# Hydrostatics

Hydrostatics is the study of objects submerged in fluids, having zero relative motion between them and the fluid. We are specifically interested in finding the resultant forces acting on a submerged object and it's corresponding line of action.

## Contents |

## Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 87 - 102.

## Basics

From week one, we know that P = ρgh and that the total pressure is given by P = P_{0} + ρgh. For most applications, since atmospheric pressure acts everywhere, and hence on both sides of a submerged body, it can generally be ignored and therefore subtracted.

It should also be noted that the force acting on a body is given by F = PA.

## Resultant Forces on Submerged Plane Surfaces

^{[1]}For a submerged plane in a fluid, the pressure acting on it would be P = P_{0} + ρgh if it was flat (parallel to the surface of liquid). However in most cases the plane surface would be inclined by some angle θ. By defining the height, h, as h = ysinθ, where y is the distance from the plane surface to the fluid surface (in the direction of θ). The equation becomes:

- P = P
_{0}+ ρgysinθ

- P = P

Now the resultant force is given by integrating the pressure profile over the entire area of the surface:

But, from mechanics, we know that . Hence the equation becomes:

In other words, the resultant force is always equal to the pressure acting at the centroid of the plane multiplied by its height to the free surface of the liquid.

## Center of Pressure

^{[2]}The next value we are interested in is where that resultant force acts. Since pressure increases with depth, it is logical to assume that the line of action of the force is at a depth lower than the centroid of the plane.

The analysis is not complicated, it involves taking moments about the centroid, but it is omitted here since it is unnecessary. In general, the center of pressure, or line of action of the resultant force acting on a submerged plane surface is given by:

And if we ignore atmospheric pressure, the equation reduces further to

Where I_{xx} = Second Moment of Area. This property is generally given.

Notice that this value, y_{p} is **not** the vertical distance from the submerged plane to the surface. It is the distance from the center of pressure of the plane to the free surface of the liquid, **in the direction of θ**.

## Resultant Forces on Submerged Curved Surfaces

^{[3]}Submerged curved surfaces are a bit more difficult than plane surfaces, since the pressure continuously changed magnitude and direction of application. To make things easier, we split the resultant force into its horizontal (F_{H})and vertical (F_{V}) components. By noticing that the pressure is equal wherever the height is equal, and that the weight of the volume of the liquid acts downward as well, it is simple to come up with the relation:

- F
_{V}= F_{y}+ W = PA + W= ρghA + ρgV

- F

Whereas the horizontal force is simply the resultant force as if on a submerged plane surface with θ = 90°:

- F
_{H}= F_{x}= P_{0}A + ρgh_{C}A

- F

The resultant force is then given by:

The angle of action can easily be calculated by:

- tanθ = F
_{V}/F_{H }

- tanθ = F

## Buoyancy

^{[4]}An object immersed in a fluid will usually feel lighter than when outside of the fluid. The reason for this is that there exists an upward force on the body due to pressure differences between its top surface and its bottom surface.

Consider a submerged block in a fluid. The pressure at the top of its surface is simply P_{Top} = ρgh, while the pressure on its bottom surface is P_{Bottom} = ρg(h+a) where a is the extra height of the block. The forces are just those pressures multiplied by the applied area. The force difference is then given by:

- F
_{R}= P_{Bottom}- P_{Top}- = ρg(h+a)A - ρghA
- = ρgaA

- But a.A is the Area times the block height, hence aA = V = Volume. Therefore:
- = ρgV

- F

Since this is the density of the fluid that is used, it can be concluded that the upward force, or the **buoyant** force acting on a body is equal to the weight of the fluid displaced by the object. This is also known as **Archimedes' Principle**.

## References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).