Hydrodynamic stability
The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence.
[1] The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century.
James Clerk Maxwell expressed the qualitative concept of stable and unstable flow nicely when he said:[1] "when an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the system is said to be unstable.
"That means that for a stable flow, any infinitely small variation, which is considered a disturbance, will not have any noticeable effect on the initial state of the system and will eventually die down in time.
[3] A key tool used to determine the stability of a flow is the Reynolds number (Re), first put forward by George Gabriel Stokes at the start of the 1850s.
[4] In a physical sense, this number is a ratio of the forces which are due to the momentum of the fluid (inertial terms), and the forces which arise from the relative motion of the different layers of a flowing fluid (viscous terms).
The essential problem is modeled by nonlinear partial differential equations and the stability of known steady and unsteady solutions are examined.
The assumption that a flow is incompressible is a good one and applies to most fluids travelling at most speeds.
If one considers a flow which is inviscid, this is where the viscous forces are small and can therefore be neglected in the calculations, then one arrives at Euler's equations:
Although in this case we have assumed an inviscid fluid this assumption does not hold for flows where there is a boundary.
To determine whether the flow is stable or unstable, one often employs the method of linear stability analysis.
Bifurcation theory is a useful way to study the stability of a given flow, with the changes that occur in the structure of a given system.
[1] Laboratory experiments are a very useful way of gaining information about a given flow without having to use more complex mathematical techniques.
Sometimes physically seeing the change in the flow over time is just as useful as a numerical approach and any findings from these experiments can be related back to the underlying theory.
[3] This motion causes the appearance of a series of overturning ocean waves, a characteristic of the Kelvin–Helmholtz instability.
Indeed, the apparent ocean wave-like nature is an example of vortex formation, which are formed when a fluid is rotating about some axis, and is often associated with this phenomenon.
The Kelvin–Helmholtz instability can be seen in the bands in planetary atmospheres such as Saturn and Jupiter, for example in the giant red spot vortex.
There have been many images captured where the ocean-wave like characteristics discussed earlier can be seen clearly, with as many as 4 shear layers visible.
[5] Weather satellites take advantage of this instability to measure wind speeds over large bodies of water.
The computers on board the satellites determine the roughness of the ocean by measuring the wave height.
[6] Once a small amount of heavier fluid is displaced downwards with an equal volume of lighter fluid upwards, the potential energy is now lower than the initial state,[7] therefore the disturbance will grow and lead to the turbulent flow associated with Rayleigh–Taylor instabilities.
[6] This instability also explains the mushroom cloud which forms in processes such as volcanic eruptions and atomic bombs.
This process acts as a heat pump, transporting warm equatorial water North.
[6] The presence of colloid particles (typically with size in the range between 1 nanometer and 1 micron), uniformly dispersed in a binary liquid mixtures, is able to drive a convective hydrodynamic instability even though the system is initially in a condition of stable gravitational equilibrium (hence opposite to the Rayleigh-Taylor instability discussed above).
If a liquid contains a heavier molecular solute the concentration of which diminishes with the height, the system is gravitationally stable.
It has been shown, however, that this mechanism breaks down if the binary mixture contains uniformly dispersed colloidal particles.
[8] The key phenomenon to understand this instability is diffusiophoresis: in order to minimize the interfacial energy between colloidal particle and liquid solution, the gradient of molecular solute determines an internal migration of colloids which brings them upwards, thus depleting them at the bottom.
In order words, since the colloids are slightly denser than the liquid mixture, this leads to a local increase of density with height.
This instability, even in the absence of a thermal gradient, causes convective motions similar to those observed when a liquid is heated up from the bottom (known as Rayleigh-Bénard convection), where the upward migration is due to thermal dilation, and leads to pattern formation.
[8] This instability explains how animals get their intricate and distinctive patterns such as colorful stripes of tropical fish.
