1. dynamic viscosity

force = S x U x L,

where = viscosity (dynamic viscosity)

Fundamental units: M / LT

SI unit: Pascal-secondcgs unit: Poise

7 fundamental concepts of fluid mechanics

Area = S

L

velocity, U

force ~ S x U x L

force

Lecture #19: Fluid Dynamics I

2. density

density, = mass / volume

Fundamental units: M / L3

3. kinematic viscosity

dynamic viscosity density

= Ratio of the fundamental properties of a fluid

fundamental units: L2 / Tcgs unit: Stokes

Air 18 x 10-6 1.2 15 x 10-6

Water 1 x 10-3 1 x 103 1 x 10-6

Air 18 x 10-6 1.2 15 x 10-6

Water 1 x 10-3 1 x 103 1 x 10-6

dynamicviscosity(Pa-s)

kinematicviscosity(kg m-3)

density(kg m-3)

diffusion

diffusion

convection

4. no slip condition(boundary layer)

solid surface

Velocity, u =u (mean stream flow)oo

u = 0(no slip condition) boundary

layer

Laminarflow oversolid surface

laminar flowthrough pipe

x0.95 x uinf

uinf

x

y

flat plate with upstream edge

u

x 5 Ground effect

S1U1 s2u2

5. Law of continuity(conservation of mass)

mass flux =

s1u1

mass flux

= s2u2

For incompressible fluid

1 =

Thus, s1u1=s2u2

gravity accelerateswater downward

= s2u2 s1u1

by conservationof mass:

rightventricle

lung

rightventricle

body

flow tank

pump

top view

dyeinjector

streamline

fish

smallfluid

volumeconstrictingstreamlines

= highervelocity

6. Bernoulli Equation(conservation of energy)

consider flowin one stream tube:

Apply conservation of energy within tube:

mghPVmuC 221

pressureenergy

kinetic energy

gravitationalenergy

PuC 221 divide by volume:

consider two pointsalong tube: )()( 12

21

222

1 PPuu

u1

u2

_

+

7. Reynolds number

Force sensoru

length, L

S

two sources of force:

1) viscosity force= u S / L

2) inertial force= u2 S

Ratio = Re (Reynolds number)= u L / = u L / v

Re is a dimensionless number thatcaptures relative importance of viscous

and inertial forces:

Low Re Hi Re

Force ~ u2Force ~ uFlows are reversible Flows are irreversible

Low Reynolds numbers

Forces proportional to u

Sperm whaleRe = 300,000,000

TunaRe = 30,000,000

DuckRe = 300,000

DragonflyRe = 30,000

CopepodRe = 300

ThripsRe = 30

BacteriumRe = 0.00003

Whale spermRe = 0.00003