L-14 Fluids [3] Fluids at rest Why things float  Archimedes’ Principle Fluids in Motion  Fluid Dynamics –Hydrodynamics –Aerodynamics

Buoyancy – why things float The trick is to keep the water on the outside of the ship, and to avoid hitting icebergs (which also float), and are easy to miss since 90 % of it is submerged. TITANIC

The area of the hole in the Titanic’s bow was just over a square meter. It sank in 160 minutes.

Buoyant Force F B W P Top A P Bottom A submerged object that has a mass density ρ O The density of the water is ρ W h P = F/A, so F = P  A Pressure increases with depth

Archimedes’ Principle the pressure difference  P= P bottom – P top, times A must equal the weight, W  P x A = W A buoyant force F B =  P x A equal to the weight of displaced water is exerted on a submerged object. The object sinks to the level where F B = W Water weighs 10 N per liter, so every liter of water displaced produces a buoyant force of 10 N W FBFB

Archimedes principle The buoyant force on an object in a fluid equals the weight of the fluid which it displaces. – this works for objects in water – helium balloons (density of He = 0.18 kg/m 3, about 7 times less dense than air) – hot air balloons  the density of hot air is lower than the density of cool air so the weight of the cool air that is displaced is higher than the weight of the balloon

Will it float? The buoyant force is always there whether the object floats or not The object will float if the buoyant force is big enough to support the object’s weight The object will displace just enough water so that the buoyant force = its weight If it displaces as much water as possible and this does not equal its weight, it will sink. Objects that have a density less than water will float- when fully submerged, they weigh less than the water, so the water supports them water weighs about 10 N per liter

Floating objects lighter object heavier object too heavy The weight of displaced water is less than the weight of the object

example problem An object having a volume of 6 liters and weighing W = 30 N is placed in a tank of water. What will happen? Will it sink? Will it float? What fraction of its volume will be submerged if it floats? If the object were completely submerged, the buoyant force would be F B, max = 10N/liter x 6 liters = 60 N thus, the object will float with half of its volume submerged, so that F B = W = 30 N W FBFB

Oil Tankers empty tanker full tanker

Floating in a cup of water Only a thin layer of water around the hull is needed for the ship to float!

Why does ice float? Water, the most plentiful substance on earth is also one of the most unusual in its behavior in that it expands when it freezes. Since it expands, the density of ice is slightly less than the density of water (917 kg/ m 3 as compared to 1000 kg/ m 3 for water). So the part of the iceberg above the surface is less than 10% of the total volume.

Place your bets! When the ice cube melts will: 1)the water spill out, or 2)the water level stay the same, or 3)the level go down ???????? Answer: The level stays the same. Ice is less dense than water, so that the volume occupied by the ice is exactly big enough to hold the volume of melted water that was not submerged! ice cube

Fluid Flow The physics of fluid flow was worked out by Daniel Bernoulli He was born in Switzerland in 1700 He was one of 5 brothers and came from a large family of mathematicians and scientists.

Pressure increases with depth, so the speed of water leaking from the bottom hole is larger than that from the higher ones. fluid flow example – leaky cup

How do we measure fluid flow? We see how much comes out in some time interval Time how long it takes to fill the bucket, say 30 seconds the flow rate is then 1 bucket say per 30 seconds in other words volume per unit time gallons per min (gpm), liters/s, cubic feet per min (cfm), gpf, or m 3 /s  volume flow rate

Volume flow rate If the water comes out of a tube of cross sectional area A with a flow speed u the volume flow rate is volume flow rate = u  A (m/s  m 2 ) To measure u just see how long it takes to fill a gallon jug from a hose and measure the diameter of the hose. m 3 /s

Mass flow rate We could also measure how much mass comes out per unit time – kg/s for example if you are using a fluid of density  coming out of a hose of cross sectional area A with speed v the mass flow rate is mass flow rate =   u  A

What makes water flow? gravity by placing the water up high the pressure at the bottom is high enough to supply water to all parts of town that are lower than the tower Stanton, IA Montgomery Co.

Pressure differences P2P2 P1P1 a pressure difference must be established across the ends of the pipe to push the water along.  P 2 must be greater than P 1 This pressure difference can be set up by a water pump.

Water does not disappear! If water goes in one end of a pipe it must come out the other end (if there are no leaks of course. Sounds obvious, but it has a number of interesting consequences! v 1, A 1 v 2, A 2 This applies to pipes that have constrictions also.

Continuity of flow since whatever goes in must come out we have that the incoming flow rate – outgoing flow rate or v 1 A 1 = v 2 A 2 thus the fluid in the narrow part of the tube must flow FASTER that the fluid on the left. Cardiologists use this to determine if arteries might be clogged.

Other examples - the nozzle effect you use this principle whenever you hold your finger over the end of the hose to make the water spray farther.

An amazing thing about moving fluids The pressure in a moving fluid is less than the pressure in a fluid at rest!  this is Bernoulli's principle. Where a fluid moves faster its pressure is lower, where it moves slower, its pressure is higher. As we will see, this is the principle that makes airplanes work.

The Venturi Meter

wind Bernoulli applies to household plumbing too! When the wind is really blowing, watch the water level in the toilet go up and down air vent sewer roof How toilets work

“atomizers” fine droplets of liquid (not atoms) are sprayed from this device using the Bernoulli effect

Prairie dogs know how to use Bernoulli's principle