Transcript Stability of an immersed body

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)
FB  F2  F1  W
F2  F1   (h2  h1 ) A
FB  (h2  h1 ) A  (h2  h1 ) A  V ]
FB  V
FB y  F2 y1  F1 y1  wy2
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 completely
immersed 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
V
V
  Fs   mas   mV  s   VV '  s
 V '   s n y
 0Ws   0W sin    V 'sin 
p s
ps 
s 2
p
p
 Fs  ( p  ps ) n y  ( p  ps ) n y  2ps n y   s n y   
s
s
p
0
  Fs  Ws   Fps  ( sin   s ) V '
p
V
 sin  
 V
  as
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.