Basics of Fluid Mechanics, 2014a

4.6. BUOYANCY AND STABILITY 131
Thus, the distance BG is (see Figure 4.38)
3.0
Stability of Square Block
α =0.1
h {}}{
1
BG = h 2 − ρ s
h 1 ρ l 2 = h (
1 − ρ )
s
2 ρ l
(4.162)
ĜM
h
2.5
2.0
1.5
1.0
0.2
0.1
0.0
-0.1
-0.2
-0.3
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
α =0.2
α =0.3
I { }}
xx
{
✓La 3
GM = ✁gρ l
12 − h
✁gρ s }
ah✓L
{{ }
2
V
(
1 − ρ s
ρ l
)
0.5
0.0
-0.5
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
a
h
April 16, 2008
Simplifying the above equation provides
Fig. -4.41. Stability of cubic body infinity long.
GM
h = 1 ( a
) 2 1 − (1 − α) (4.163)
12 α h 2
where α is the density ratio. Notice that
GM/h isn’t a function of the depth, L.
This equation leads to the condition where the maximum height above which the body
is not stable anymore as
a
h ≥ √ 6(1− α)α (4.164)
One of the interesting point for the
above analysis is that there is a point
above where the ratio of the height to the
body width is not stable anymore. In cylindrical
shape equivalent to equation (4.164)
can be expressed. For cylinder (circle) the
moment of inertia is I xx = πb 4 /64. The
distance BG is the same as for the square
shape (cubic) (see above (4.162)). Thus,
the equation is
GM
h = g ( ) 2 b
− 1 (1 − α)
64 α h 2
End Solution
And the condition for maximum height for stability is
b
h ≥ √ 32 (1 − α) α
a
h
2.5
2.0
1.5
1.0
0.5
Stability of Solid Blocks
α =0.4
=0.5
α =0.9 =0.8 =0.7 =0.6
3.0 square
circle
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
April 16, 2008
Fig. -4.42. The maximum height reverse as a
function of density ratio.
This kind of analysis can be carried for different shapes and the results are shown for
these two shapes in Figure 4.42. It can be noticed that the square body is more stable
than the circular body shape.
α