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Fluid Mechanics Chapter 10

Density Recall that the density of an object is its mass per unit volume (SI unit is kg/m3) The specific gravity of a substance is its density expressed in g/cm3

Pressure in Fluids Fluids exert a pressure in all directions
A fluid at rest exerts pressure perpendicular to any surface it contacts The pressure at equal depths within a uniform fluid is the same SI Unit for Pressure is Pa 1 Pa= 1 N/m2 1 atm= kPa=760 mm-Hg

Pressure in Fluids Gauge Pressure is a measure of the pressure over and above the atmospheric pressure i.e. the pressure measured by a tire gauge is gauge pressure. If the tire gauge registers 220 kPa then the absolute pressure is 321 kPa because you have to add the atmosphere pressure (101 kPa) If you want the absolute pressure at some depth in a fluid then you have to add atmosphere pressure

Pressure in Fluids Pascal’s Principle: Pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container

Buoyancy Buoyant force is the force acting on an object that is immersed in a fluid Archimedes Principle: The buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the object Since the buoyant force acts opposite of gravity, an object seems to weigh less in a fluid Apparent Weight= Fg-FB Fb = Buoyant Force Fg = Gravity

Sinking vs Floating Think back to free body diagrams
If the net external force acting on an object is zero then it will be in equilibrium FB If Fb=Fg then the object will be in equilibrium and will FLOAT! Fg

Cubes floating in a fluid
70% Submerged 20% Submerged 100% Submerged

Density determines depth of submersion
This equation gives the percent of the object’s volume that is submerged Vf is the volume of fluid displaced Vo is the total volume of the object ρo is the density of the object ρf is the density of the fluid

Summary of Floating

Continuity Equation Continuity tells us that whatever the volume of fluid in a pipe passing a particular point per second, the same volume must pass every other point in a second. If the cross-sectional area decreases, then velocity increases The quantity Av is the volume rate of flow

Bernoulli’s Principle
The pressure in a fluid decreases as the fluid’s velocity increases. Fluids in motion have kinetic energy, potential energy and pressure

How do planes fly?

Bernoulli’s Equation The kinetic energy of a fluid element is:
The potential energy of a fluid element is:

Bernoulli’s Equation This equation is essentially a statement of conservation of energy in a fluid. Notice that volume is missing. This is because this equation is for energy per unit volume.

Applications of Bernoulli’s Equation
If a hole is punched in the side of an open container, the outside of the hole and the top of the fluid are both at atmospheric pressure. Since the fluid inside the container at the level of the hole is at higher pressure, the fluid has a horizontal velocity as it exits.

Applications of Bernoulli’s Equation
If the fluid is directed upwards instead, it will reach the height of the surface level of the fluid in the container.

Sample Problem p. 306 #40 What is the lift (in newtons) due to Bernoulli’s principle on a wing of area 80 m2. If the air passes over the top and bottom surfaces at speeds of 350 m/s and 290 m/s, respectively. Let’s make point 1 the top of the wing and point 2 the bottom of the wing The height difference between the top of the wing and the bottom is negligible

Sample Problem p.306 #40 The net force on the wing is a result of the difference in pressure between the top and the bottom. P1 is exerted downward, P2 is exerted upward If we know the difference in pressure we can use that to find the force

Sample Problem p.306 #40 P2-P1=20318 Pa

P.306 #43 Water at a pressure of 3.8 atm at street level flows into an office building at a speed of 0.60 m/s through a pipe 5.0 cm in diameter. The pipes taper down to 2.6 cm in diameter by the top floor, 20 m above street level. Calculate the flow velocity and the pressure in such a pipe on the top floor. Ignore viscosity. Pressures are gauge pressures.

Find the flow velocity at the top
A1 is area of first pipe= πr2 = 1.96x10-3 m2 A2 is area of second pipe= πr2 = 5.31x10-4 m2 V1= 0.6 m/s

Find pressure at the top
P2= 1.86 x 105 Pa= 1.8 atm

Sample Problem p.305 #37 What gauge pressure in the water mains is necessary if a fire hose is to spray water to a height of 12.0 m? Let’s make point 1 as a place in the water main where the water is not moving and the height is 0 Point 2 is the top of the spray, so v=0 , P= atmospheric pressure, height = 12m

Sample Problem p.305 #37 Remember that Gauge Pressure is the pressure above atmospheric pressure. So to get gauge pressure, we need to subtract atmospheric Pressure from absolute pressure.

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