Download BE105_19_fluid_dynamics_I

Lecture #19: Fluid Dynamics I
7 fundamental concepts of fluid mechanics
1. dynamic viscosity
velocity, U
force
Area = S
L
force ~ S x U x L
force = m S x U x L,
where m = viscosity (dynamic viscosity)
Fundamental units: M / LT
SI unit: Pascal-second
cgs unit: Poise
2. density
density, r = mass / volume
Fundamental units: M / L3
3. kinematic viscosity
n
dynamic viscosity
=
density
Ratio of the fundamental
properties of a fluid
fundamental units: L2 / T
cgs unit: Stokes
dynamic
viscosity
(Pa-s)
Air
Water
18 x 10-6
1 x 10-3
density
(kg m-3)
1.2
1 x 103
kinematic
viscosity
(kg m-3)
15 x 10-6
1 x 10-6
4. no slip condition
(boundary layer)
Laminar
flow over
solid surface
Velocity, u =
u oo (mean stream flow)
u=0
(no slip condition)
boundary
layer
solid surface
laminar flow
through pipe
diffusion
convection
diffusion
uinf
d (x) = 0.95 x uinf
y
x
flat plate with upstream edge
d =5
xn
u
Ground effect
5. Law of continuity
(conservation of mass)
mass flux
mass flux =
= r2 s2u2
r1 s1u1
S1U1
s2u2
gravity accelerates
water downward
by conservation
of mass:
r1 s1u1 = r2 s2u2
For incompressible fluid
r1 = r2
Thus, s1u1=s2u2
right
ventricle
lung
body
right
ventricle
flow tank
pump
dye
injector
small
fluid
volume
top view
fish
stream
line
constricting
streamlines
= higher
velocity
6. Bernoulli Equation
(conservation of energy)
consider flow
in one stream tube:
Apply conservation of energy within tube:
C = mu  PV  mgh
2
1
2
kinetic
energy
divide by volume:
consider two points
along tube:
1
2
pressure gravitational
energy
energy
C = ru  P
2
1
2
r (u  u ) = ( P2  P1 )
2
2
2
1
u1
_
u2
+
7. Reynolds number
two sources of force:
1) viscosity force
=muS/L
2) inertial force
= r u2 S
u
Force sensor
S
Ratio = Re (Reynolds number)
=ruL/m=uL/v
Re is a dimensionless number that
captures relative importance of viscous
and inertial forces:
length, L
Low Re
Force ~ u
Flows are reversible
Hi Re
Force ~ u2
Flows are irreversible
Low Reynolds numbers
Forces proportional to u
Copepod
Re = 300
Sperm whale
Re = 300,000,000
Thrips
Re = 30
Tuna
Re = 30,000,000
Duck
Re = 300,000
Dragonfly
Re = 30,000
Whale sperm
Re = 0.00003
Bacterium
Re = 0.00003