In physics, fluid dynamics is a subdiscipline of fluid
mechanics that deals with fluid flow—the natural science of fluids (liquids and
gases) in motion. It has several subdisciplines itself, including aerodynamics
(the study of air and other gases in motion) and hydrodynamics (the study of
liquids in motion). Fluid dynamics has a wide range of applications, including
calculating forces and moments on aircraft, determining the mass flow rate of petroleum
through pipelines, predicting weather patterns, understanding nebulae in interstellar
space and reportedly modelling fission weapon detonation. Some of its
principles are even used in traffic engineering, where traffic is treated as a
continuous fluid.

Fluid dynamics offers a systematic structure—which
underlies these practical disciplines—that embraces empirical and
semi-empirical laws derived from flow measurement and used to solve practical
problems. The solution to a fluid dynamics problem typically involves
calculating various properties of the fluid, such as velocity, pressure, density,
and temperature, as functions of space and time.

Before the twentieth century, hydrodynamics was
synonymous with fluid dynamics. This is still reflected in names of some fluid
dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of
which can also be applied to gases.[1]

The foundational axioms of fluid dynamics are the conservation
laws, specifically, conservation of mass, conservation of linear momentum (also
known as Newton's Second Law of Motion), and conservation of energy (also known
as First Law of Thermodynamics). These are based on classical mechanics and are
modified in quantum mechanics and general relativity. They are expressed using
the Reynolds Transport Theorem.

In addition to the above, fluids are assumed to obey
the continuum assumption. Fluids are composed of molecules that collide with
one another and solid objects. However, the continuum assumption considers
fluids to be continuous, rather than discrete. Consequently, properties such as
density, pressure, temperature, and velocity are taken to be well-defined at infinitesimally
small points, and are assumed to vary continuously from one point to another.
The fact that the fluid is made up of discrete molecules is ignored.

For fluids which are sufficiently dense to be a
continuum, do not contain ionized species, and have velocities small in
relation to the speed of light, the momentum equations for Newtonian fluids are
the Navier–Stokes equations, which is a non-linear set of differential
equations that describes the flow of a fluid whose stress depends linearly on
velocity gradients and pressure. The unsimplified equations do not have a
general closed-form solution, so they are primarily of use in Computational
Fluid Dynamics. The equations can be simplified in a number of ways, all of
which make them easier to solve. Some of them allow appropriate fluid dynamics
problems to be solved in closed form.

In addition to the mass, momentum, and energy
conservation equations, a thermodynamical equation of state giving the pressure
as a function of other thermodynamic variables for the fluid is required to
completely specify the problem. An example of this would be the perfect gas
equation of state:

where p is pressure, ρ is density, Ru is the gas
constant, M is the molar mass and T is temperature.

Terminology in fluid dynamics

The concept of pressure is central to the study of
both fluid statics and fluid dynamics. A pressure can be identified for every
point in a body of fluid, regardless of whether the fluid is in motion or not.
Pressure can be measured using an aneroid, Bourdon tube, mercury column, or
various other methods.

Some of the terminology that is necessary in the
study of fluid dynamics is not found in other similar areas of study. In
particular, some of the terminology used in fluid dynamics is not used in fluid
statics.

Terminology in incompressible fluid dynamics

The concepts of total pressure and dynamic pressure
arise from Bernoulli's equation and are significant in the study of all fluid
flows. (These two pressures are not pressures in the usual sense—they cannot be
measured using an aneroid, Bourdon tube or mercury column.) To avoid potential
ambiguity when referring to pressure in fluid dynamics, many authors use the
term static pressure to distinguish it from total pressure and dynamic
pressure. Static pressure is identical to pressure and can be identified for
every point in a fluid flow field.

In Aerodynamics, L.J. Clancy writes:[9] To
distinguish it from the total and dynamic pressures, the actual pressure of the
fluid, which is associated not with its motion but with its state, is often
referred to as the static pressure, but where the term pressure alone is used
it refers to this static pressure.

A point in a fluid flow where the flow has come to
rest (i.e. speed is equal to zero adjacent to some solid body immersed in the fluid
flow) is of special significance. It is of such importance that it is given a
special name—a stagnation point. The static pressure at the stagnation point is
of special significance and is given its own name—stagnation pressure. In
incompressible flows, the stagnation pressure at a stagnation point is equal to
the total pressure throughout the flow field.

Terminology in compressible fluid dynamics

In a compressible fluid, such as air, the
temperature and density are essential when determining the state of the fluid.
In addition to the concept of total pressure (also known as stagnation pressure),
the concepts of total (or stagnation) temperature and total (or stagnation)
density are also essential in any study of compressible fluid flows. To avoid
potential ambiguity when referring to temperature and density, many authors use
the terms static temperature and static density. Static temperature is
identical to temperature; and static density is identical to density; and both
can be identified for every point in a fluid flow field.

The temperature and density at a stagnation point
are called stagnation temperature and stagnation density.

A similar approach is also taken with the
thermodynamic properties of compressible fluids. Many authors use the terms
total (or stagnation) enthalpy and total (or stagnation) entropy. The terms
static enthalpy and static entropy appear to be less common, but where they are
used they mean nothing more than enthalpy and entropy respectively, and the
prefix "static" is being used to avoid ambiguity with their 'total'
or 'stagnation' counterparts. Because the 'total' flow conditions are defined
by isentropically bringing the fluid to rest, the total (or stagnation) entropy
is by definition always equal to the "static" entropy.

Hydrodynamic stability

In fluid dynamics, hydrodynamic stability is the field
which analyses the stability and the onset of instability of fluid flows.
Instabilities may develop further into turbulence.[1] The foundations of
hydrodynamic stability, both theoretical and experimental, were laid by —
notably — Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth
century.[1]

Rayleigh–Taylor instability

The Rayleigh–Taylor instability, or RT instability
(after Lord Rayleigh and G. I. Taylor), is an instability of an interface
between two fluids of different densities that occurs when one of the fluids is
accelerated into the other.[1][2] Examples include supernova explosions in
which expanding core gas is accelerated into denser shell gas,[3][4]
instabilities in plasma fusion reactors,[5] and the common terrestrial example
of a denser fluid such as water suspended above a lighter fluid such as oil in
the Earth's gravitational field.[2]

To model the last example, consider two completely
plane-parallel layers of immiscible fluid, the more dense on top of the less
dense one and both subject to the Earth's gravity. The equilibrium here is
unstable to any perturbations or disturbances of the interface: if a parcel of
heavier fluid is displaced downward with an equal volume of lighter fluid
displaced upwards, the potential energy of the configuration is lower than the
initial state. Thus the disturbance will grow and lead to a further release of potential
energy, as the more dense material moves down under the (effective)
gravitational field, and the less dense material is further displaced upwards.
This was the set-up as studied by Lord Rayleigh.[2] The important insight by G.
I. Taylor was, that he realised this situation is equivalent to the situation
when the fluids are accelerated, with the less dense fluid accelerating into
the more dense fluid.[2] This occurs deep underwater on the surface of an
expanding bubble and in a nuclear explosion.[6]

Instability will occur even though an idealized,
perfectly static interface between the fluid layers would remain static
forever. For example, the Earth's atmospheric pressure could easily support a
layer of water spread evenly across the flat ceiling of an air filled room.
However, small perturbations of the interface between the water and the air
always set in. These small perturbations grow due to the energetically
favorable displacements described above. It is the RT instability that causes
the layer of ceiling water to inevitably end up on the floor.[1]

As the RT instability develops, the initial
perturbations progress from a linear growth phase into a non-linear or
"exponential" growth phase, eventually developing "plumes"
flowing upwards (in the gravitational buoyancy sense) and "spikes"
falling downwards. In general, the density disparity between the fluids
determines the structure of the subsequent non-linear RT instability flows
(assuming other variables such as surface tension and viscosity are negligible
here). The difference in the fluid densities divided by their sum is defined as
the Atwood number, A. For A close to 0, RT instability flows take the form of
symmetric "fingers" of fluid; for A close to 1, the much lighter
fluid "below" the heavier fluid takes the form of larger bubble-like
plumes.[1]

This process is evident not only in many terrestrial
examples, from salt domes to weather inversions, but also in astrophysics and electrohydrodynamics.
RT instability structure is also evident in the Crab Nebula, in which the
expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected
material from the supernova explosion 1000 years ago.[7] The RT instability has
also recently been discovered in the Sun's outer atmosphere, or solar corona,
when a relatively dense solar prominence overlies a less dense plasma bubble.[8]
This latter case is an exceptionally clear example of the magnetically
modulated RT instability.[9][10]

Note that the RT instability is not to be confused
with the Plateau-Rayleigh instability (also known as Rayleigh instability) of a
liquid jet. This instability, sometimes called the hosepipe (or firehose)
instability, occurs due to surface tension, which acts to break a cylindrical
jet into a stream of droplets having the same volume but lower surface area.

Many people have witnessed the RT instability by
looking at a Lava lamp, although some might claim this is more accurately
described as an example of Rayleigh–Bénard convection due to the active heating
of the fluid layer at the bottom of the lamp

Richtmyer–Meshkov instability

The Richtmyer–Meshkov instability (RMI) occurs when
an interface between fluids of differing density is impulsively accelerated,
e.g. by the passage of a shock wave. The development of the instability begins
with small amplitude perturbations which initially grow linearly with time.
This is followed by a nonlinear regime with bubbles appearing in the case of a
light fluid penetrating a heavy fluid, and with spikes appearing in the case of
a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached
and the two fluids mix.

During the implosion of an inertial confinement
fusion target, the hot shell material surrounding the cold D-T fuel layer is
shock-accelerated. Mixing of the shell material and fuel is not desired and
efforts are made to minimize any tiny imperfections or irregularities which
will be magnified by RMI.

At an entirely different scale, stellar core
materials (e.g. Cobalt-56) from Supernova 1987A were observed earlier than
expected, giving evidence of turbulent mixing due to Richtmyer–Meshkov and Rayleigh–Taylor
instabilities.

Between these two extremes, supersonic combustion in
a Scramjet may benefit from RMI as the fuel-oxidants interface is enhanced by
the breakup of the fuel into finer droplets.

The study of deflagration to detonation transition
(DDT) processes show that RMI-induced spontaneous flame acceleration can result
in detonation.

RMI can be considered the impulsive-acceleration
limit of the Rayleigh–Taylor instability.

R. D. Richtmyer provided a theoretical prediction in
"Taylor instability in a shock acceleration of compressible fluids", Communications
on Pure and Applied Mathematics 13, 297-319 (1960).

E. E. Meshkov (Евгений Евграфович Мешков) provided
experimental verification in "Instability of the Interface of Two Gases
Accelerated by a Shock Wave", Soviet Fluid Dynamics 4,101-104 (1969).

Kelvin–Helmholtz instability

The Kelvin–Helmholtz instability (after Lord Kelvin
and Hermann von Helmholtz) can occur when there is velocity shear in a single continuous
fluid, or where there is a velocity difference across the interface between two
fluids. An example is wind blowing over water: The instability manifests in
waves on the water surface. More generally, clouds, the ocean, Saturn's bands, Jupiter's
Red Spot, and the sun's corona show this instability.

The theory predicts the onset of instability and
transition to turbulent flow in fluids of different densities moving at various
speeds. Helmholtz studied the dynamics of two fluids of different densities
when a small disturbance, such as a wave, was introduced at the boundary
connecting the fluids.

For some short enough wavelengths, if surface
tension is ignored, two fluids in parallel motion with different velocities and
densities yield an interface that is unstable for all speeds. Surface tension
stabilises the short wavelength instability however, and theory predicts
stability until a velocity threshold is reached. The theory with surface
tension included broadly predicts the onset of wave formation in the important
case of wind over water.

Kelvin-Helmholtz billows 500m deep in the Atlantic
Ocean

In gravity, for a continuously varying distribution
of density and velocity (with the lighter layers uppermost, so that the fluid
is RT-stable), the dynamics of the KH instability is described by the Taylor–Goldstein
equation and its onset is given by a Richardson number, Ri. Typically the layer
is unstable for Ri<0.25. These effects are common in cloud layers. The study
of this instability is applicable in plasma physics, for example in inertial
confinement fusion and the plasma–beryllium interface.

Numerically, the KH instability is simulated in a
temporal or a spatial approach. In the temporal approach, experimenters
consider the flow in a periodic (cyclic) box "moving" at mean speed
(absolute instability). In the spatial approach, experimenters simulate a lab
experiment with natural inlet and outlet conditions (convective instability).

he Plateau–Rayleigh instability, often just called
the Rayleigh instability, explains why and how a falling stream of fluid breaks
up into smaller packets with the same volume but less surface area. It is
related to the Rayleigh–Taylor instability and is part of a greater branch of
fluid dynamics concerned with fluid thread breakup. This fluid instability is
exploited in the design of a particular type of ink jet technology whereby a
jet of liquid is perturbed into a steady stream of droplets.

The driving force of the Plateau–Rayleigh
instability is that liquids, by virtue of their surface tensions, tend to
minimize their surface area. A considerable amount of work has been done
recently on the final pinching profile by attacking it with self similar
solutions.[1][2]

History

The Plateau–Rayleigh instability is named for Joseph
Plateau and Lord Rayleigh. In 1873, Plateau found experimentally that a
vertically falling stream of water will break up into drops if its wavelength
is greater than about 3.13 to 3.18 times its diameter.[3] Later, Rayleigh
showed theoretically that a vertically falling column of non-viscous liquid
with a circular cross-section should break up into drops if its wavelength
exceeded its circumference.[4]

Theory

Intermediate stage of a jet breaking into drops.
Radii of curvature in the axial direction are shown. Equation for the radius of
the stream is , where is
the radius of the unperturbed stream, is
the amplitude of the perturbation, is
distance along the axis of the stream, and is
the wave number

The explanation of this instability begins with the
existence of tiny perturbations in the stream.[5][6] These are always present,
no matter how smooth the stream is. If the perturbations are resolved into sinusoidal
components, we find that some components grow with time while others decay with
time. Among those that grow with time, some grow at faster rates than others.
Whether a component decays or grows, and how fast it grows is entirely a
function of its wave number (a measure of how many peaks and troughs per
centimeter) and the radius of the original cylindrical stream. The diagram to
the right shows an exaggeration of a single component.

By assuming that all possible components exist
initially in roughly equal (but minuscule) amplitudes, the size of the final
drops can be predicted by determining by wave number which component grows the
fastest. As time progresses, it is the component whose growth rate is maximum
that will come to dominate and will eventually be the one that pinches the
stream into drops.[7]

Although a thorough understanding of how this
happens requires a mathematical development (see references[5][7]), the diagram
can provide a conceptual understanding. Observe the two bands shown girdling
the stream—one at a peak and the other at a trough of the wave. At the trough,
the radius of the stream is smaller, hence according to the Young–Laplace
equation (discussed above)[where?] the pressure due to surface tension is
increased. Likewise at the peak the radius of the stream is greater and, by the
same reasoning, pressure due to surface tension is reduced. If this were the
only effect, we would expect that the higher pressure in the trough would
squeeze liquid into the lower pressure region in the peak. In this way we see
how the wave grows in amplitude over time.

But the Young-Laplace equation is influenced by two
separate radius components. In this case one is the radius, already discussed,
of the stream itself. The other is the radius of curvature of the wave itself.
The fitted arcs in the diagram show these at a peak and at a trough. Observe
that the radius of curvature at the trough is, in fact, negative, meaning that,
according to Young-Laplace, it actually decreases the pressure in the trough.
Likewise the radius of curvature at the peak is positive and increases the
pressure in that region. The effect of these components is opposite the effects
of the radius of the stream itself.

The two effects, in general, do not exactly cancel.
One of them will have greater magnitude than the other, depending upon wave
number and the initial radius of the stream. When the wave number is such that
the radius of curvature of the wave dominates that of the radius of the stream,
such components will decay over time. When the effect of the radius of the
stream dominates that of the curvature of the wave, such components grow
exponentially with time.

When all the math is done, it is found that unstable
components (that is, components that grow over time) are only those where the
product of the wave number with the initial radius is less than unity ().
The component that grows the fastest is the one whose wave number satisfies the
equation:[7]

Examples

Water dripping from a faucet/tap

Water dropping from a tap.

A special case of this is the formation of small droplets
when water is dripping from a faucet/tap. When a segment of water begins to
separate from the faucet, a neck is formed and then stretched. If the diameter
of the faucet is big enough, the neck doesn't get sucked back in, and it
undergoes a Plateau–Rayleigh instability and collapses into a small droplet.

Urination

Further information: Urination

Another everyday example of Plateau–Rayleigh
instability occurs in urination, particularly standing male urination.[8][9]
The stream of urine experiences instability after about 15 cm (6 inches),
breaking into droplets, which causes significant splash-back on impacting a
surface. By contrast, if the stream contacts a surface while still in a stable
state – such as by urinating directly against a urinal or wall – splash-back is
almost completely eliminated.

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