Chapter 14 – Fluid Mechanics
I. States of Matter
Solid > Liquid > Gas > Plasma
Fluids = matter that can flow = liquids, gases, and plasmas
II. Density of a homogeneous material - density refers to how massive and tightly packed the atoms and molecules are.
A. Density of homogeneous materials:
1. density =
, in kg/m3, g/cm3 and slugs/ft3
2. weight density =
W.D. = , in lbs/ft3
3. Specific Gravity = S.G. = , no units
4. Some densities:
material / r(x 103 kg/m3) / r
(g/cm3) / W.D.
(lbs/ft3) / S.G.
Water / 1.00 / 1.00 / 62.4 / 1.00
Ice / 0.917
Aluminum / 2.70
Gold / 19.3
Mercury / 13.6
Air / 0.00128
B. General Definition of Density:
III. Pressure
In general, when a force is exerted on a surface, the force can be resolved into a normal component and a shear component.
- for solids both forces exist
- for fluids the normal force exists and the shear force will exist if the fluid is viscous.
A. Average Pressure,
units: N/m2 = Pascal (Pa), dynes/cm2, lb/ft2
other units: psi, atmospheres, bar, millibar, mm or cm or inches of mercury
B. Pressure, p
, this represents the pressure at a specific point
C. Constant Pressure
If the pressure is constant, then
IV. Variation of Pressure with Depth
A. How does the pressure in a fluid vary with depth?
Imagine a thin disk of the fluid of thickness dy and area A:
In equilibrium, SFy = 0
Apply the pressure depth differential equation to an incompressible fluid (r = constant)
If point 2 is moved to the surface and the pressure there is p0 , then the pressure at a depth h is
p = p0 + rgh
If the container is open to the atmosphere, then p0 = atmospheric pressure = patm or pa , where
patm = 1.013 x 105 Pa = 14.7 lb/in2 = 1 atm 76 cm of Hg,
and the pressure at a depth h is written as
p = patm + rgh
C. Shape of container is unimportant.
D. Gauge pressure = p - patm = rgh.
e.g., tire pressures are in terms of gauge pressure.
E. Pascal's Principle - a change in pressure applied to a fluid is transmitted equally to all parts of the fluid. Example: hydraulic lift
F. Examples: Remember that the pressure at a particular depth within a continuous fluid is the same
1. U-tube: two immiscible fluids are in a U-tube. For the levels of the fluids given in the u-tube, what is the density of the unknown fluid?
2. barometer
V. Buoyant Force and Archimede's Principle
A. From your own experience, you know that a person seems to weigh considerably less when carried in water. Why?
The fluid exerts a netupward force on the person called a buoyant force. The buoyant force is caused by the difference in the pressures between the top and the bottom of the object. It is the net force the fluid exerts on the object.
net upward force = pBA - pTA
= [(patm + rghB) - (patm + rghA)]A
= rg(hB - hA)A
= rVg = weight of the fluid displaced
B. Archimede's Principle: The buoyant force acting on an object in a fluid equals in magnitude the weight of the fluid displaced by the object.
Wfluid displaced = mfluid displacedg = rfluid displacedVfluid displacedg
C. Examples:
1. A 0.5 kg block of aluminum is suspended from a string. The block of aluminum is completely immersed in water. What is the tension (apparent weight) in the string?
2. What percentage of an iceberg (r = 0.92 g/cm3) is above the surface of seawater (r = 1.024 g/cm3)?
3. How does a submarine work?
4. A cubical block of wood (r = 0.855 g/cm3), 15 cm on a side, floats in sea water (r = 1.034 g/cm3).
a. How far is it from the top of the block to the water?
b. What mass of aluminum must be placed on top of the block so that the top of the block is even with the water level?
c. What mass of aluminum must be tied to the bottom of the wooden block so that the top of the block is even with the water level?
VI. Hydrodynamics
A. Streamline flow - every particle traveling through a point takes the same path. Turbulent flow otherwise.
B. Ideal fluid - nonviscous, incompressible, streamline flow
Continuity Equation: A1v1 = A2v2
Bernoulli's Equation: p1 + rv12 + rgy1 = p2 + rv22 + rgy2
C. Toricelli’s Theorem: A container filled with water has a large surface area. The water fills the container to a height H above the bottom of the container. A small hole is punched in the bottom. At what speed will the water emerge from the hole? What is the flow rate?
D. Examples
1. How far from the edge of the container will the water hit the surface? The cross section of the container is large compared to the size of the hole.
2. A large container holds water to a depth H. At the bottom of the container is a narrow, horizontal pipe with a cross-sectional area A. The horizontal tube narrows to a cross-sectional area of A/2, from which the water flows. A small vertical tube is inserted into the larger cross-section horizontal pipe. To what level, h, does the water rise in the vertical tube?
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