1. Viscosity, Poiseuille’s Equation, Coanda Effect
Fluid Dynamics
Viscosity, Poiseuille’s Equation, Coanda Effect

2. 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.

3. 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

4. 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

5. Measuring viscosity 𝐹= 𝐴𝑣 𝑙 Pulled with force F Moving plate, speed v
Plate separation, l
Stationary plate
𝐹= 𝐴𝑣 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

6. Coefficient of viscosity
Fluid
 (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
Coefficient of viscosity, , varies for different substances
= 𝐹𝑙 𝑣𝐴
Units of 𝑁𝑚 𝑚 𝑠 𝑚2 = 𝑁 𝑚2 𝑠=𝑃𝑎 ∙ 𝑠

7. For situations with laminar flow, Poiseuille’s equation
𝑉 𝑡 = 𝜋𝑅4 𝑃1−𝑃2 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

8. Turbulent flow
The onset of turbulence occurs when the Reynolds number, Re >2000.
Reynolds number is defined as
𝑅𝑒= 2𝑣𝑎𝑣𝑒𝑟𝜌 
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

9. Solids traveling through viscous fluids
Lift
Coandă Effect
Demo: cylindrical object in stream of water
Coanda planes, proof of concept physics

10. 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.

11. Drag
𝐹𝑑𝑟𝑎𝑔= 1 2 𝐶𝐷𝜌𝐴𝑣2
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

12. 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
CD=0.5
A = 3.0 m2 U
Fdrag=? E
𝐹𝑑𝑟𝑎𝑔= 1 2 𝐶𝐷𝜌𝐴𝑣2 S
𝐹𝑑𝑟𝑎𝑔= 𝑘𝑔 𝑚 𝑚2 27 𝑚 𝑠 2
𝐹𝑑𝑟𝑎𝑔=180 𝑁
For more experimentally determined values of coefficient of drag, check Engineering Toolbox and Wikipedia (yea, science nerds!)

13. 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
CD=0.6
A = 1.5 m2 U
𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙=? E
Terminal velocity = no acceleration, therefore Fweight = Fdrag,
Where 𝐹𝑑𝑟𝑎𝑔= 1 2 𝐶𝐷𝜌𝐴𝑣2 and 𝐹𝑤𝑒𝑖𝑔ℎ𝑡=𝑚𝑔
So 1 2 𝐶𝐷𝜌𝐴𝑣2=𝑚𝑔
So 𝑣= 2𝑚𝑔/(𝐶𝐷𝜌𝐴) S
𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙= 𝑘𝑔 9.8 𝑚 𝑠 𝑘𝑔 𝑚3 (1.5𝑚2)
𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙=55 𝑚/𝑠