Download Fluid Dynamics

Fluid Dynamics
Viscosity, Poiseuille’s Equation, Coanda Effect
From ideal fluids to real fluids
So far, we have considered ideal fluids:
• They coast along with no difference in pressure
• An ideal milk shake would be as easy to drink as a watery soda
• The primary difference between ideal fluids and real fluids is their viscosity.
Viscosity
Honey and water have almost identical densities,
but their flow properties are dramatically
different.
Viscosity: measure of a fluid’s resistance to flow
blood flow
flight
curve ball
Factors that affect flow of fluids
• Pressure difference
• How hard is fluid being pushed forward
minus how hard fluid is being pushed back
• Radius of tube
• Harder to push fluids through narrower
tubes
• Length of tube
• Longer tubes offers more resistance
• Viscosity of fluid
• Water flows more easily than molasses
Measuring viscosity
Pulled with force F
Moving plate, speed v
Plate separation, l
Stationary plate
𝐴𝑣
𝐹=
𝑙
Where
F is the force required to pull a plate across the fluid
 (Greek letter, eta) is the coefficient of viscosity and is
determined experimentally.
A is the area of the fluid in contact with each plate
v is the speed of the moving plate
l is the distance between the plates
Coefficient of viscosity
Fluid
Coefficient of viscosity, , varies for
different substances
=
𝐹𝑙
𝑣𝐴
Units of
𝑁𝑚
𝑚
𝑠
𝑚2
= 𝑁
𝑚
2 𝑠 = 𝑃𝑎 ∙ 𝑠
 (Ps)
Air (20C)
1.8x10-5
Water (20C)
1.0x10-3
Water (40C)
0.7*10-3
Water (60C)
0.5*10-3
Blood (37C)
2.5*10-3
Motor oil (-30C)
3.0*105
Motor oil (40C)
0.07
Motor oil (100C)
0.01
Honey (15C)
600
Honey (40C)
20
For situations with laminar flow,
Poiseuille’s equation
𝑉 𝜋𝑅4 𝑃1 − 𝑃2
=
𝑡
8𝑙
The flow rate is proportional to…
• the radius of the tube (to the 4th power!)
• pressure difference
The flow rate is inversely proportional to…
• the length of the tube
• the coefficient of viscosity
Turbulent flow
The onset of turbulence occurs when the Reynolds number, Re >2000.
Reynolds number is defined as
2𝑣𝑎𝑣𝑒𝑟𝜌
𝑅𝑒 =

Where vave is the average speed of the fluid
r is the radius of the tube through with the fluid is flowing
 is the density of the fluid
 is the coefficient of viscosity of the fluid
Solids traveling through viscous fluids
• Lift
• Coandă Effect http://en.wikipedia.org/wiki/Coand%C4%83_effect
• Demo: cylindrical object in stream of water
• Coanda planes, proof of concept physics
Lift
• When air passes over a wing,
viscosity of air creates
“downwash”
• Coandă effect creates a
boundary layer next to surface
of wing
• A change in direction requires
a force.
• If the wing exerts a downward
force on air, then air exerts an
upward force on wing.
Drag
1
𝐹𝑑𝑟𝑎𝑔 = 𝐶𝐷𝜌𝐴𝑣2
2
where CD is the dimensionless number related to shape
 is the density of the fluid
A is the cross-sectional area exposed to fluid
v is the speed of the solid through the fluid
Example
Estimate the drag on a car traveling at 27 m/s (60 mph). Assume the
drag coefficient for a well-designed car is 0.5, air = 1.3 kg/m3, and the
frontal area of the car is 3.0 m2.
G
v= 27 𝑚/𝑠
𝜌𝑎𝑖𝑟 = 1.3 𝑘𝑔/𝑚3
U
E
Fdrag=?
CD=0.5
A = 3.0 m2
1
𝐹𝑑𝑟𝑎𝑔 = 𝐶𝐷𝜌𝐴𝑣2
2
S
1
𝐹𝑑𝑟𝑎𝑔 = 0.5
2
S
𝐹𝑑𝑟𝑎𝑔 = 180 𝑁
𝑘𝑔
1.3 3
𝑚
3.0
𝑚2
27
𝑚
𝑠
2
For more experimentally
determined values of
coefficient of drag, check
Engineering Toolbox and
Wikipedia (yea, science
nerds!)
Example
Estimate the terminal velocity of a 60-kg skydiver who has a surface
area of 1.5 m2 and an assumed CD of 0.6.
G
𝑚 = 60 𝑘𝑔
𝜌𝑎𝑖𝑟 = 1.3 𝑘𝑔/𝑚3
U
E
𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 =?
CD=0.6
A = 1.5 m2
Terminal velocity = no acceleration, therefore Fweight = Fdrag,
1
Where 𝐹𝑑𝑟𝑎𝑔 = 2 𝐶𝐷𝜌𝐴𝑣2 and 𝐹𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑔
1
So 2 𝐶𝐷𝜌𝐴𝑣2 = 𝑚𝑔
So 𝑣 =
S
S
2𝑚𝑔/(𝐶𝐷𝜌𝐴)
𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 =
2 60 𝑘𝑔
0.6
𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 = 55 𝑚/𝑠
𝑚
9.8 𝑠2
𝑘𝑔
1.3 𝑚3 (1.5𝑚2)