PHYSICS 231 Lecture 22 fluids and viscous flow PHY 231 1 Pressure at depth h P P0 fluidgh h distance between liquid surface and the point where you measure P P0 h P Buoyantt force B f f for submerged b d object bj t B fluidVobjectg Mfluidg wfluid The buoyant y force equals q the weight g of the amount of water that can be put in the volume taken by the object object or fluid If object is not moving B wobject object fluid b b fl d Buoyant force for floating object The buoyant force equals the weight of the amount of water that can be put in the part of the volume of the object that is under water B wobject objectVobject waterVdisplaced h h objectVobject waterA object fluid PHY 231 B h w 2 Bernoulli s equation P1 v12 gy1 P2 v22 gy2 P v2 gy constant The sum of the pressure P P the kinetic energy per unit volume v2 and the potential energy per unit volume gy is constant at all points along a path of flow Note that for an incompressible fluid A1v1 A2v2 This is called the equation of continuity PHY 231 3 hole in a tank P0 y h Pdepth h Pdepth 0 gh If h 1m y 3m what is x Assume that the holes are small and the water level doesn t drop noticeably x PHY 231 4 If h 1 m and y 3 m what is X P0 B y h Use Bernoulli s law A PA vA2 gyA PB vB2 gyB At A PA P0 vA yA y 3 At B Pb P P0 vB 0 0 yB y h 4 y h 4 P0 vA2 g3 P0 g4 vA g 2 2 2 2 2 2 m s x1 PHY 231 5 vA 3m 0 Each water element of mass m has the same velocity y vA Let s look at one element m vA g 2 2 2 m s In the horizontal direction x t x0 v0xt at x t x t t2 2 2t 2 2t x1 In the vertical direction y t y0 v0yt at2 3 0 5gt2 0 when the water hits the ground so tt 0 0 78 78 s so x 0 x 0 78 2 2 0 78 1 72 78 2 2 0 78 1 72 m PHY 231 6 Viscosity Viscosity stickiness of a fluid a measure of its resistance to flow One layer of fluid feels a large resistive force when sliding along another one or along a surface of for example a tube Any real fluid requires a pressure difference between the h ends d of f a tube b to keep k the h fluid fl d moving at constant velocity PHY 231 7 Viscosity Consider a viscous fluid flowing with a constant average speed Vavg through a tube of length L and cross section area A p 8 Lvavg A coefficient of viscosity unit Ns m2 or poise 0 1 Ns m2 The pressure difference causes a volume flow rate Q vavg A Poiseuille s equation PHY 231 8 Buoyant forces When submerged in water an object weighs 1 6N At the same time the water level in the water container with A 0 01 m2 rises 0 01 A 0 01 0 01 m m What is the specific gravity sg of the object water 1 0x103 kg m3 Use the fact that the Buoyant force on a submerged object equals the A 0 01 m2 weight of the displaced water water W Fg B Mobject g Mwater displaced g j p objectVobjectg waterVobjectg Vobjectg object water 6N 0 01 0 01 g 0 0 0 0 g object water 1 0x10 0x 0 4 9 8 9 8 water sg sg 1 1 6N sg 2 63 PHY 231 9 Keep it coming A plastic l ti b bag contains t i a glucose l solution The part of the bag that is not filled is under vacuum If the pressure in a blood vein is 1 33x104 Pa how high must one hang the bag to make sure the solution specific gravity 1 02 enters the body w 1 0x103kg m3 P P0 gh 1 33x104 0 1 02 1 0x103 9 8 h h 1 33 m PHY 231 10 Titanic After the Titanic sunk 10 people manage to seek refuge on a 2x4m wooden raft It is still 11 0 0 cm above water water A heavy debate follows when another person 60 kg wants to board as well Fortunately a PHY231 student is among the 10 Can she h convince i the th others th that th t it is i safe f to t pull ll th the person on board without the whole raft sinking w 1 0x103 kg m g 3 With 10 people Fg B Mraft wg f M10 g Vdisplaced before p f With 11 people Fg Mraft M10 M1 g B Vdisplaced before Vextra wg stationary if Fg B B Mraft M10 g g M M1g g V Vdisplaced before Vextra r ft M displ c d b f r V xtr wg M1g Vextra wg so Vextra M1 w Vextra 60 1 0x103 0 06m3 Vextra LxWxH 8H 0 06 LxWxH 8H 0 06 so H 0 H 0 75cm 75cm so there is still 0 25cm to spare 11 PHY 231 A 5cm2 Bernoulli 1 0x103 kg m3 2m A 2cm2 P 1 atm Water flows over a height of 2m through an oddly shaped pipe A If the fluid velocity is 1 m s at the bottom what is it at the top B What is the water pressure at the top A Use the equation of continuity A1v1 A2v2 5 vvtop 5 2 11 vtop 0 4 m s t 2 t 0 4 B Use Bernoulli Ptop vtop2 ghtop Pbot vbot2 ghbot Ptop 0 5 1E 03 0 42 1E 3 9 8 2 1E 05 0 5 1E 03 12 Ptop 80820 80820 Pa P PHY 231 12 question Two cups are filled to the same level with water One of the two cups has ice cubes floating in it Wh the When h iice cubes b melt l iin which hi h cup is i the h level l l of water higher a The cup without ice cubes b The glass with ice cubes c It is the same in both Before the ice melts B Fg so wgVdisp icegVice Vdispp iceVice water After it melts Mmelted water Mice the same waterVmelted iceVice Vmelted iceVice water PHY 231 global warming 13
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