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Transcript of Bio-fluid Mechanics I -...
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2007 IntroBiomechanics-- W5 2007/10/8
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Bio-fluid Mechanics IViscosity, Re, Vortices, Forces of Flow,
and Bernoullis Principles
Introduction to Biomechanics 2008/10/22
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I. Viscosity
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Contrast b/w solid and fluid
Bernoullis principleCons. of energy
Principle of continuityCons. of mass
DragFrictional force
Momentum flux changeForce
Velocity gradientShearing plane
StreamlineInterface
ViscosityElastic modulus
DensityMass
FluidsSolids
resist to deform defining boundary
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Important properties
- Density (m/V): air ~ 1.2 kg/m3, water ~ 1000 kg/m3
- Viscosity: Viscous dense, e.g. motor oil < water
tkF =Fluids: how fast sheared; shape not returned
dzdv
SF == (unit: kg/ms or Pas)
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Viscosity: Measuring kinematic viscosity
e.g. Ostwald (capillary) viscometer
www.greentree.com.tw
www.oil.net.tw/lbg2006/chapter/3-1.htm
=
''
=tt
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=
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In this class, assuming.
1. Incompressible fluids
2. No-slip condition
3. Newtonian fluids (i.e. constant viscosity)
4. Steady flow (i.e. same v @ same location)
5. No fluid-fluid interfaces
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II. Flow Regimes
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Flow regimes
Laminar flow(predictable, orderly)
Turbulent flow
Major difference NOT the presence of vortex
When fluids pass through an obstacle
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Transition: laminar turbulent
Two forces involve: inertial & viscous forces
lSvma
FF
viscous
inertial
=
(individuality) (groupiness)
Re===
lv
lSvtSlv
Reynolds number
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lv
=Re, of the fluid (medium)
l Characteristic lengthConvention: mostly in the direction of the flow
Exception:
Osborne Reynolds (1883): In circular pipe of certain length, turbulent occurs when
Re > 2000 (l = d=2r) or Re > 1000 (l = r) (our circulatory system, still laminar when Re ~ 4000)
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=
In water vs. air
for a given size, same velocity: Rewater > Reairbut usually vair > vwater, Rebird ~ Refish if similar size
lv
=Re
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Note: Re is just a crude number, so only look at its order of magnitude (usually significant digits < 2)
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A useful tool for making models:
For two geometrically similar situations, equality of Re Equality of the patterns of flow, whatever the individual
values of length, speed, density, and viscosity
Other use for Re
Examples
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Re & Consequences of the No-slip condition
@ low Re, viscous force dominates; with no-slip condition shallow velocity gradient
(semi-stagnant fluid surrounding object)
Daphnia
rake or paddle?
by Koehl
Male moths antenna
Leakiness: 8~18% (Vogel, 1983) 16
III. Vortices
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Streamlines and the Appearance of Flows
S1 S2dl1
dl2
1v2v
At the same time, volume in = volume out
dtdlS
dtdlS 2211 =
2211 vSvS = Principle of continuity
Conservation of mass
Incompressible flow in a rigid pipe
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Path of each fluid particle
(pathline)
Particles wont cross Virtual wall
3D ~ in tube (Principle of Continuity holds)
2211 vsvs =
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Fig. 6.7: Flow around a circular cylinder
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Asymmetrical vortices shedding subject vibrates
Some problems
Antenna of the car vibrates use spiral around to break the vortices
Chimney, bridge, tree problematic if vibrating frequency caused by vortices shedding = natural frequency
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Some vortex phenomena & examples
Various vortices
Fig. 6.10
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Von Karman trail vs. vortices behind a swimming fish
Removes momentum from the fluid
Impart momentum to the fluid
[Some vortex phenomena & examples]
23 24
Ground-Level Vortices
Ground
Wind What happens behind the chimney?
Viscous entrainment
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e.g. A black fly larva
Use of Ground-Level Vortices
Fig. 6.12 (a)Fig. 6.12 (a)
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[Use of Ground-Level Vortices]
e.g. Phalarope
Fig. 6.12 (b)
Obst, et al. 1996. Kinematics and fluid mechanics of spinning in phalaropes. Nature (+ cover photo). 384: 121.
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http://content.ornith.cornell.edu/UEWebApp/images/fig2_1305.gif
http://www.mbari.org/seminars/1998/jan28_hamner.html
[More on phalarope]
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Shear motion within a velocity gradient rotation in solid body or fluid itself
Fig. 6.13
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IV. Forces of Flow
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Force as Momentum Change
Main forces: Drag ()(Ch7), Lift ()(Ch12), Thrust ()(Ch13)
Force:
maF =
tmv
tvmF
=
=
)(dt
dvmF =
rate of momentum change
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2SvtvSl
tmv == Eq. 4.9
For flowing fluids, the dimension
211
222 vSvSt
mvF =
=
Fig. 7.1a
If v2 > v1,
by Principle of Continuity(S1v1 = S2v2)
S2 < S1
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For a propeller or fan Force exerted on the mounting
For a propeller on a craft Thrust
For a passive body Drag (v1 > v2)
211
222 vSvSF =
From amount of constriction or v:
But in real case, v2 is rarely uniform across the flow
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How to measure drag?
Fig. 7.1b
211
222 vSvSF =
To get vFlow sensorNeutrally buoyant particles (photographs, videorecording)
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(Digital) Particle Image Velocimetry
(PIV or DPIV)
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Example: Liao et al., 2003, Science.
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V. Pressure in the Flow & Bernoullis Principle
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Static + Dynamic pressure ( = total head) sd ppH +=
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21 22 vp
Vmv d
==
Kinetic energy per unit volume
dynamic pressure
Dynamic Pressure
Incompressible fluid stop at the wall exerts pressure
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Two Principles
Principle of Continuity applied all the time (even viscosity)
Bernoullis Principle some assumptions:
1. Viscosity negligible (i.e. Ideal fluid)
2. Steady flow
3. Incompressible fluid
4. Gravity can be ignored
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Bernoullis Principle (Conservation of Energy)
SSdlm =
dp
dzdtdvSdlma
gdzSSdpF
==
=
Divided by mass & rearrange 0=++dtdv
dldzg
dldp
Newtons 2nd Law:
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0=++dtdv
dldzg
dldp
to get rid of t, assuming steady flow vdldt =
0=++dl
vdvdldzg
dldp
.21 2 constvgzp =++
assuming const. density
Integrate
.21 2 constvgzp =++ OR
0)(21 2
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2 =++ vvzgp
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Flow-induced pressure difference (as ventilation system)
Fig. 7.5
Ex1: Prairie dog burrow
(Work done by Vogel use geometrically similar model (i.e. same Re) (p. 126 of text)
Applications of Bernoullis Principle
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Ex2: Keyhole limpet
[Applications of Bernoullis Principle]
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Ex3: Fish nose How does water get in?
Ventral view
Shark:
[Applications of Bernoullis Principle]
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Dye released upstream of the nares in a dead shark
Experiment setup:
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PPv
v
Unidirectional flow
A slope exists between incurrent and excurrent nares:
Velocity gradient creates a pressure difference (Bernoullis Principle)
Results:
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VI. Drag
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Pressure & Drag
Fluid flows from high P to low P
At some point, upstream
flow occurs
Fig. 7.8
Separation of flow
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2
21 v
p
v
p
=
48Fig. 6.7
Flow separation vs. Drag
Flow separation indirect cause
of high drag (asym. pressure)
At higher velocity (Re), more
momentum to turn
Separation point further aft
(less pressure difference)
Narrow wake
Less drag
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Two types of drag
1.Skin friction: from viscosity (Re < 10), Total surface area
Dominates @ low Re Streamlining does no good
2. Pressure drag: due to pressure distribution
Drag Dynamic pressure ( )
Depends on frontal area, shape, Re ( )
2
21 v
v
lv
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Drag coefficient
2
2vpCp
= 2
2vSDCd
=pSD =Q
For a given shape,
Cd varies only with Re
(graphs available to look up)
Fig. 7.9
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III III
vD 2vD
11 Re
vCvD
d
I
00
2
Re
vCvD
d
II
5.05.0
5.1
Re
vCvD
d
Streamlining:
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Drag coefficients for various shapes (from Denny, 1988)
sphere hollow hemisphere solid hemisphere
47.0=dC 38.0 42.1 42.0 17.1
cylinder hollow half-cylinder half-rectangular solid
17.1=dC 20.1 30.2 55.1 98.1
long, flat plate
v
v
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2
21 SvCD d =
Area S Different convention for different shapes
High-drag bodies frontal (projected) area
Streamlined bodies total surface (wetted) area
Lift-producing bodies plan form area
Blimps & organisms (eg. fish) Volume2/3
Same drag different Cd if different S is used
Have to be consistent54
Tricks for dropping ones drag
Streamlining
Dimpling golf balls go further than smooth ones
(permit fluid to flow further around before
separation narrower wake less drag)
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Flow-wise ribbing e.g. shark skin, scallop shells,
Riblets for racing boats
by breaking up cross-flow vortices near the surface
skin friction at high Re56
Mucus long-chain polymeric molecules
(gentler velocity gradient less surface shear)
The issues
(1) Well documented in pipes and for flow over test
body: kinds of molecules (fishes & some animals seem to
have the right kind)
(2) Drag reduction by mucus in Nature is difficult to
demonstrate: cost of production + other functions of
mucus just complicate the situations
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Compliant surfaces, e.g. dolphin skin (a long story)
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Other mechanisms:
Surface heating by skin friction, but calculations suggest it negligible
Injection of fast water along the surface from
opercula, drag 10% for a trout at top speed
Surface roughening in critical regions seems possible
for some fast fishes
Active control pressure detectors allow a fish to fine-
tune its head turning to drag
[Good reviews see Bushnell & Moore (1991) and Fish (1998)]
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TO BEAR IN MIND
A claim of drag reduction in a biol. system should be viewed
with skepticism until
(1) plausible physical mechanism
(2) shown to work on physical models under biol. relevant
conditions
(3) shown to work by some direct test on real organisms
under controlled and reproducible conditions
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http://web.mit.edu/fluids/www/Shapiro/ncfmf.html
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Stanislav Gorb
Leader of Evolutionary Biomaterials Group
Max-Planck-Institute for Metals Research
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Paper written by Professor Stanislav Gorb:
1. Arzt, E., Gorb, S. and Spolenak, R. (2003). From micro to nanocontacts in biological attachment devices. PNAS 100, 10603-10606.
2. Arzt, E., Gorb, S. and Spolenak, R. (2003). From micro to nanocontacts in biological attachment devices. PNAS 100, 10603-10606.
3. Gorb, S., Varenberg, M., Peressadko, A. and Tuma, J. (2007). Biomimetic mushroom-shaped fibrillar adhesive microstructure. J. R. Soc. Interface 4, 271-275.
4. Gorb, S. N. and Varenberg, M. (2007). Mushroom-shaped geometry of contact elements in biological adhesive systems. J. Adh. Sci. & Tech. 21, 1175-1183.
5. Jiao, Y., Gorb, S. and Scherge, M. (2000). Adhesion measured on the attachment pads of Tettigonia viridissima (Orthoptera, insecta). JEB 203, 1887-1895.
3-page report on 1 paper due on 12/3
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1. Read this paper
paper reading worksheet due on 11/26
2. Do motion analyses on 12/3
present on 12/10