Chapter 1 Introduction to stability

To understand the concept of symmetry breaking, consider a flat elastic plate, such as a postcard, subject to an oncoming flow aligned with the surface of the card. Such a situation is shown in fig. 1.1(a). The plate is cantilevered at the downstream end. Despite the configuration of this system being perfectly symmetric along the up-down axis, by virtue of interacting with the surrounding flow, the plate spontaneously bends along a direction perpendicular to the flow, as shown in fig. 1.1(b). This is an example of a system spontaneously breaking symmetry, in this case symmetry of reflection. The governing physics is symmetric under reflection about the dash-dotted line. This symmetry is also referred to as an invariance of the system to reflection, i.e. this system looks and behaves identically when reflected about the dash-dotted line. Yet the system spontaneously evolves towards a state that is not symmetric.

Now consider a different kind of symmetry, that of time-translation invariance. For this symmetry, the example is that of blowing a raspberry. Relax your lip and slightly pucker them. Now gently blow through the lips taking care that you force the flow to be steady. With a little trial and error, you can get the lips to vibrate spontaneously. This is an example where the time-translation symmetry is broken. Here by time-translation symmetry, we mean the equivalence of every instance of time with every other. Because you are blowing steadily, the aerodynamic force on the lips is expected to be unchanging, or invariant, with time. The physical laws governing the motion of the lips are also invariant with time, i.e. they do not change with time. Yet, the lips spontaneously respond by executing time-dependent motion. In this manner, the invariance under time-translation is broken.

Spontaneous appearance of oscillations despite a steady forcing is a common occurence in nature and engineering. Examples include onset of phonation in our vocal cords, the production of sound in many musical instruments such as organ pipes, flutes, clarinets, and the onset of oscillations that preceded disastrous collapse of the Tacoma Narrows Bridge in 1940.

Other examples of the symmetry are:

1. 1.

   Spatial translation symmetry: A infinitely long thread of liquid exhibits spatial translation symmetry along its axis. Every location along its axis is equivalent to every other location. As you have learned, such a thread is susceptible to the Rayleigh-Plateau instability and breaks down into droplets. Different locations along the thread are no longer equivalent, because the droplets pinch off at some locations, while others form the centre of these droplets.

2. 2.

   Axisymmetry: Wine forms tears when swirled in a glass. The shape of the glass is axisymmetric. When swirled, the wine coats the surface of the glass in nearly an axisymmetric manner. But as it falls down, the tear pattern that forms is not axisymmetric. This phenomenon is caused by gradients of surface tension through a mechanism known as the Marangoni effect.

What causes symmetries in nature to be broken? The general explanation for broken symmetries is a positive feedback. Imagine that, for whatever reason, the system starts in a state that is not perfectly symmetric but there is a slight deviation from it. Perhaps the elastic plate is not perfectly aligned with the flow, or perhaps there is a slightly asymmetric gust of wind. Sources of perturbations are omnipresent, which cause the slightest of asymmetries to be introduced in the system. Such sources of asymmetry may be so weak that they are essentially invisible to us. However, the asymmetry causes the fluid dynamic force on the plate to also be slightly asymmetrical. If the nature of the dynamics is such that the resulting force on the system tends to amplify the asymmetry, then the asymmetry will naturally grow until it is noticeable and visible. Such is the nature of the positive feedback which breaks symmetry.

We will further elaborate on the various components of this mechanism using the following example, which is posed as an exercise for the reader.

In question 1.1, the variable $\theta$ represents the degree of asymmetry. When the left-hand side of eq. 1.2, which depends only on the system parameters, exceeds the right-hand-side, which depends only on the system configuration, the feedback on the system asymmetry is positive, and the asymmetry will continue to grow spontaneously.

It is believed that Samuel Langley’s attempts at flight were hindered by the problem of torsional aeroelastic divergence of wings[7]. The aircraft wing near the tips is susceptible to torsion along its span, which acts similar to the torsional spring in question 1.1. When the air flow speed exceeds a certain threshold, or when the torsional stiffness of the wing is small, in close analogy with the analysis in eq. 1.2, twisting of the wing increases the local angle of attack, which further increases the lift force causing the torsional deformation to grow. This is the mechanism of divergence. Torsional divergence is also a serious hindrance to the adoption of forward-swept wing design in subsonic and supersonic flight.

Section 1.2 presents an outline of the mechanism of symmetry breaking. How this type of mechanism can be analyzed systematically in general is presented in this chapter using the formalism of stability theory. The theory applies equally well to fluid dynamic stability as it does to fluid-structure instability, and indeed also extends to the realm beyond fluid or solid mechanics.

From the physical system is carefully curated a set of quantities that are essential for the examination and analysis of the phenomenon under consideration. These quantities are termed as the state variables, which we will formally denote $𝐱$ for the remainder of this chapter. For the examination of a purely hydrodynamic instability, the fluid Eulerian velocity field is usually a part of the state variables. The shape of the interface between two fluids is also a part of the state variables when the motion of the interface is invoked, such as in the case of Rayleigh-Plateau and Rayleigh-Taylor instabilities. The state of temperature of the fluid could also be a part of the state variables for cases where the temperature influences the dynamics, such as in the case of Rayleigh-Benard convection or Marangoni convection.

The state variables are distinct from parameters in the sense that parameters are generally considered to be fixed in a given realization of the system, while the state variables are allowed to evolve with time $t$. Therefore, we tacitly consider $𝐱(t)$.

A specification of the state variables uniquely identifies the relevant portion of the state of the system necessary for understanding the mechanism of the instability. In the example of question 1.1, the state is defined by a single variable $\theta (t)$.

The state evolves in time according to the laws of nature, which govern the system under consideration. The governing evolution may depend on some system parameters, formally denoted as $p$. We formally write the governing physics limited to the phenomenon under consideration as

$$\frac{d𝐱}{dt}=𝓕(𝐱;p),$$

(1.3)

where $𝓕$ is the function that determines the time-rate of change of the state. As stated earlier, note that $p$ are treated as given and constant, so they do not evolve with time. If it is necessary for the system parameters to evolve with time, consider including them as part of the state.

For the example in question 1.1, let us further assume that the plate has negigible mass and its rotational motion about the hinge damps according to a damper with angular damping constant $b$. In other words, a torque $-bd\theta /dt$ about the hinge applies on the plate because of an angular damper. Then the motion of the plate is governed by the torque balance

$$b\frac{d\theta }{dt}=\frac{1}{2}{C}_L\rho U^{2}cwd\sin \theta \cos \theta -\kappa \theta .$$

(1.4)

Here $\rho$, $U$, $c$, $w$, $d$, $\kappa$ and $b$ are system parameters.

Once the governing evolution equations for the state variable are established, we seek a steady state of these equations, especially one which obeys the symmetry of the system under investigation for being broken. Here are some examples:

1. 1.

   For systems obeying time-translation symmetry, which is what we will consider exclusively in the rest of this chapter, we consider a state which is independent of time. This state is invariant under time translation transformation and thus obeys time-translation symmetry. For the case in question 1.1, we consider the possibility that the state $\theta$ is a constant in time.

2. 2.

   For systems obeying, spatial translation symmetry, such as in the case of Rayleigh-Plateau instability, one seeks a state that is constant along that spatial dimension. For Rayleigh-Plateau instability, the liquid thread radius is taken to be uniform along its axis to conform to the spatial translation symmetry along the axis. (Also note that for Rayleigh-Plateau, which also obeys time-translation symmetry, a state chosen is also steady.)

3. 3.

   For systems with axisymmetry, an axisymmetric steady state is chosen.

For the case of question 1.1, which obeys a reflection symmetry and time-translation symmetry, we choose a steady reflection-symmetric state. The only such state is $\theta ={\theta }_0=0$. This state is identical to its reflection.

Formally, let us denote this state as ${𝐱}_0$. This state satisfies

$$𝓕({𝐱}_0;p)=0.$$

(1.6)

That $\theta =0$ obeys eq. 1.4 is readily verified.

Our goal now is to determine the fate of any perturbation away from the steady state we determined in section 1.5. For this purpose, we define the perturbation variable ${𝐱}^{\prime }$ as

$$𝐱={𝐱}_0+{𝐱}^{\prime },$$

(1.7)

and eliminate $𝐱$ in favour of ${𝐱}^{\prime }$. Substituting in eq. 1.3 yields the evolution equation that governs the fate of the perturbation.

$$\frac{d{𝐱}^{\prime }}{dt}=𝓕({𝐱}_0+{𝐱}^{\prime };p),$$

(1.8)

where we have used the steady nature of ${𝐱}_0$ so that $d{𝐱}_0/dt=0$.

For our example in question 1.1, the perturbation is $\theta ={\theta }_0+{\theta }^{\prime }=0+{\theta }^{\prime }={\theta }^{\prime }$. Therefore, ${\theta }^{\prime }$ satisfies

$$b\frac{d{\theta }^{\prime }}{dt}=\frac{1}{2}{C}_L\rho U^{2}cwd\sin {\theta }^{\prime }\cos {\theta }^{\prime }-\kappa {\theta }^{\prime }.$$

(1.9)

In rare cases, such as for eq. 1.9, it is possible to examine the evolution equation for the perturbation and deduce the fate of the perturbation – see the Advanced tip below.

However, in the general case, especially when the governing evolution is not as simple and clean as in question 1.1, this type of analysis is not feasible. There are techniques called “Energy stability”, which can sometimes provide useful information about the stability characteristics. These are outside the scope of this text.

While in the general case, the inference about the stability, positive feedback and symmetry breaking may not be deduced from the equivalent of eq. 1.8, much can be gained by considering a perturbation of infinitesimal size. The rationale for considering infinitesimal perturbations is that it provides a so-called mathematically sufficient condition for instability and symmetry breaking. In the best and most careful experiments, sources of noise and perturbations cannot be completely eliminated and thus are unavoidable. Eliminating perturbations becomes more and more difficult as the perturbation size becomes smaller and smaller. Thus, a system free of small perturbations is an idealization that cannot be practically realized. At the very least, thermal fluctuations are always present. If the natural evolution of the system is such that the tiniest of the perturbations grow in magnitude, then that state is impossible to maintain. Thus, if we can determine that infinitesimal perturbations necessarily grow, then we can safely conclude that such a steady state cannot be maintained. In this manner, we can deduce a sufficient condition for instability and symmetry breaking.

The other reason for considering infinitesimal perturbations is that the arsenal of mathematical techniques that can be deployed for analysis expands. Powerful techniques based on linear algebra can be brought to bear upon the resulting analysis. We will examine these techniques in the next section.

In this section, let us make pursue the consequences of assuming that the perturbations are infinitesimal, i.e. the magnitude of ${𝐱}^{\prime }$ approaches zero. In this case, the right-hand side of eq. 1.8 may be simplified using a Taylor expansion as

$$𝓕({𝐱}_0+{𝐱}^{\prime };p)=𝓕({𝐱}_0;p)+\frac{\delta 𝓕}{\delta 𝐱}({𝐱}_0;p){𝐱}^{\prime }+\text{higher order terms}.$$

(1.10)

Here, by construction of the steady state ${𝐱}_0$, $𝓕({𝐱}_0;p)=0$, and therefore the first nontrivial term that remains on the right-hand side is the linear term. In the limit as the magnitude of ${𝐱}^{\prime }$ approaches zero, the higher order terms limit to zero, and are formally neglected.

With the Taylor expansion of $𝓕$, the evolution of the infinitesimal perturbation obeys the linear equation

$$\frac{d{𝐱}^{\prime }}{dt}=𝓛({𝐱}_0;p){𝐱}^{\prime },\text{where}𝓛({𝐱}_0;p)=\frac{\delta 𝓕}{\delta 𝐱}({𝐱}_0;p)$$

(1.11)

is formally the linear operator that acts on the infinitesimal perturbation.

For the case in question 1.1, the perturbation evolution eq. 1.9 simplifies upon assuming and infinitesimal perturbation ${\theta }^{\prime }$ to

$$b\frac{d{\theta }^{\prime }}{dt}=\left(\frac{1}{2}{C}_L\rho U^{2}cwd-\kappa \right){\theta }^{\prime }.$$

(1.12)

Here we have used $\sin {\theta }^{\prime }\approx {\theta }^{\prime }$ and $\cos {\theta }^{\prime }\approx 1$, and deduced that the higher order terms in this approximation do not enter the linear expression. We shall use this equation in further analysis.

This stage is ripe for examining the dimensions of the perturbation variables, the terms in their evolution equation eq. 1.11, and the dependence on the set of parameters $p$. It is often insightful to use physical intuition to identify scales of mass, length, time and any other relevant dimensions using the parameters $p$, and construct any dimensionless variables that govern the linear system.

In the example presented in question 1.1, the plate perturbation relaxes back to equilibrium on a time scale $b/\kappa$ in the absence of flow. We can choose this scale to non-dimensionalize time as $\tilde{t}=t\kappa /b$, and eliminate $t$ in favour of $\tilde{t}$. In doing so, we also introduce $\alpha ={\displaystyle \frac{C_L\rho U^{2}cwd}{2\kappa }}$ as the only dimensionless parameter. The perturbation angle now satisfies the dimensionless governing equation

$$\frac{d{\theta }^{\prime }}{d\tilde{t}}=\left(\alpha -1\right){\theta }^{\prime }.$$

(1.13)

Note that this step is optional, but if taken, the convenience of the analysis, the quality of result, and the level of insight obtained generally improves greatly.

As mentioned before, the linear nature of eq. 1.11 allows us to unleash the full potential of linear algebra to bear upon this topic. The first property to exploit is based on the definition of linearity itself. If ${𝐱}_1^{\prime }(t)$, ${𝐱}_2^{\prime }(t)$, …, ${𝐱}_n^{\prime }(t)$ all satisfy eq. 1.11, then so does their linear combination

$${𝐱}^{\prime }(t)=c_1{𝐱}_1^{\prime }(t)+c_2{𝐱}_2^{\prime }(t)+\mathrm{\cdots }+c_n{𝐱}_n^{\prime }(t),$$

(1.14)

where $c_1$, $c_2$, …, and $c_n$ are arbitrary constants.

Secondly, we can exploit the time-translation invariance of eq. 1.11 and assert an exponential growth for the perturbation in time. Mathematically,

$${𝐱}^{\prime }(t)=\widehat{𝐱}{e}^{st},$$

(1.15)

where $\widehat{𝐱}$ is a constant vector of the same dimensions as ${𝐱}^{\prime }$, and $s$ is the so-called growth rate associated with $\widehat{𝐱}$. Substituting in eq. 1.11 reveals the relation between $\widehat{𝐱}$ and $s$ as

$$s\widehat{𝐱}=𝓛\widehat{𝐱},$$

(1.16)

which is an eigenvalue equation for $s$. According to eq. 1.16, $s$ must be an eigenvalue of $𝓛$ and $\widehat{𝐱}$ the corresponding eigenvector (or eigenfunction, as the case may be). Note that, here we suppress explicitly writing the arguments of $𝓛({𝐱}_0;p)$. Supposing $𝓛$ has a complete spectrum $\{{\widehat{𝐱}}_1,{\widehat{𝐱}}_2,\mathrm{\dots },{\widehat{𝐱}}_n\}$, i.e. a set of eigenvectors that spans the linear space defined by ${𝐱}^{\prime }$, with corresponding eigenvalues $s_1,s_2,\mathrm{\dots },s_n$, respectively, arranged in descending order of their real parts (i.e. $\mathrm{\Re }(s_1)\ge \mathrm{\Re }(s_2)\ge \mathrm{\cdots }\ge \mathrm{\Re }(s_n)$) the general solution for eq. 1.11 may be written as

$${𝐱}^{\prime }(t)=c_1{\widehat{𝐱}}_1{e}^{s_{1}t}+c_2{\widehat{𝐱}}_2{e}^{s_{2}t}+\mathrm{\cdots }+c_n{\widehat{𝐱}}_n{e}^{s_{n}t}.$$

(1.17)

Here by completeness, we mean that ${\widehat{𝐱}}_1$, ${\widehat{𝐱}}_2$, …, ${\widehat{𝐱}}_n$ form a basis for the linear space of perturbations ${𝐱}^{\prime }$. This is the formal presentation of the method for solving a linear first order differential equation of the form of eq. 1.11. Here the constants $c_1,c_2,\mathrm{\dots },c_n$ are to be determined by the initial condition given as ${𝐱}^{\prime }(t=0)={𝐱}_i^{\prime }$, so that

$${𝐱}_i^{\prime }=c_1{\widehat{𝐱}}_1+c_2{\widehat{𝐱}}_2+\mathrm{\cdots }+c_n{\widehat{𝐱}}_n,$$

(1.18)

which yields unique values for $c_1$, $c_2$, …, $c_n$ owing to the completeness of the spectrum.

In the case of question 1.1, eq. 1.13 is a scalar equation, i.e. the linear space of perturbation ${\theta }^{\prime }$ is one-dimensional. The solution to eq. 1.13 is

$${\theta }^{\prime }(\tilde{t})={\theta }_i^{\prime }{e}^{s\tilde{t}},\text{where}s=\left(\alpha -1\right).$$

(1.19)

At this stage, we interpret the result comprising of eqs. 1.17 and 1.18 in terms of the sufficient condition for instability. We invoked earlier our inability to control the source of perturbations as the inspiration for undertaking this analysis. We now state that relation between the external source of perturbation and the mathematically formulation as follows. The external perturbation source forces an initial condition ${𝐱}_i^{\prime }$, whose subsequent evolution is determined by the preceding analysis. While the source of perturbation is omnipresent, we only need to concern ourselves with the response of our system at a time to one perturbation that causes the initial condition ${𝐱}_i^{\prime }$. It is so because if this one perturbation grows and disrupts the state of the systems, then it is unlikely that the other perturbations conspire in a way to exactly negate the disruption. Underlying this assumption is the impossibility of eliminating infinitesimal perturbations, and thus ${𝐱}_i^{\prime }$ can not be zero.

Similarly, we have no control over the shape of the perturbation. This lack of control implies an absence of knowledge of the exact shape of ${𝐱}_i^{\prime }$, and a tacit treatment of the initial condition as a random variable drawn from a suitable distribution. While this should take us into the realm of probability and stochastic differential equations, hydrodynamic stability avoids following such a route. Instead, the theory concludes that the randomness of ${𝐱}_i^{\prime }$ and the decomposition eq. 1.18 implies that none of the constants $c_1$, $c_2$, …, $c_n$ may be assumed to vanish. This, in turn implies in the light of eq. 1.17, that whether the perturbation grows or decays depends solely on the growth rates $s_1$, $s_2$,…,$s_n$, especially their real parts, and in this way, circumvents the uncertainty of the perturbation shape.

Since we arranged the growth rates in descending order, we infer that all infinitesimal perturbations decay if . Such modes are called linearly stable and when all the modes are stable, the system is said to be linearly stable. This by itself is not a strong inference because it only applies to infinitesimal perturbations. It is silent about perturbations of a finite amplitude. However, if $\mathrm{\Re }(s_1)>0$, then at least one mode must grow despite being infinitesimal in amplitude. The exponentially growth of this mode prevents the system from approaching the steady state ${𝐱}_0$, and the steady state ${𝐱}_0$ becomes impossible to maintain. Such a mode ${\widehat{𝐱}}_1$ is called linearly unstable. Furthermore, if ${\widehat{𝐱}}_1$ is the only unstable eigenvector, i.e. one with an eigenvalue such that $\mathrm{\Re }(s_1)>0$, then we also expect the shape of this mode to be observed in the system. When an unstable mode is present, the system on the whole is considered to be linearly unstable.

As the parameters $p$ are changed, the modes of the system may transition from being stable to unstable. The threshold of stability is determined by the condition that the eigenvalue of $𝓛$ with the largest real part is either zero or purely imaginary, i.e. $\mathrm{\Re }(s)=0$. Such modes are called neutral modes, which neither grow nor decay, and the parameter values at which these modes appear is the threshold of linear instability or the critical condition for the onset of linear instability. Much of linear stability analysis concerns itself with classifying parameter space into linearly stable and unstable regions.

For the example in question 1.1, there is only one growth rate, $s=\alpha -1$. This growth rate is negative when and positive when $\alpha >1$. Thus linear instability ensues when $\alpha >1$, which is the critical condition for the onset of instability. In dimensional terms, the parameter measures the relative strength of the aerodynamic force to the elastic force. When the destabilizing aerodynamic force exceeds the stabilizing elastic force, instability ensues. The critical speed, called the divergence speed, for this structure can be determined from the threshold condition to be

$$U=\sqrt{\frac{2\kappa }{C_L\rho cdw}}.$$

(1.20)

The purpose of the analysis is to gain insight. This motivates making simplifying assumptions so that we can understand the influence of and interaction between various physical effects. While it is difficult to ascertain the fate of arbitrary perturbations made to a system, the framework of linear algebra facilitates the analysis of infinitesimal perturbations. We exploit this to the fullest in linear stability analysis and arrive at a sufficient condition for the disruption of a steady state.

The preceding formalism applies with little modification to cases where the degrees of freedom are finite. In systems that contain an infinite number of degrees of freedom, such as for a continuum fluid, and especially one which is infinite in spatial extent, some of these considerations need to be revisited, the precise meaning of spectrum, eigenvectors and stability need to be analyzed separately and carefully on a case-by-case basis. Some of the modern developments in stability theory, including spatial versus temporal stability, non-normality of modes, resolvent analysis, transient growth of perturbations have arisen from such considerations.

We conclude by noting that linear stability theory is silent about ultimate the fate of the unstable perturbation mode. In some instances, a weakly nonlinear theory may be constructed to determine this fate, but such a treatment is outside the scope of this chapter.