Basics of Fluid Mechanics, 2014a

4.5. FLUID FORCES ON SURFACES 107
In cases where average density cannot be represented reasonably 13 , the integral has be
carried out. In cases where density is non–continuous, but constant in segments, the
following can be said
∫
F total = gρhdA=
A
∫
gρ 1 hdA+
A 1
∫
gρ 2 hdA+ ···+
A 2
∫
gρ n hdA (4.131)
A n
As before for single density, the following can be written
⎡
x c1 A 1
x
{ }} {
c2 A
∫
{
2
∫ }} {
F total = g sin β ⎢
⎣ ρ 1 ξdA+ρ 2 ξdA+ ···+ ρ n
A 1 A 2
x cn A
{ }}
n
⎤
∫ {
ξdA⎥
⎦ (4.132)
A n
Or in a compact form and in addition considering the “atmospheric” pressure can be
written as
Total Static Force
n∑
F total = P atmos A total + g sin β ρ i x ci A i
(4.133)
where the density, ρ i is the density of the layer i and A i and x ci are geometrical
properties of the area which is in contact with that layer. The atmospheric pressure can
be entered into the calculation in the same way as before. Moreover, the atmospheric
pressure can include all the layer(s) that do(es) not with the “contact” area.
The moment around axis y, M y under the same considerations as before is
∫
M y = gρξ 2 sin βdA (4.134)
A
After similar separation of the total integral, one can find that
n∑
M y = g sin β ρ i I x ′ x ′ i
(4.135)
If the atmospheric pressure enters into the calculations one can find that
i=1
i=1
Total Static Moment
n∑
M y = P atmos x c A total + g sin β ρ i I x ′ x ′ i
i=1
In the same fashion one can obtain the moment for x axis as
Total Static Moment
n∑
M x = P atmos y c A total + g sin β ρ i I x ′ y ′ i
i=1
(4.136)
(4.137)
13 A qualitative discussion on what is reasonably is not presented here, However, if the variation of
the density is within 10% and/or the accuracy of the calculation is minimal, the reasonable average
can be used.