1. Chapter 10: Fluids

2. Don’t Be Dense
Fluid = any state of matter with the ability to flow. Mostly liquids and gasses (also plasma and colloids, but we will stick to liquid and gas)
Density (ρ) = a ratio of mass per volume
ρ = m/v
Specific gravity (SG) the ratio of the density of a substance to that of water at 4.0oC (which is 1.00g/cm3)

3. Under Pressure Pressure = force per unit area pressure = P = F/A
The units: N/m2 named the Pascal (Pa) 1Pa = 1N/m2
Example: A 60kg person whose feet have a total area of 500cm2 exerts a pressure on the Earth of F/A = mg/A = (60kg)(9.8m/s2) / (0.050m2) = 12,000N/m2. If he stood on one foot, the force is the same, but the area is cut in half, so the pressure is doubled to 24,000N/m2.

4. Pressure Cont. A fluid exerts a pressure in all directions.
The force due to fluid pressure always acts perpendicular to any surface it is in contact with.
Pressure depends on depth. Imagine a point in the ocean at a depth h. The pressure on this point is caused by the force of the weight of the column of water above it.

5. 20,000 Leagues Here’s the math:
F = mg = ρAhg, coming from if ρ = m/v, then m = ρv and v = Ah.
The pressure, P = F/A = (ρAhg)/A = ρgh
P = ρgh
This means that the pressure at equal depths in a uniform fluid are equal.

6. Incompressible Fluid
The last equation can be used for a fluid that does not change in density with depth.
This is actually quite an appropriate approximation for liquids (except at really high pressures) but this is not very true for gasses.
ΔP = ρgΔh

7. Atmospheric Pressure The pressure at sea level is 1.013 x 105 N/m2
This is referred to as one atmosphere (atm)
1 atm = 1.013E5N/m2 = 101.3kPa
Another unit of pressure is the bar (often used in meteorology)
1 bar = 100kPa
How on Earth does a human body survive 100,000 N of force on every square meter of its body?

8. Gauge Pressure
The answer is that every cell in the human body has an internal pressure close to that of 1 atm, so the forces balance out.
A tire gauge measures the pressure above that of atmospheric, which is called the gauge pressure.
If one wants the absolute pressure, P, one must add atmospheric pressure and gauge pressure together.

9. Pascal’s Principal
“Pressure applied to a confined fluid increases the pressure throughout by the same amount”.
Basically Pascal says that liquids in confined spaces are socialist, equally distributing the pressure.
This fact is incredibly useful.

10. In and Out
Pascal’s Principal means that the pressure put into the system at one end is equal to the pressure experienced at the other end or
Pout = Pin or
Fout /Aout = Fin/Ain or
Fout/Fin = Aout /Ain
Where Fout/Fin is called the mechanical advantage.
For example, of the output piston is 20 times the in put piston, then force gets multiplied by 20.

11. Bouyancy Why do submerged objects appear to weigh less?
Remember, weight is a force. So to diminish weight is to diminish the force by applying a counter force.
To place an object into a fluid would be to provide it another surface, which will supply a force perpendicular to the surface. Newton’s third then dictates that the fluid will exert an opposite perpendicular force (i.e in the upward direction.)

12. Buoyant Force
For a submerged cylinder, the fluid will exert a pressure down on the top of the cylinder and a pressure up on the bottom of the cylinder.
The downward pressure is Pdown = ρfgh1, where h1 is the depth at which the top of the cylinder resides and ρf is the density of the fluid.
Using P = F/A, we get Fdown = PdownA = ρfgh1A
Similarly, Fup = PupA = ρfgh2A

13. Buoyant Force Cont
The buoyant force, FB, is the difference between F2 and F1.
FB = F2 – F1 = ρfgA(h2 – h1) = ρfgAh = ρfgv, where v is the volume of the cylinder.

14. Archimedes‘ Principle
From the last slide we had ρfgv
But wait, ρ = m/v and m = ρv, therefore, ρfgv = mfg, which is the weight of the fluid, which takes up the same volume as the object.
The cool thing is this works regardless of the shape of the object. This was first found by Archimedes and is thus named the Archimedes’ Principle: the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by that object.

15. Apparent weight
As we stated earlier, submerged objects appear to weigh less.
w‘ = the apparent weight of the object
w‘ = F’T = w – FB = ρOgv – ρfgv
This yields a useful proportion

16. Fluid dynamics
Everything we have done thus far would be called fluid statics.
Now we will explore a fluid that is moving.
To begin with, there are two types of fluid motion:
Laminar flow = smooth flow
Turbulent flow = has erratic, small whirlpools called “eddy” currents. Much energy is removed from the system in these eddy currents.

17. Inertia analog
Not every fluid flows at the same rate. Water and ketchup do not flow the same.
The difference is internal friction, or resistance to flow, called viscosity.
We usually think of viscous fluids as being “thick”
Several factors affect viscosity: density, pressure, temperature, electrostatic factors, etc.

18. Flow rate
The mass flow rate is defined at the Δm of fluid that passes a given point per unit time Δt, or Δm/Δt.
Picture a fluid flowing in a pipe that reaches a choke point.
A1v1 = A2v2 this is called the equation of continuity

19. Bernoulli’s Principle
Bernoulli’s Principle: where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high.