Faegrel Book ratings by Goodreads. Herron, Mathematical Reviews show more. This modern introduction is written for any student, researcher, or practitioner working in the area, for whom an understanding of hydrodynamic instabilities is essential. The theory of dynamical systems provides the basic structure of the exposition, together with asymptotic methods. Nonlinear dynamics of systems with few degrees of freedom; 9.

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The presence of a boundary causes some viscosity at the boundary layer which cannot be neglected and one arrives back at the Navier—Stokes equation. Finding the solutions to these governing equations under different circumstances and determining their stability is the fundamental principle in determining the stability of the fluid flow itself.

Linear stability analysis[ edit ] To determine whether the flow is stable or unstable, one often employs the method of linear stability analysis. In this type of analysis, the governing equations and boundary conditions are linearized.

For such disturbances, it is reasonable to assume that disturbances of different wavelengths evolve independently. A nonlinear governing equation will allow disturbances of different wavelengths to interact with each other. Analysing flow stability[ edit ] Bifurcation theory[ edit ] 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.

Hydrodynamic stability is a series of differential equations and their solutions. A bifurcation occurs when a small change in the parameters of the system causes a qualitative change in its behavior, [1].

The parameter that is being changed in the case of hydrodynamic stability is the Reynolds number. It can be shown that the occurrence of bifurcations falls in line with the occurrence of instabilities.

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. Experimental analysis is also useful because it allows one to vary the governing parameters very easily and their effects will be visible.

When dealing with more complicated mathematical theories such as Bifurcation theory and Weakly nonlinear theory, numerically solving such problems becomes very difficult and time consuming but with the help of computers this process becomes much easier and quicker. Since the s computational analysis has become more and more useful, the improvement of algorithms which can solve the governing equations, such as the Navier—Stokes equation, means that they can be integrated more accurately for various types of flow.

Kelvin—Helmholtz instability[ edit ] This is an image, captured in San Francisco, which shows the "ocean wave" like pattern associated with the Kelvin—Helmholtz instability forming in clouds. The Kelvin—Helmholtz instability KHI is an application of hydrodynamic stability that can be seen in nature.

It occurs when there are two fluids flowing at different velocities. The difference in velocity of the fluids causes a shear velocity at the interface of the two layers. 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. Waves are generated by the wind, which shears the water at the interface between it and the surrounding air. The computers on board the satellites determine the roughness of the ocean by measuring the wave height. This is done by using radar , where a radio signal is transmitted to the surface and the delay from the reflected signal is recorded, known as the "time of flight".

From this meteorologists are able to understand the movement of clouds and the expected air turbulence near them. Rayleigh—Taylor instability[ edit ] This is a 2D model of the Rayleigh—Taylor instability occurring between two fluids. In this model the red fluid — initially on top, and afterwards below — represents a more dense fluid and the blue fluid represents one which is less dense.

The Rayleigh—Taylor instability is another application of hydrodynamic stability and also occurs between two fluids but this time the densities of the fluids are different.

If this is the case then both fluids will begin to mix. It is pushed out of the Galactic plane by magnetic fields and cosmic rays and then becomes Rayleigh—Taylor unstable if it is pushed past its normal scale height. Winds that come from the coast of Greenland and Iceland cause evaporation of the ocean surface over which they pass, increasing the salinity of the ocean water near the surface, and making the water near the surface denser.

This then generates plumes which drive the ocean currents. This process acts as a heat pump, transporting warm equatorial water North. Without the ocean overturning, Northern Europe would likely face drastic drops in temperature.

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## Hydrodynamic stability

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