Elementary Mechanics of Fluids

1. Elementary Mechanics of Fluids CE 319 F Daene McKinney Viscosity

2. y Moving plate u=V V Fluid B x Fixed plate u=0 Some Simple Flows • Flow between a fixed and a moving plate Fluid in contact with the plate has the same velocity as the plate u = x-direction component of velocity

3. r R x V Fluid Some Simple Flows • Flow through a long, straight pipe Fluid in contact with the pipe wall has the same velocity as the wall u = x-direction component of velocity

4. y Moving plate u=V t0 t1 t2 Fluid x Fixed plate u=0 Fluid Deformation • Flow between a fixed and a moving plate • Force causes plate to move with velocity V and the fluid deforms continuously.

5. y dL Moving plate u=V+dV da t t+dt dy dx Fluid x Fixed plate u=V Fluid Deformation Shear stress on the plate is proportional to deformation rate of the fluid

6. Shear in Different Fluids • Shear-stress relations for different typesof fluids • Newtonian fluids: linear relationship • Slope of line (coefficient of proportionality) is “viscosity”

7. V+dv V Viscosity • Newton’s Law of Viscosity • Viscosity • Units • Water (@ 20oC) • m = 1x10-3N-s/m2 • Air (@ 20oC) • m = 1.8x10-5N-s/m2 • Kinematic viscosity

8. y Moving plate u=V V Force acting ON the plate Fluid B x Fixed plate u=0 Flow between 2 plates Force is same on topand bottom Thus, slope of velocity profile is constant and velocity profile is a st. line

9. Shearon fluid Flow between 2 plates Shear stress anywherebetween plates y Moving plate u=V V t B t x Fixed plate u=0

10. r x y x Flow between 2 plates • 2 different coordinate systems B V

11. Example: Journal Bearing • Given • Rotation rate, w = 1500 rpm • d = 6 cm • l = 40 cm • D = 6.02 cm • SGoil = 0.88 • noil = 0.003 m2/s • Find: Torque and Power required to turn the bearing at the indicated speed.

12. Example: cont. • Assume: Linear velocity profile in oil film

13. Example: Rotating Disk • Assume linear velocity profile: dV/dy=V/y=wr/y • Find shear stress