Bio-fluid Mechanics I - 國立中興大學ezphysics.nchu.edu.tw/chi/2008f_IntroBiomechanics/20… · · 2009-02-24Bio-fluid Mechanics I Viscosity, Re, ... fluid mechanics of spinning

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

6

=


2

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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


3

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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 =


4

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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]

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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 +=

21

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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63 64

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

Documents

Bio-fluid Mechanics I - 國立中興大學ezphysics.nchu.edu.tw/chi/2008f_IntroBiomechanics/20… · · 2009-02-24Bio-fluid Mechanics I Viscosity, Re, ... fluid mechanics of spinning