1 Class #4 Retarding forces Stokes Law (viscous drag) 2-D motions with viscous drag Newton’s Law (inertial drag) Reynolds number Plausibility of Stokes law Plausibility of Newton’s Law Projectile motions with inertial drag Worked Problems :10

2 2-D Motion with viscosity z x :60

3 Velocity Dependent Force  Forces are generally dependent on velocity and time as well as position  Fluid drag force can be approximated with a linear and a quadratic term = Linear drag factor (Stokes Law, Viscous or “skin” drag) = Quadratic drag factor ( Newton’s Law, Inertial or “form” drag) :15

4 The Reynolds Number R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent D v :20

5 Reynolds Number Regimes R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent

6 The Reynolds Number II :20 D “D”= “characteristic” length

7 The Reynolds Number III R < 10 – Linear drag 1000< R < 300,000 – Quadratic R > 300,000 – Turbulent D v

8 Inertial Drag I Plate with area “ A n ” moves a distance through fluid with density The mass of the fluid displaced is Mass “M” must acquire a velocity “v” to move out of the way of the plate. The moving plate is causing Rearranging we get AnAn :35

9 Inertial Drag II – A sphere  Previously demonstrated  “A n ” means “A normal to velocity”  Form factor for sphere  Plug ‘n’ play :40

10 Falling raindrops redux II 1) Newton 2) On z-axis 3) Rewrite in terms of v 4) Rearrange terms 5) Separate variables z x :45

11 Falling raindrops redux III :50

12 Tanh and sinh and cosh :55

13 Falling raindrops L4-1 A small raindrop falls through a cloud. It has a 1 mm radius. The density of water is 1 g/cc. The viscosity of air is 180  Poise. The density of air is 1.3 g/liter at STP. a) What is the Reynolds number of this raindrop? (assume 10 m/s fall velocity) b) Based on “a”, which type of drag should be more important? c) What should be the terminal velocity of the raindrop, using quadratic drag? d) What should be the terminal velocity of the raindrop, using linear drag? e) Which of the previous of two answers should we use and why? :70

14 Falling raindrops L4-2 A small raindrop is given an initial horizontal velocity of and subsequently falls through a cloud. It has a 10  m radius. The density of water is 1 g/cc. The viscosity of air is 180  Poise. a) Quantify the viscous force on the drop for a velocity of 10 mm/sec as well as the inertial force. b) Should this drop be analyzed with linear or quadratic drag? c) What is the Reynolds number of this raindrop? d) Write a formula for the position vector of the raindrop as a function of time (set the origin to zero at point where it is released) :50

15 Pool Ball L4-3 A pool ball 6 cm in diameter falls through a graduated cylinder. The density of the pool ball is 1.57 g/cc. The viscosity of water is approximately 1 CentiPoise. a) Quantify the force on the ball for a velocity of 100 mm/sec. b) What should be the terminal velocity of the ball? c) Quantify the force if we assume quadratic drag :50

16 Lecture #4 Wind-up. :72  Linear Drag (Sphere)  Quadratic Drag (Sphere)