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.