buoyancy, flotation and stability when a stationary body is completely submerged in a fluid, or...
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Buoyancy, Flotation and Stability
• When a stationary body is completely submerged in a fluid, or floating (partially submerged), the resultant fluid force on the body is the buoyant force.• A net upward force results because • Buoyant force has a magnitude equal to the weight of the fluid displaced by body and is directed vertically upward.• Archimedes’ principle (287-212 BC)
2 1BF F F W
2 1 2 1( )F F h h A
2 1 2 1( ) ( ) ]BF h h A h h A V
BF V
2 1 1 1 2BF y F y F y wy
Buoyant force passes through the centroid of the displaced volume
Figure 2.24 (p. 70)
Buoyant force on submerged and floating bodies.
Example 1
A spherical buoys has a diameter of 1.5 m, weighs 8.50 kN
and is anchored to the seafloor with a cable. What is the
tension on the cable when the buoy is completely immersed?
Example 2
• Measuring specific gravity by a hydrometer
Stability of Immersed and Floating Bodies
• Centers of buoyancy and gravity do not coincide
• A small rotation can result in either a restoring or overturning couple.
• Stability is important for floating bodies
Stability of an immersed body
Stability of a completelyimmersed body – center of gravity below entroid.
Stability of a completely immersed body – center of gravity above centroid.
Stability of a floating body
Elementary Fluid Dynamics
• Newton’s second law
• Bernoulli equation (most used and the most abused equation in fluid mechanics)
• Inviscid flow- flow where viscosity is assumed to be zero; viscous effects are relatively small compared with other effects such as gravity and pressure differences.
• Net pressure force on a particle +net gravity force in particle
• Two dimensional flow (in x-z plane)
• Steady flow (shown in Figure 3.1)
Figure 3.1 (p. 95)(a) Flow in the x-y plane. (b) flow in terms of streamline and normal coordinates.
Streamlines
• Velocity vector is tangent to the path of flow
• Lines that are tangent to the velocity vectors throughout
the flow field are called streamlines
• Equation for a streamline:
dr dx dy dz
V u v w
Force balance on a Streamline
's s
V VF ma mV VV
s s
'V s n y
0 0 sin 'sinsW W V
2s
p sp
s
( ) ( ) 2s s s s
p pF p p n y p p n y p n y s n y
s s
0 ( sin ) 's s ps
pF W F V
s
sin s
p VV a
s s
Figure 3.3 (p. 97)
Free-body diagram of a fluid particle for which the important forces are those due to pressure and gravity.
• The physical interpretation is that a change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle weight along the streamline.
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