Fluid pressure Fluid pressure is most often the compressive stress at some point within a
fluid. (The term
fluid refers to both liquids and gases – see below for more information specifically about
liquid pressure or
gas pressure.) Fluid pressure occurs in one of two situations: • An open condition, called "open channel flow", e.g. the ocean, a swimming pool, or the atmosphere. • A closed condition, called "closed conduit", e.g. a water line or gas line. Pressure in open conditions usually can be approximated as the pressure in "static" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. Such conditions conform with principles of
fluid statics. The pressure at any given point of a non-moving (static) fluid is called the
hydrostatic pressure. Closed bodies of fluid are either "static", when the fluid is not moving, or "dynamic", when the fluid can move as in either a pipe or by compressing an air gap in a closed container. The pressure in closed conditions conforms with the principles of
fluid dynamics. The concepts of fluid pressure are predominantly attributed to the discoveries of
Blaise Pascal and
Daniel Bernoulli.
Bernoulli's equation can be used in almost any situation to determine the pressure at any point in a fluid. The equation makes some assumptions about the fluid, such as the fluid being ideal and incompressible. \frac{p}{\gamma} + \frac{v^2}{2g} + z = \mathrm{const}, where: •
p, pressure of the fluid, •
{\gamma} =
ρg, density × acceleration of gravity is the (volume-)
specific weight of the fluid, Microscopically, the molecules in solids and liquids have attractive interactions that overpower the thermal kinetic energy, so some tension can be sustained. Thermodynamically, however, a bulk material under negative pressure is in a
metastable state, and it is especially fragile in the case of liquids where the negative pressure state is similar to
superheating and is easily susceptible to
cavitation. In certain situations, the cavitation can be avoided and negative pressures sustained indefinitely, Negative liquid pressures are thought to be involved in the
ascent of sap in plants taller than 10 m (the atmospheric
pressure head of water). • The
Casimir effect can create a small attractive force due to interactions with
vacuum energy; this force is sometimes termed "vacuum pressure" (not to be confused with the negative
gauge pressure of a vacuum). • For non-isotropic stresses in rigid bodies, depending on how the orientation of a surface is chosen, the same distribution of forces may have a component of positive stress along one
surface normal, with a component of negative stress acting along another surface normal. The pressure is then defined as the average of the three principal stresses. • The stresses in an
electromagnetic field are generally non-isotropic, with the stress normal to one surface element (the
normal stress) being negative, and positive for surface elements perpendicular to this. • In
cosmology,
dark energy creates a very small yet cosmically significant amount of negative pressure, which accelerates the
expansion of the universe.
Stagnation pressure Stagnation pressure is the pressure a fluid exerts when it is forced to stop moving. Consequently, although a fluid moving at higher speed will have a lower
static pressure, it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by: p_{0} = \frac{1}{2}\rho v^2 + p where • p_0 is the
stagnation pressure, • \rho is the density, • v is the flow velocity, • p is the static pressure. The pressure of a moving fluid can be measured using a
Pitot tube, or one of its variations such as a
Kiel probe or
Cobra probe, connected to a
manometer. Depending on where the inlet holes are located on the probe, it can measure static pressures or stagnation pressures.
Surface pressure and surface tension There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force. Surface pressure is denoted by π: \pi = \frac{F}{l} and shares many similar properties with three-dimensional pressure. Properties of surface chemicals can be investigated by measuring pressure/area isotherms, as the two-dimensional analog of
Boyle's law, , at constant temperature.
Surface tension is another example of surface pressure, but with a reversed sign, because "tension" is the opposite to "pressure".
Gas pressure In an
ideal gas, molecules have no volume and do not interact. According to the
ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume: p = \frac{nRT}{V}, where: •
p is the absolute pressure of the gas, •
n is the
amount of substance, •
T is the absolute temperature, •
V is the volume, •
R is the
ideal gas constant.
Real gases exhibit a more complex dependence on the variables of state.
Vapour pressure Vapour pressure is the pressure of a
vapour in
thermodynamic equilibrium with its condensed
phases in a closed system. All liquids and
solids have a tendency to
evaporate into a gaseous form, and all
gases have a tendency to
condense back to their liquid or solid form. The
atmospheric pressure boiling point of a liquid (also known as the
normal boiling point) is the temperature at which the vapor pressure equals the ambient atmospheric pressure. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and lift the liquid to form vapour bubbles inside the bulk of the substance.
Bubble formation deeper in the liquid requires a higher pressure, and therefore higher temperature, because the fluid pressure increases above the atmospheric pressure as the depth increases. The vapor pressure that a single component in a mixture contributes to the total pressure in the system is called
partial vapor pressure.
Liquid pressure When a person swims under the water, water pressure is felt acting on the person's eardrums. The deeper that person swims, the greater the pressure. The pressure felt is due to the weight of the water above the person. As someone swims deeper, there is more water above the person and therefore greater pressure. The pressure a liquid exerts depends on its depth. Liquid pressure also depends on the density of the liquid. If someone was submerged in a liquid more dense than water, the pressure would be correspondingly greater. Thus, we can say that the depth, density and liquid pressure are directly proportionate. The pressure in a liquid of uniform density is represented by the following formula: p = \rho gh, where: •
p is liquid pressure, •
ρ is
density of the liquid, •
g is the acceleration due to gravity, •
h is depth within the liquid. Another way of saying the same formula is the following: p = \text{density} \times \text{gravitational acceleration} \times \text{depth}. The pressure a liquid exerts against the sides and bottom of a container depends on the density and the depth of the liquid. If atmospheric pressure is neglected, liquid pressure against the bottom is twice as great at twice the depth; at three times the depth, the liquid pressure is threefold; etc. Or, if the liquid is two or three times as dense, the liquid pressure is correspondingly two or three times as great for any given depth. Liquids are practically incompressible – that is, their volume can hardly be changed by pressure (water volume decreases by only 50 millionths of its original volume for each atmospheric increase in pressure). Thus, except for small changes produced by temperature, the density of a particular liquid is practically the same at all depths. Atmospheric pressure pressing on the surface of a liquid must be taken into account when trying to discover the
total pressure acting on a liquid. The total pressure of a liquid, then, is
ρgh plus the pressure of the atmosphere. When this distinction is important, the term
total pressure is used. Otherwise, discussions of liquid pressure refer to pressure without regard to the normally ever-present atmospheric pressure. The pressure does not depend on the
amount of liquid present. Volume is not the important factor – depth is. The average water pressure acting against a dam depends on the average depth of the water and not on the volume of water held back. For example, a wide but shallow lake with a depth of exerts only half the average pressure that a small deep pond does. (The
total force applied to the longer dam will be greater, due to the greater total surface area for the pressure to act upon. But for a given -wide section of each dam, the deep water will apply one quarter the force of deep water). A person will feel the same pressure whether their head is dunked a metre beneath the surface of the water in a small pool or to the same depth in the middle of a large lake. If four interconnected vases contain different amounts of water but are all filled to equal depths, then a fish with its head dunked a few centimetres under the surface will be acted on by water pressure that is the same in any of the vases. If the fish swims a few centimetres deeper, the pressure on the fish will increase with depth and be the same no matter which vase the fish is in. If the fish swims to the bottom, the pressure will be greater, but it makes no difference which vase it is in. All vases are filled to equal depths, so the water pressure is the same at the bottom of each vase, regardless of its shape or volume. If water pressure at the bottom of a vase were greater than water pressure at the bottom of a neighboring vase, the greater pressure would force water sideways and then up the neighboring vase to a higher level until the pressures at the bottom were equalized. Pressure is depth dependent, not volume dependent, so there is a reason that water seeks its own level. Restating this as an energy equation, the energy per unit volume in an ideal, incompressible liquid is constant throughout its vessel. At the surface of a stationary liquid in a vessel
gravitational potential energy is large but liquid pressure is low. At the bottom of the vessel, all the gravitational potential energy is converted to pressure. The two energy components change linearly with the depth so the sum of pressure and gravitational potential energy per unit volume is constant throughout the volume of the fluid. The units of pressure are equivalent to energy per unit volume. (In the
SI system of units, the pascal is equivalent to the joule per cubic metre.) Mathematically, it is described by
Bernoulli's equation, where velocity head is zero and comparisons per unit volume in the vessel are \frac{p}{\gamma} + z = \mathrm{const}. Terms have the same meaning as in
section Fluid pressure.
Direction of liquid pressure An experimentally determined fact about liquid pressure is that it is exerted equally in all directions. If someone is submerged in water, no matter which way that person tilts their head, the person will feel the same amount of water pressure on their ears. Because a liquid can flow, this pressure is not only downward. Pressure is seen acting sideways when water spurts sideways from a leak in the side of an upright can. Pressure also acts upward, as demonstrated when someone tries to push a beach ball beneath the surface of the water. The bottom of a ball is pushed upward by water pressure (
buoyancy). When a liquid presses against a surface, there is a net force that is perpendicular to the surface. Although pressure does not have a specific direction, force does. A submerged triangular block has water forced against each point from many directions, but components of the force that are not perpendicular to the surface cancel each other out, leaving only a net perpendicular point. This is why liquid particles' velocity only alters in a
normal component after they are collided to the container's wall. Likewise, if the collision site is a hole, water spurting from the hole in a bucket initially exits the bucket in a direction at right angles to the surface of the bucket in which the hole is located. Then it curves downward due to gravity. If there are three holes in a bucket (top, bottom, and middle), then the force vectors perpendicular to the inner container surface will increase with increasing depth – that is, a greater pressure at the bottom makes it so that the bottom hole will shoot water out the farthest. The force exerted by a fluid on a smooth surface is always at right angles to the surface. The speed of liquid out of the hole is \scriptstyle \sqrt{2gh}, where
h is the depth below the free surface. As predicted by
Torricelli's law this is the same speed the water (or anything else) would have if freely falling the same vertical distance
h.
Kinematic pressure P = \frac{p}{\rho_0} is the kinematic pressure, where p is the pressure and \rho_0 constant mass density. The SI unit of
P is m2/s2. Kinematic pressure is used in the same manner as
kinematic viscosity \nu in order to compute the
Navier–Stokes equation without explicitly showing the density \rho_0.
Navier–Stokes equation with kinematic quantities \frac{\partial u}{\partial t} + (u \nabla) u = - \nabla P + \nu \nabla^2 u. ==See also==