The document discusses fluid mechanics, focusing on concepts such as hydrostatic force on inclined surfaces, the first and second moments of area, and pressure distribution. It describes how to calculate pressures on submerged surfaces, determine the center of pressure, and understand hydrostatic thrust on curved surfaces. Additionally, it elaborates on buoyancy principles, including Archimedes' principle regarding the buoyant force acting on submerged bodies.

1. Fluid Mechanics
By Samar Singhal

2. Hydrostatic Force on Inclined Surface

3. First Moment of Area
•The first moment of area of a shape, about a certain axis,
equals the sum over all the infinitesimal parts of the shape of
the area of that part times its distance from the axis .
•First moment of area is used to determine the centroid of an
area.

4. Second and Product of moment of Area
•The second moment of area Ixx (often denoted as Ix) can be
computed as
•The product moment of area is defined as

5. Product of moment of Area
•Green area need to balance the
red area.
•Automatically satisfied when x
and y are symmetry axes.

6. Hydrostatic Force on Inclined Surface
 The pressure on any submerged surface varies
linearly with depth.
 If h is the depth to any element area dA of the
plate, the pressure there is
 From the figure
 First Moment of Area
 Therefore
 θ is constant along the plate
 Force on the Plate

7. Hydrostatic Pressure on Inclined Surface
 But,
 Therefore Force on the plate can be written as
 whereas hcg the depth straight down from the surface to the
plate centroid pcg is the pressure at centroid of the plate.
 The force on one side of any plane submerged surface in a
uniform fluid equals the pressure at the plate centroid times
the plate area, independent of the shape of the plate or the
angle θ at which it is slanted.

8. Pressure Distribution
The hydrostatic pressure force on a plane surface is equal,
regardless of its shape, to the resultant of the three-
dimensional linear pressure distribution on that surface.

9. Special Cases

10. Center of Pressure
•To find the coordinates ( xCP , yCP ), we sum moments of the
elemental force p dA about the centroid and equate to the
moment of the resultant F .
•vanishes due to on centroid of centroidal axes
•Let
•Then
•Ixx is the area moment.

11. Center of Pressure
•The negative sign in Eq. shows that yCP is below the centroid
at a deeper level and, unlike F , depends on angle θ . If we
move the plate deeper, yCP approaches the centroid because
every term remains constant except pCG , which increases.
•determination of xCP
•For positive Ixy, xCP is negative because the dominant

12. Hydrostatic Force on Inclined Surface
•In most cases the ambient pressure pa is neglected because
it acts on both sides of the plate; for example, the other side
of the plate is inside a ship or on the dry side of a gate or
dam. In this case pCG = γhCG , and the center of pressure
becomes independent of specific weight:

13. Pressure Prism
•Length of prism is the linearly varying pressure.
Its volume is equal to the magnitude of the resultant
hydrostatic force acting on the plate since FR = ʃ P dA, and the
line of action of this force passes through the centroid of
this homogeneous prism. The projection of the centroid on
the plate is the pressure center.

14. Centroidal moments of inertia for various cross sections:

15. Hydrostatic thrust on a submerged
curved surface

16. Hydrostatic thrust on a submerged
curved surface
•The hydrostatic force on the elemental area dA is
•The force acts in a direction normal to the area dA.
•The components of the force dF in x, y and z directions are
•The components of the surface element dA projected on yz,
xz and xy planes are, respectively

17. Hydrostatic thrust on a submerged
curved surface
Forces can be written as:
Integrating the forces:
zc is the z cordinate of centroid of area Ax and Ay the
projected areas on yz and and xz planes respectively.

18. Center of Pressure
Point of action of Fx
Point of action of Fy
For a curved surface, the component of hydrostatic force in a
horizontal direction is equal to the hydrostatic force on the

19. Vertical component of force
Vertical component of the hydrostatic force:
where V is the volume of the body of liquid within the
region extending vertically above the submerged
surface to the free surface of the liquid.
Therefore, the vertical component of hydrostatic force
on a submerged curved surface is equal to the weight
of the liquid volume vertically above the solid surface
to the free surface of the liquid and acts through the
centre of gravity of the liquid in that volume.

20. Surface submerged in a multilayered
fluid
The hydrostatic force on a surface submerged in a
multilayered fluid can be determined by considering
parts of the surface in different fluids as different
surfaces.

21. Hydrostatic thrust on a submerged
curved surface
•We could sum the separate three components of these
elemental pressure forces, but it turns out that we need not
perform a laborious three-way integration.
•The horizontal component of force on a curved surface
equals the force on the plane area formed by the projection
of the curved surface onto a vertical plane normal to the
component.
•The vertical component of pressure force on a curved
surface equals in magnitude and direction the weight of the
entire column of fluid, both liquid and atmosphere, above
the curved surface.

22. Hydrostatic thrust on a submerged
curved surface

23. Hydrostatic thrust on a submerged
curved surface

24. Hydrostatic thrust on a submerged
curved surface
The horizontal component of the hydrostatic force acting on a curved
surface is equal (in both magnitude and the line of action) to
thehydrostatic force acting on the vertical projection of the curved
surface.
The vertical component of the hydrostatic force acting on a curved
surface is equal to the hydrostatic force acting on the horizontal
projection of the curved surface, plus (minus, if acting in the opposite
direction) the weight of the fluid block.

25. Buoyancy
When a body is either wholly or partially immersed in a
fluid, the hydrostatic lift due to the net vertical
component of hydrostatic pressure forces experienced
by the body is called the buoyant force and the
phenomenon is called buoyancy.

26. Buoyancy
•The resultant horizontal force in any direction for such a
closed surface is always zero.
•The vertical forces acting on the two ends of such a prism
•Therefore, the buoyant force (the net vertically upward
force) acting on the elemental prism is
•Integrating will give net buoyant force:

27. Line of action of the force
•The line of action of the force can be found by taking
moment of the force with respect to z-axis.
•Substituting the values:
•Which is the centroid of the displaced volume.
•Buoyant force FB equals to the weight of liquid displaced by
the submerged body of volume V and known as the
Archimedes principle.
•The principle states that the buoyant force on a submerged
body is equal to the weight of liquid displaced by the body,
and acts vertically upward through the centroid of the
displaced volume.

28. Float, Suspended and Sinking Body

29. Numericals
•A tank of oil has a right-triangular panel near the
bottom, as in Fig. Omitting pa , find the (a) hydrostatic
force and (b) CP on the panel.

30. Numericals
A long solid cylinder of radius 0.8 m hinged at point A is used
as an automatic gate, as shown in Fig. When the water level
reaches 5 m, the gate opens by turning about the hinge at point
A. Determine (a) the hydrostatic force acting on the cylinder and
its line of action when the gate opens and (b) the weight of the
cylinder per m length of the cylinder.