Class #4 Retarding forces Stokes Law (iscous drag) Newton s Law (inertial drag) Reynolds number Plausibility of Stokes law Projectile motions with iscous drag Plausibility of Newton s Law Projectile motions with inertial drag Worked Problems 1 :10
-D Motion with iscosity m mr mgˆ br Viscousdrag w/ graity : b 1 m mg b ( Term )( component ) Solution fromlasttime : y Term(1 exp( t )) + y0 exp( t ) Nowsolethex component ( whichiseensimplerthan) x m x bx x drag ' x d x dt ' x0 x mg t ln( x) ln( x0) t x x0 exp( ) x :60
Velocity Dependent orce r ( r, r, t ) ( r ) br + cr orces are generally dependent on elocity and time as well as position luid drag force can be approximated with a linear and a quadratic term Ratio quad lin is important f f b c Linear drag factor (Stokes Law, Viscous or skin drag) Quadratic drag factor ( Newton s Law, Inertial or form drag) 3 :15
The Reynolds Number density ρ iscosity η D R inertial ( quad) drag iscous ( linear) drag R < 10 Linear drag 1000< R < 300,000 Quadratic R > 300,000 Turbulent R ρd η 4 :0
Reynolds Number Regimes R < 10 Linear drag 1000< R < 300,000 Quadratic R > 300,000 Turbulent 5
R R D density iscosity ρ D η D η ρ ρ η OR where The Reynolds Number II R inertial iscous inertial iscous inertial ( quad) drag iscous ( linear) π ρ D 16 3πDη drag π ρ D 16 ρd K K R 3πDη η D characteristic length 6 :0
C D Linear Regime C D d 1 ρ kdη A ρa 1 The Reynolds Number III Dη ρd R < 10 Linear drag 1000< R < 300,000 Quadratic D R R > 300,000 Turbulent ρd η 1 (1/ Reynolds #) R QuadraticRegime C D ka ρ 1 1 ρa k 7
ρ Inertial Drag I t A n drag drag cr kρa n t Plate with area A n moes a distance through fluid with density ρ The mass of the fluid displaced is M ρa t n Mass M must acquire a elocity to moe out of the way of the plate. The moing plate is causing Rearranging we get M (ρa t p t p ) n ρa n 8 :35
Inertial Drag II A sphere drag kρa n Preiously demonstrated An k 1 π D 4 1 4 A n means A normal to elocity orm factor for sphere drag π 16 ρd ˆ Plug n play 9 :40
alling raindrops redux II mr mgˆ cr Assume ertical motion m mg c d dt g Define d (1 ) c m mg d g(1 ) c dt g dt x π c ρ D 16 drag mg 1) Newton ) On -axis 3) Rewrite in terms of 4) Rearrange terms 5) Separate ariables 10 :45
alling raindrops redux III d (1 ) g dt t ()/ 0 u d du du (1 u ) t () tanh( gt/ ) d t () t 0 0 (1 ) t g dt gt arctanh( t ()/ ) gt dx arctanh( x) (1 x ) 11 :50
Tanh and sinh and cosh ix e + e cos x ix e e sin x ix e e tan x e ix e + x e + e coshx x e e tanh x e x + e ix ix ix ix x x x f(x) 5 4.5 4 3.5 3.5 1.5 1 0.5 sin, sinh and tanh sinh( x) sin( x) 0 0 50 100 150 00 50 300 350 x [degrees] 1 sin(x) sinh(x) tanh(x) tanh( x) :55
alling 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 iscosity of air is 180 µpoise. 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 elocity) b) Based on a, which type of drag should be more important? c) What should be the terminal elocity of the raindrop, using quadratic drag? d) What should be the terminal elocity of the raindrop, using linear drag? e) Which of the preious of two answers 13 :70 should we use and why?
alling raindrops L4- A small raindrop is gien an initial horiontal elocity of 0 3 xˆ m / s and subsequently falls through a cloud. It has a 10 µm radius. The density of water is 1 g/cc. The iscosity of air is 180 µpoise. a) Quantify the iscous force on the drop for a elocity of 10 mm/sec as well as the inertial force. b) Should this drop be analyed with linear or quadratic drag? c) What is the Reynolds number of this raindrop? d) Write a formula for the position ector of the raindrop as a function of time (set the origin to ero at point where it is released) 14 :50
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 iscosity of water is approximately 1 CentiPoise. a) Quantify the force on the ball for a elocity of 100 mm/sec. b) What should be the terminal elocity of the ball? c) Quantify the force if we assume quadratic drag drag π ρ D 16 15 :50