In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. 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. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. When studying flow stability it is useful to understand more simplistic systems, e.g. incompressible and inviscid fluids which can then be developed further onto more complex flows. Since the 1980s, more computational methods are being used to model and analyse the more complex flows.
Stable and unstable flows To distinguish between the different states of fluid flow one must consider how the fluid reacts to a disturbance in the initial state. These disturbances will relate to the initial properties of the system, such as velocity, pressure, and density. James Clerk Maxwell expressed the qualitative concept of stable and unstable flow nicely when he said: "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. For a fluid flow to be considered stable it must be stable with respect to every possible disturbance. This implies that there exists no mode of disturbance for which it is unstable. On the other hand, for an unstable flow, any variations will have some noticeable effect on the state of the system which would then cause the disturbance to grow in amplitude in such a way that the system progressively departs from the initial state and never returns to it. This means that there is at least one mode of disturbance with respect to which the flow is unstable, and the disturbance will therefore distort the existing force equilibrium.
Determining flow stability
Reynolds number 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. Associated with Osborne Reynolds who further developed the idea in the early 1880s, this dimensionless number gives the ratio of inertial terms and viscous terms. 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 equation for this is
R e = inertial viscous = ρ u 2 μ u L = ρ u L μ = u L ν {\displaystyle R_{e}={\frac {\text{inertial}}{\text{viscous}}}={\frac {\rho u^{2}}{\frac {\mu u}{L}}}={\frac {\rho uL}{\mu }}={\frac {uL}{\nu }}}
where
The Reynolds number is useful because it can provide cut off points for when flow is stable or unstable, namely the Critical Reynolds number R c {\displaystyle R_{c}} . As it increases, the amplitude of a disturbance which could then lead to instability gets smaller. At high Reynolds numbers it is agreed that fluid flows will be unstable. High Reynolds number can be achieved in several ways, e.g. if μ {\displaystyle \mu } is a small value or if ρ {\displaystyle \rho } and u {\displaystyle {\text{u}}} are high values. This means that instabilities will arise almost immediately and the flow will become unstable or turbulent.
Navier–Stokes equation and the continuity equation In order to analytically find the stability of fluid flows, it is useful to note that hydrodynamic stability has a lot in common with stability in other fields, such as magnetohydrodynamics, plasma physics and elasticity; although the physics is different in each case, the mathematics and the techniques used are similar. The essential problem is modeled by nonlinear partial differential equations and the stability of known steady and unsteady solutions are examined. The governing equations for almost all hydrodynamic stability problems are the Navier–Stokes equation and the continuity equation. The Navier–Stokes equation is given by:
∂ u ∂ t + ( u ⋅ ∇ ) u − ν ∇ 2 u = − ∇ p 0 + b , {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla p_{0}+\mathbf {b} ,}
where
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