Download Physics 141 Mechanics Yongli Gao Lecture 4 Motion in 3-D

Physics 141
Mechanics
Lecture 20
Fluid Dynamics
Yongli Gao
• A fluid is a substance that can flow. In contrast to a
solid, a fluid has no shape, and it takes the form of
its container.
• Gases and liquids are fluids. A gas is compressible.
A liquid is almost incompressible. From a
molecular point of view, there's a lot empty space
between the molecules in a gas, and almost none in a
liquid. The molecules in a fluid can move around
everywhere at random, whereas the molecules in a
solid can only vibrate slightly around the
equilibrium positions.
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Density and Pressure
m
The density is the mass of unit volume  
, or ,
V
for a non-uniform object,
dm
(r) 
dV
The unit of density in SI is kg/m3.
The density of solids and liquids are almost
constant, but the density of gases depend on the
pressure and temperature.
F
Pressure is the normal force on unit area, p 
,
A
dF
or, for a non-uniform situation,
p(r) 
dA
The unit of pressure in SI is N/m2, or pascal (Pa).
Hydrostatic Pressure
• What should be the pressure a diver has to bare ?
Why is it difficult to build deep ocean submarines?
• The pressure increases as we go deeper into the
water, described by the equation for hydrostatic
pressure
p  p0  gh
• The equation applies to all incompressible fluids.
For gasses, we'd have to include the compressibility
and the resulting changes in .
• If you consider the free body diagram of a column
of water, you'd see how it works.
Pascal's Principle
• Pascal's principle:
A change in the pressure applied to an enclosed
incompressible fluid is transmitted undiminished to
every portion of the fluid.
• This is the principle behind all the hydraulic
F1 F2
A2
machines since
p  
 F2  F1
A1 A2
A1
• You may have a huge mechanical advantage by
enlarge the ratio of the areas. You don't gain in term
of work since the volume is constant,
A
V  A1 d1  A2 d2  d2  1 d1
A2
A2
A1
 W2  F2 d2  F1  d1  F1d1  W1
A1
A2
Achimedes Principle
• Due to the difference of pressures at the top and the
bottom, any body submerged in a fluid experiences
an upward force termed as buoyancy.
• Achimedes principle:
The force of buoyancy acting on a body is the
weight of the fluid it displaces.
• Consider if the body is made of the same fluid, it'd
be in equilibrium => the total force by the pressure
from all directions equals to the weight of the fluid
displaced.
• This is the principle to all the ships and balloons.
Acting Point of Buoyancy
• As we've seen in statics for rigid bodies, the action
of point of a force is one of the three elements of a
force.
• The acting point of buoyancy is the CM of the
replaced fluid. You can understand it by realizing
that the replaced fluid was at equilibrium before
being replaced by the submerged object. The weight
of the replaced fluid is at its CM, so the total force it
experiences by the surrounding fluid must also at the
CM.
• If the submerged object is uniform, it may float or
sink, but it'll not tilt since its CM is the same as the
displaced fluid. Otherwise, it'll tilt such that the two
CM's are along the same vertical line.
Continuity of Flow
• Let's consider an incompressible fluid flowing in a
tube of varying cross section. In the same time
interval, the quantity of fluid flowing through any
cross section must be the same along the tube,
V  A1v1t  A2 v2 t
 A1v1  A2 v2
• This is the equation of continuity, which links the
speed and cross section of the flow of an
incompressible fluid inside a tube, without any
source or sink in between.
Bernoulli's Equation
• If an incompressible fluid is not in equilibrium, the
pressure and flow velocity may change along the
tube. The relation is the conservation of energy: the
work done by the pressure is converted to the kinetic
energy of the fluid.
1
1
p1V  mgy1  mv12  p2 V  mgy2  mv2 2
2
2
m   V
1 2
1
 p1  gy1  v1  p2  gy2  v2 2
2
2
1 2
 p  gy  v  constant
2