production ( ~ flux
•
us (x) = u(x, y = h) + c =
Chemical cost
in
hλ 2
17 hλ 2
frame
moving
with
=
!σ |σ|" + ...
!σ " − #
mucus flow rate in
the frame
moving
withsnail)
the gastropod isE given
4µ by
96 µσ
Q=
#
0
h
C.
∗
gh2
us (x)dy =
2µ
!
Solution
"
!
22 S
3 − !|σ|"!σ |σ|"
%g
h
h
79
185
∗4
∗ 4 !σ " ∗3
∗
−s V=s h# −
−y Q
!σ[y
+ ...
|σ|" +1(h
+ #− y ) − 4y h].
3
43212µσ
µσ∗ ∗
8532σ∗
!σ |σ|"
"
At this order we have to enforce the constraints
the average rate of energy dissipation (equal to the rate of mechanical
work done !
by the gastropod to cr
! λ
λ
by
g2 dx = 0,
[g0 y2 + g2 y0 +
# #
λ
E=
h
σ γ̇ dy dx.
0
0
16
Averaging 0Eq.0 (17c)
we find the second order correction to
=
(y − y ) − %
(y − y ) ,
y≥y .
(
Substituting
µσ∗ this
dx result
solution
g 2order
S
hg
∗ 3
∗3
∗
Vs ,
(13)
(h − 2y ) − % II. [(h
y = h) + c =
g1∗ =ASYMPTOTIC
3g0 y1 /2h, leading
to
2µ
3µσ
µ dx
Brief Article
1
!
"
(!
g1 =
2
2
%g
S
h
gh
to give
The Author
144 hσ∗
[y ∗4 + (h − y ∗ )4 − 4y ∗3 h].
(14)
− y ∗ − Vs h −
=
2µ 3
12µσ∗
g2S
g
∗first
3
∗3
∗ the mechanica
∗
March
28,
2006
Finally,
we
find
the
order
correction
to
[(y
−
y
)
+
y
]
−
c
−
V
,
y
≤
y
(
y(y
−
2y
)
+
%
u(x,
y)
=
s
Mechanical
work done
in∗ done
crawling
(=
rate of toviscous
ssipation (equal to
work
by the
gastropod
crawl)
3µσ
dissipation)
g
g2S
Brief Article
17 hλ(
∗
u(x, y) =
y(y − 2y ) − %
[(y − y ∗ )3 − y ∗3 ] −>c 0− Vs , y ≥ y ∗
E1 = −
# λ # h 2µ
3µσ∗
96 µσ∗
hλ 2
17 hλThe2 Author
E=
!σ |σ|" + ...
!σ " − #
E = dp
σ γ̇ dy dx.
(15)
|σ̄(x)|
≡
g(x)
and
≡
S(x)
in
order
to
simplify
the
notation.
e have defined dx
4µ
96
µσ
∗
0
0
σ̄(x)
Marchstress
28, 2006
3
•
79 h
#185 !σ " − !|σ|"!σ |σ|"
" >Qs0,= that
isshear
when
the
mucus
!σ |σ|"thickening
+ ...
#
1+
2
g S 8532σ∗ ∗ 3 !σ∗3
hg
432 µσ∗ ∗
|σ|"
[(h − y ) − y ] − Vs ,
(h − 2y ) − %
associated
with mucus
2µ
•
us (x) = u(x, y = h) + c =
Chemical cost
in
hλ 2
17 hλ 2
frame
moving
with
=
!σ |σ|" + ...
!σ " − #
mucus flow rate in
the frame
moving
withsnail)
4µ by
96 µσ
Q=
#
0
h
C.
∗
gh2
us (x)dy =
2µ
!
Solution
"
!
22 S
3 − !|σ|"!σ |σ|"
%g
h
h
79
185
∗4
∗ 4 !σ " ∗3
∗
−s V=s h# −
−y Q
!σ[y
+ ...
|σ|" +1(h
+ #− y ) − 4y h].
3
43212µσ
µσ∗ ∗
8532σ∗
!σ |σ|"
"
work done !
! λ
λ
by
g2 dx = 0,
[g0 y2 + g2 y0 +
# #
λ
E=
h
σ γ̇ dy dx.
0
0
16
=
(y − y ) − %
(y − y ) ,
y≥y .
(
Substituting
µσ∗ this
dx result
solution
g 2order
S
hg
∗ 3
∗3
∗
Vs ,
(13)
(h − 2y ) − % II. [(h
y = h) + c =
g1∗ =ASYMPTOTIC
3g0 y1 /2h, leading
to
2µ
3µσ
µ dx
Brief Article
1
!
"
(!
g1 =
2
2
%g
S
h
gh
to give
The Author
144 hσ∗
[y ∗4 + (h − y ∗ )4 − 4y ∗3 h].
(14)
− y ∗ − Vs h −
=
2µ 3
12µσ∗
g2S
g
∗first
3
∗3
∗ the mechanica
∗
March
28,
2006
Finally,
we
find
the
order
correction
to
[(y
−
y
)
+
y
]
−
c
−
V
,
y
≤
y
(
y(y
−
2y
)
+
%
u(x,
y)
=
s
Mechanical
work done
in∗ done
crawling
(=
rate of toviscous
ssipation (equal to
work
by the
gastropod
crawl)
3µσ
dissipation)
g
g2S
Brief Article
17 hλ(
∗
u(x, y) =
y(y − 2y ) − %
[(y − y ∗ )3 − y ∗3 ] −>c 0− Vs , y ≥ y ∗
E1 = −
# λ # h 2µ
3µσ∗
96 µσ∗
hλ 2
17 hλThe2 Author
E=
!σ |σ|" + ...
!σ " − #
E = dp
σ γ̇ dy dx.
(15)
|σ̄(x)|
≡
g(x)
and
≡
S(x)
in
order
to
simplify
the
notation.
e have defined dx
4µ
96
µσ
∗
0
0
σ̄(x)
Marchstress
28, 2006
3
•
79 h
#185 !σ " − !|σ|"!σ |σ|"
" >Qs0,= that
isshear
when
the
mucus
!σ |σ|"thickening
+ ...
#
1+
2
g S 8532σ∗ ∗ 3 !σ∗3
hg
432 µσ∗ ∗
|σ|"
[(h − y ) − y ] − Vs ,
(h − 2y ) − %
associated
with mucus
2µ
•
us (x) = u(x, y = h) + c =
Chemical cost
in
hλ 2
17 hλ 2
frame
moving
with
=
!σ |σ|" + ...
!σ " − #
mucus flow rate in
the frame
moving
withsnail)
4µ by
96 µσ
Q=
#
0
h
∗
gh2
us (x)dy =
2µ
!
> 0 C.
Solution
"
!
22 S
3 − !|σ|"!σ |σ|"
%g
h
h
79
185
∗4
∗ 4 !σ " ∗3
∗
−s V=s h# −
−y Q
!σ[y
+ ...
|σ|" +1(h
+ #− y ) − 4y h].
3
43212µσ
µσ∗ ∗
8532σ∗
!σ |σ|"
"
work done !
! λ
λ
by
g2 dx = 0,
[g0 y2 + g2 y0 +
# #
λ
E=
h
σ γ̇ dy dx.
0
0
16
=2 − α (y
−βy ) − %
(y − y ) , y ≥ y .
(
185givenσby
+ ∂y α µ
Substituting
dx +
µσ∗ this
dx result
herefore
="
I ·
(40)
4266
1
α 2order solution for y0 and Eq (23) leads to g0 y1 = 5(!σ̄|σ̄
g S
hg
∗ 3
∗3
−
y
)
−
y
] −SOLUTION
Vs ,
(13)
(h − 2y ∗ ) − % II. [(h
y = h) + c =
g1∗ =ASYMPTOTIC
3g0 y1 /2h, leading
to
2µ
3µσ
of the other terms on the right hand side of Eq. (40)
Brief Article
e
moving
with
the
gastropod
is
given
by
positive
if 4βcondition
> 1 or u(x,
equivalently,
> Ithe
. This
1
1 I2fact
the boundary
y = 0) = −cif− 2I
Vs ,3 and
that the velocity profile must be 5continuou
!
"
(!
g1 =
2
2
%g
S
h
gh
to give
The Author
144 hσ∗
[y ∗4 + (h − y ∗ )4 − 4y ∗3 h].
(14)
− y∗ − V h −
=
2µ
σ∗
3
s
12µσ∗
g2S
g
∗first
3
∗3
∗ the mechanica
∗
March
28,
2006
Finally,
we
find
the
order
correction
to
[(y
−
y
)
+
y
]
−
c
−
V
,
y
≤
y
(
y(y
−
2y
)
+
%
u(x,
y)
=
s
Mechanical
work done
in∗ done
crawling
((41)
=
rate of toviscous
ssipation (equal to
work
by the
gastropod
crawl)
3µσ
dissipation)
g
g2S
Brief Article
17 hλ(
∗
$ y) =
# 1− 2y ) − %%
#
u(x,
y(y
[(y − y ∗ )3 − y ∗3 ] −>c 0− Vs , y ≥ y ∗
E1 = −
# λ # h 2µ
3µσ∗
2
96 µσ∗
hλ 2 (42)
17 hλThe2 Author
w
(x)
2w(x)
−
w(u)du
dx
R
E=
!σ |σ|" + ...
!σ " − #
E = dp
σ γ̇ dy dx.
(15)
|σ̄(x)|
≡
g(x)
and
≡
S(x)
in
order
to
simplify
the
notation.
e2w≥
have 01defined
w
0
4µ
96
µσ
∗
0
dx0
σ̄(x)
Consequently,
shear
'
Marchstress
28, 2006
%
# 1by 2 for% a!given
#
to the moving
gastropod
is $therefore given
79 h
#185 !σ 3 " − !|σ|"!σ |σ|"
" >Q−
isshear
when
the
mucus is shear-thickening.
!σ |σ|"thickening
+ ...
= that
# w(u)du
1+
2 (43)
s0,
2w(x)
w(u)du dx +
hg
R1
432 µσ∗ ∗ dx g S
8532σ∗ ∗ 3 !σ∗3
|σ|"
[(h − y ) − y ] − Vs ,
us (x)
2w≥= 0u(x,
w y = h) + c = 0 (h − 2y ) − %
Chemical
cost associated
with mucus
2µ
•
•
hλ 2
17 hλ 2
frame
moving
with
snail)
E
=
!σ |σ|" + ...
!σ
"
−
#
4µ by
96 µσ
Q=
#
h
gh2
us (x)dy =
2µ
(44)
!
∗
> 0 C.
Solution
"
!
22 S
3 − !|σ|"!σ |σ|"
%g
h
h
79
185
∗4
∗ 4 !σ " ∗3
∗
−s V=s h# −
−y Q
!σ[y
+ ...
|σ|" +1(h
+ #− y ) − 4y h].
3
43212µσ
µσ∗ ∗
8532σ∗
!σ |σ|"
"
0
At this
we have
to enforce the constraints
hat the sign of ∆Q/Q is the same
as order
the sign
of ", for
work done !
!
λ
λ
t flow rate, we need to pick " < 0, i.e. a shear-thinning
shear thinning
by
g dx = 0,
[g0 y2 + g2 y0 +
st associated with locomotion, this criteria
# λ # h trumps the 2
0
0
σ γ̇ dy for
dx. the
id will decrease the overall cost Eof=locomotion
16
Cost of Locomotion
“The high cost is primarily due to
the cost of mucus production, which
alone is greater than the total cost
of movement for a mammal or
reptile of similar weight, ...”
shear thinning
Mark Denny, Science, 208, No. 4449 (1980)
17
Cost of Locomotion
“The high cost is primarily due to
the cost of mucus production, which
alone is greater than the total cost
of movement for a mammal or
reptile of similar weight, ...”
shear thinning
Mark Denny, Science, 208, No. 4449 (1980)
8mm plate with sandpaper
100 micron gap, T=22ºC
Pedal mucus from common garden
snail, Helix aspera is strongly shearthinning.
17
Final Comments
•
3-link (and n-link) swimmer (low Reynolds number)
Optimizing kinematics
Trade-off between efficiency and robustness in biological
systems?
•
Snails
Rely on the nonlinear response of pedal mucus to crawl
We can “tune” viscous material properties to find which
weakly nonlinear response is energetically favorable 
shear thinning
Mechanical wall-climber
18
Acknowledgments
Collaborators:
•
•
•
•
•
Daniel Tam (GS, MIT)
Brian Chan (GS, MIT)
Eric Lauga (MIT, Dept of Math)
- Optimizing 3-link swimmer
- Robosnails + mechanical swimmer
- Optimizing crawling
Funding:
Schlumberger-Doll Research (SDR)
NSF
19
Acknowledgments
Collaborators:
•
•
•
•
•
Daniel Tam (GS, MIT)
Brian Chan (GS, MIT)
Eric Lauga (MIT, Dept of Math)
- Optimizing 3-link swimmer
- Robosnails + mechanical swimmer
- Optimizing
crawling
Tuesday
10:45-11:10
Slip, Swim, Mix, Pack: Fluid Mechanics at
the Micron Scale
Funding:
Schlumberger-Doll Research (SDR)
NSF
19

Optimizing Locomotion - American Physical Society

Transcription

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