Fluid Mechanics/Fluid Statics/Fundamentals of Fluid Statics

Hydrostatic Equilibrium

Hydrostatic equilibrium or hydrostatic balance is the defining condition of fluids studied in fluid statics. Hydrostatic equilibrium is the condition in which a volume of fluid is at rest or moves with constant velocity. Although individual molecules in a fluid are not at rest with respect to each other, hydrostatic equilibrium mandates that the system as a whole must be stationary in some inertial reference frame. In other words, a system in hydrostatic equilibrium contains fluid that is not accelerating.

Static Pressure

This figure shows pressure exerted by particle collisions inside a closed container. The collisions that exert the pressure are highlighted in red.

Static Pressure or simply 'pressure' is the force per unit area applied in the direction perpendicular to a surface. It has to be noted that this definition is only applicable to static fluids (dynamic pressure for fluid in motion). The idea of pressure is as 'stress' in solid mechanics. Mathematically, pressure is defined as

p={\frac {dF}{dA}}

where:

p is Static pressure

F is the component of force perpendicular to the surface

A is the area of the surface

When a force is constant over an area, the pressure acting on that area is simply

p={\frac {F}{A}}

Pressure is a scalar quantity, thus it acts in all directions at any given point. It is like a proportionality constant that relates the vector area element with the normal force acting on it as:

d\mathbf {F} _{n}=-p\,d\mathbf {A} =-p\,\mathbf {n} \,dA.

The minus sign comes from the fact that the force is considered towards the surface element, while the normal vector points outward. In order for pressure to create a force, the pressure must be integrated over some area.

Units of pressure

In 1971, the SI unit for pressure became known as the pascal (symbol: Pa), equal to one newton per square meter (N/m² or kg·m⁻¹·s⁻²), in honor of the French physicist Blaise Pascal. Since the pascal is a relatively small amount of pressure for many engineering purposes, the kilopascal (1 kPa = 1,000 Pa) and the megapascal (1 MPa = 1,000,000 Pa) are often used in its place. The bar (symbol: bar) is defined as 100 kPa, or 100,000 Pa, which has the same order of magnitude as atmospheric pressure. However, atmospheric pressure is most closely equivalent to the standard atmosphere (symbol: atm), defined as 101,325 Pa.

The English unit for pressure is the pound per square inch (symbol: psi, lbf/in², or lbf/sq in). It is the pressure resulting from a force of one pound-force applied to an area of one square inch. 1 psi is approximately equal to 6894.757 Pa.

Another non-SI unit of pressure is the torr (Symbol: Torr), which is defined to be 760 atm. The torr was chosen to be approximately equal to the pressure exerted by one millimeter of mercury (symbol: mm_Hg). The torr was named in honor of Evangelista Torricelli, an Italian physicist who discovered the use of the mercury barometer in 1643.

Since a system under pressure has the potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to energy density and may be expressed in units such as joules per cubic metre (J/m³, which is equal to Pa).

What follows is a table listing the conversions between common units of pressure.

Pressure units
	pascal	bar	standard atmosphere	torr	pound per square inch
	Pa	bar	atm	Torr	psi
1 Pa	≡ 1 N/m²	10⁻⁵	9.8692×10⁻⁶	7.5006×10⁻³	145.04×10⁻⁶
1 bar	10⁵	≡ 10⁵ Pa	0.98692	750.06	14.5037744
1 at	0.980665 ×10⁵	0.980665	0.96784	735.56	14.223
1 atm	1.01325 ×10⁵	1.01325	≡ p₀	760	14.696
1 Torr	133.322	1.3332×10⁻³	1.3158×10⁻³	≈ 1 mm_Hg	19.337×10⁻³
1 psi	6.895×10³	68.948×10⁻³	68.046×10⁻³	51.715	≡ 1 lbf/in²

Absolute pressure and gauge pressure

Absolute pressure is zero-referenced against a perfect vacuum, so it is equal to gauge pressure plus atmospheric pressure. Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric pressure. Negative signs are usually omitted.

Hydrostatics

The pressure distribution in a fluid under gravity is given by the relation

${\frac {dp}{dz}}=-\rho g$

where dz is the change in the direction of the gravitational field (usually in the vertical direction). Note that it is quite straightforward to get the relations for arbitrary fields too, for instance, the pseudo field due to rotation.

The pressure in a fluid acts equally in all directions. When it comes in contact with a surface, the force due to pressure acts normal to the surface. The force on a small area dA is given by p dA where the force is in the direction normal to dA. The total force on the area A is given by the vector sum of all these infinitesimal forces.

Mass Density

It is the ratio of mass per unit volume. It is denoted by symbol rho ρ=mass/volume. And it’s S.I. unit is kg/m³

Compressibility

In thermodynamics and fluid mechanics, compressibility (also known as the coefficient of compressibility or isothermal compressibility) is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change.