Lec 4: Fluid statics, buoyancy and stability, pressure

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Lec 4 Fluid statics, buoyancy and stability,
pressure
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• For next time
• Read 3-1 to 3-4
• HW 2
• Outline
• Zeroth law of thermodynamics
• Pressure and resulting forces
• Buoyancy and stability
• Important points
• How to calculate pressure force
• How to calculate application point of pressure
  force
• How to analyze stability

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Fluid statics

• Fluid statics deals with non-flow
  situations--fluids at rest.
• It is particularly applicable with pressure
  measurements in terms of fluid column heights.

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TEAMPLAY

• You accidentally drive your car into a lake and
  it submerges but does not admit a significant
  amount of water into the passenger compartment.
• A. Can you open a door?
• B. How will you get out?

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Fluid statics

• The car door may be regarded as a plane surface
  of area about 10 square feet.
• In order to study the force on the submerged car
  door resisting attempts to open it, we must delve
  into
• Force magnitude
• Force application point, known as center of
  pressure.

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Fluid statics

• Consider the effect of a constant pressure at the
  top of the liquid. This could be Patm or some
  other pressure P0.
• We can neglect P0 as long as it acts on both
  sides.

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Fluid statics

• Consider an arbitrary flat shape and orientation

The pressure at any point on the shape
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Fluid statics

• The resultant force FR is given by

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Fluid statics

• The integral is related to the y coordinate of
  the centroid (center)

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TEAMPLAY

• Your pickup, named Bigfoot, has a door which is 4
  ft high by 3.5 ft wide and all windows are stuck
  in the closed position. The bottom of the door
  is 4 ft off the ground. You accidentally drive
  into a stock tank where it comes to rest on its
  wheels in water 10 ft deep. Assume the bottom of
  the stock tank is flat. What is the force on the
  door? Can you open it?

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Fluid statics

• Now that we know the resultant force on a
  submerged plane body is
• where yc is the y-coordinate of the centroid.
• it is necessary to know where the center of
  pressure is, that is, the point through which it
  acts.

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Fluid statics

• In general the location yP of the center of
  pressure isbelow the location of the centroid
  yC because the pressure increases with depth.

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Fluid statics

• Equate the moment of the resultant force FR to
  the moment of the distributed pressure force
  about the x-axis.
• Where is the second moment
  of area (area moment of inertia).

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Fluid statics

• Most area moments of inertia are given about the
  centroid of the shape (IXX,C).
• They are relate to the moment IXX,0 about the
  x-axis by
• Area moments of inertia about the centroid are in
  Fig. 10-5 for some common shapes. Centroids are
  also given there.

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TEAMPLAY

• Solve Problem 10-13

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Buoyancy

• A buoyant force FB is caused by increasing
  pressure with depth, so

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Buoyancy

• The upward force from the bottom is obviously
  greater, and so the net buoyancy force is
• where is the density of the fluid, not the
  body, and V is the volume of the body.

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TEAMPLAY

• The previous equation does not depend on the
  density of the submerged body.
• What changes about a submarine as it goes up and
  down (with zero propulsive thrust)?
• What is the upward force on a submarine as it
  holds a constant depth?
• Does this force change as it changes depth (with
  zero thrust)?

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Buoyancy and stability

• The buoyant force for a constant volume system is
  equal to the weight W of the displaced fluid.

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Buoyancy

• The gravity force downward on a submerged body
  acts through the centroid.
• Similarly, the buoyant force upward must act
  through the centroid or there would be a rolling
  moment.
• Thus, we have Archimedes Principle
• The buoyant force acting on a body immersed in a
  fluid is equal to the weight of the fluid
  displaced by the body, and it acts upward through
  the centroid of the displaced volume.

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Buoyancy

• For floating bodies, the buoyant force is given
  by the weight of the displaced fluid, or

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Stability

• Immersed bodies must be bottom-heavy to be
  stable. Thus the center of gravity G must below
  the center of buoyancy B so that any disturbance
  will provide a restoring moment about G.

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Stability

• Model a submarine as a horizontal tube with the
  top half empty and the bottom half filled with
  engines, crew quarters, and weaponry. Neglect
  the mass of the shell (tube). Where is the
  center of gravity G? Where is the center of
  buoyancy B? Do the two forces act to restore the
  sub to an upright condition if it starts to roll,
  or increase its rolling tendency?

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Stability

• Rotational stability criteria are similar for
  floating bodies.
• However, if the center of buoyancy shifts during
  rolling motion, it may be possible to have the
  center of gravity G above the center of buoyancy
  and still achieve stability.

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Stability

• The metacenter M is required to be above G. The
  metacenter height is the vertical distance
  between G and M.
• For many hull shapes the metacenter is almost a
  fixed point for rolling angles up to about 20.

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Pressure

• The normal force exerted on a (small) area.
  Small enough that changes over the area are
  unimportant, and large enough that molecular
  effects also are unimportant.

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Pressures

• For pressures above atmospheric

• For pressures below atmospheric

P1
P1
Pgage
Patm
Patm
Pvac
P2
Pabsolute
Pabsolute
P0
P0
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In the SI system we use

• 1 Pa 1 N/m2
• 1 kPa 1,000 N/m2
• 1 bar 100,000 N/m2
• 1 MPa 1,000,000 N/m2

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In the USCS system we use

• lbf/in2 or psi
• psi is usually written with an asuffix (psia)
  or a g suffix, for absolute or gage (psig)

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Atmospheric pressure is
1 atm 14.696 psia 101.325 kPa 1.01
bar 0 psig 14.696 psia
Absolute pressure (Pabs) gage pressure (psig)
atmospheric pressure (Patm)
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For example
A gage pressure of 20.0 psig is an absolute
pressure at standard sea level conditions of
Pabs Patm Pg 14.7 psia 20.0 psig
34.7 psia
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TEAMPLAY

• Consider the bicycle tire, which produces a
  pressure reading of 30 psig. What is the
  absolute pressure at sea level and at 10,000 ft
  altitude where Patm 10 psia?

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Pressure Measurement
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TEAMPLAY
What pressure (above atmospheric) is exerted on
your ears at the bottom of a 12 foot deep
swimming pool?
Assume the density of water is 62.4 lbm/ft3