Viscosity, Poiseuille’s Equation, Coanda Effect 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 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

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

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

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

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 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

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= 1 2 0.5 1.3 𝑘𝑔 𝑚3 3.0 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!)

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= 2 60 𝑘𝑔 9.8 𝑚 𝑠2 0.6 1.3 𝑘𝑔 𝑚3 (1.5𝑚2) 𝑣𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙=55 𝑚/𝑠