Fluid statics

Fluid statics is the science of fluids at rest. The term usually refers to the mathematical treatment of the subject. It embraces study of the conditions under which fluids are at rest in a stable equilibrium.
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Pressure in fluids
Fluids are characterised by their inability to sustain a shear stress in equilibrium. This is a property of gases, liquids and, to some extent, plastic solids, and leads to their behaviour in tending to take the shape of the containing vessel.
A fluid, then, can only exert stress normal to any contact surface, and the accompanying pressure in the fluid is the same in all directions for a static fluid.
Pascal's Law asserts that for an enclosed fluid, any force applied to the fluid is dispersed as equal pressure throughout the fluid.
Hydrostatic pressure
Considering a small cube of water at rest below a free surface, the weight of the water above must be balanced by a pressure in this small cube. For a infinitely small cube, this weight or equivalent hydrostatic pressure can be expressed as
 <math>\ P = \rho g h<math>
where
P is the hydrostatic pressure (in Pascals)
ρ is the water density (in kilograms per cubic metre)
g is gravitational acceleration
h is the height of fluid above (in metres)
Density, ρ, of a fluid is, in general, a function of temperature and, in the case of compressible fluids, of pressure.
Atmospheric pressure
The MaxwellBoltzmann distribution predicts that, for a gas of constant temperature, T, its density, ρ, will vary with height, h, as:
 <math>\rho (h)=\rho(0) e^{gh/kT}<math>,
 where k is Boltzmann's constant and g is the acceleration due to gravity.
more to be done
Buoyancy
A solid body immersed in a fluid will have an upward buoyant force acting on it equal to the weight of displaced fluid. This is due to the hydrostatic pressure in the fluid.
In the case of a container ship, for instance, its weight force is balanced by a buoyant force from the displaced water, allowing it to float. If more cargo is loaded onto the ship, it would sit lower in the water  displacing more water and thus receive a higher buoyant force to balance the increased weight force.
Discovery of the principle of buoyancy is attributed to Archimedes.
Stability
A floating object is stable if it tends to restore itself to an equilibrium position after a small displacement. For example, floating objects will generally have vertical stability, as if the object is pushed down slightly, this will create a greater buoyant force, which, unbalanced against the weight force will push the object back up.
Rotational stability is of great importance to floating vessels. Given a small angular displacement, the vessel may return to its original position (stable), move away from its original position (unstable), or remain where it is (neutral).
Rotational stability depends on the relative lines of action of forces on an object. The upward buoyant force on an object acts through the centre of buoyancy, being the centroid of the displaced volume of fluid. The weight force on the object acts through its centre of gravity. An object will be stable if an angular displacement moves the line of action of these forces to set up a 'righting moment'.
Liquids with free surfaces
Liquids can have free surfaces at which they interface with gases, or with a vacuum. In general, the lack of the ability to sustain a shear stress entails that free surfaces rapidly adjust towards a equilibrium. However, on small length scales, there is an important balancing force from surface tension.
Surface tension effects
Capillary action
When liquids are constrained in vessels whose dimensions are small, compared to the relevant length scales, surface tension effects become important leading to the formation of a meniscus through capillary action.
Drops
Without surface tension, drops would not be able to form. The dimensions and stability of drops are determined by surface tension.