![[Graphics:../Images/linstab_gr_8.gif]](../Images/linstab_gr_8.gif)
It has been observed in nature, that the steady state solutions for different systems can become unstable to infinitesimal disturbances which should be expected to always be present, (the ground is always vibrating, buildings breath and bend, etc. ...) and possibly because of molecular motions. A common example is the formation of waves on bodies of water owing to the action of wind. The "Taylor- Couette Flow" instability is a popular laboratory instabilility that arises due to centrifugal force, and Rayleigh-Benard convection, which arises because of density differences is important both in nature and in laboratories.
Each of these instabilities has a precise, although not necessarily well understood, physical mechanism. The common feature of an instability is that infinitesimal velocity or density perturbances are amplified (by the base flow or global forces) and thus grow to finite size. Growth of distrubances could be algebraic or exponential. Typical analysis (such as those shown below) assume an exponential growth because it is expected that this would overwhelm any algebraic growth. However, algebraic analyses have been used in some situations where exponential models did not match data. It is not clear that these have matched any better, but this discussion is beyond the point of this introductory module.
Infinitesimal perturbations are expected to be in the form of noise. The question is, how to represent this when we want to model instability. Fortunately, the noise is infinitesimal, which means its amplitude is small compared to any length scale such as its wavelength. This allows the nonlinear governing equations to be well-approximated by linearized versions. The linear equations are amenable to a Fourier mode analysis that can be used to represent any noice signal as a linear combination of independent modes.
If we assume exponential growth (because is the strongest possible and what is observed in nature), and if the growth is in time (which could be how it occurs) an equation for the amplitude, a, of some disturbance, a is a = a0 Exp[ ω t], where a0 is the initial amplitude of the disturbance and ω is the temporal growth rate. Even is a0 is of molecular dimensions (i.e. 10^-10 cm) , and the growth rate is a very reasonable (for common systems), ω = 1 /s. We find that it would take only 23 seconds for the disturbance to reach an amplitude of 1 cm! We thus expect a linearly unstable flow to show some evidence of growing disturbances, unless the residence time is short.
Here is the time calculation
Note that if the amplitude is growing exponentially, at some point nonlinear processes will become important. Nonlinear analysis is beyond the scope of this module.
To do mathematical analysis of an instability, we need to chose a "basestate" that is the base flow in the absence of an instability. This could be 0 velocity or it could be a falling film with no waves or a stratified flow with no waves, etc..
Since we expect that linear equations should govern the initial growth of instability, we will linearize the complete governing equations around the base flow. This is done by taking the baseflow, say u0 and allowing a small perturbation, say [. Thus the complete velocity field would be u = u0 + [ u1. Note this this is for a one dimensional problem. For higher dimensions, you would have order [ components in the other directions even if the base flow in that direction was 0.
The magnitude of [ is small (<<1) because it is, for example, the amplitude to wavelength of the noise signal.
We proceed by substituting u = u0 + [ u1into the governing equations and the boundary conditions and collecting powers of [. Because we desire the analysis to be valid for any arbitrary [, we can separate the system into powers of [.
The ![[Graphics:../Images/linstab_gr_10.gif]](../Images/linstab_gr_10.gif){kind=small}
equations will be the equations for the base state and should be identically 0.
The ![[Graphics:../Images/linstab_gr_11.gif]](../Images/linstab_gr_11.gif){kind=small}
equations, should give the behavior of the very small amplitude disturbance and will contain u0, which we know and u1, the distrubance that we wish to study. These equations are necessarily linear (we just linearized them with this procedure). Any higher powers of [ will be ignored and saved for when we want to do nonlinear analysis.
To determine the response of the equation to an arbitrary noise signal, we choose a mode that represents the kind of disturbance that we expect to see. If the domain is fixed and there is no flow through, (e.g., a solid beam), we might expect to use fixed spatially periodic modes that grow in time. For waves on water, we would use (traveling) spatially and temporally periodic disturbances that could grow in space and/or time.
Since the system is linear, we can examine the response of any separate mode without worrying about the effect of other modes. This linearity allows us to decompose an arbitrary disturbance into a sum of Fourier modes; each mode will ultimately satisfy the equations and boundary conditions. By scanning the entire frequency or wavenumber range, we can be sure that we understand the effect of any initial (infintesimal) disturbance.
![[Graphics:../Images/linstab_gr_9.gif]](../Images/linstab_gr_9.gif){kind=small}