Linear stability of parallel flows

Consider a steady incompressible flow $U(y)\mathbf e_x$ between parallel boundaries or in an unbounded shear layer. Write the total velocity as this base state plus a disturbance multiplied by a small bookkeeping parameter. Subtract the base-flow equation from Navier–Stokes. Terms linear in the disturbance remain; products of disturbances are discarded. The pressure disturbance is retained because it enforces incompressibility and redistributes momentum between components, even when it does no net work in a closed energy budget.

The result is an approximation to the early evolution of sufficiently small disturbances, not an alternative fluid law. A disturbance eventually amplified to the scale of the base flow invalidates its own linearization. Likewise, an externally maintained base profile requires its actual forcing and boundary conditions to be stated. A calculation about an ideal parallel profile can guide interpretation of a slowly varying experiment, but it does not automatically describe a rapidly developing jet or a separated flow.

For a two-dimensional inviscid disturbance, introduce a streamfunction with components $u'=\partial_y\psi'$ and $v'=-\partial_x\psi'$. A normal mode has the form $\phi(y)e^{i\alpha(x-ct)}$, with real positive streamwise wavenumber $\alpha$ and complex phase speed $c=c_r+ic_i$. Its real physical part propagates at $c_r$ while its envelope multiplies by $e^{\alpha c_i t}$. With this convention, positive imaginary phase speed means temporal growth. Write that envelope explicitly before comparing a formula using a different time convention.

Eliminating pressure and neglecting viscosity gives Rayleigh’s equation. Impermeable walls impose zero streamfunction for a nonzero streamwise wavenumber; disturbances in an unbounded domain must satisfy the appropriate decay condition. An admissible eigenvalue therefore belongs to the differential equation together with its boundary conditions. Arbitrarily choosing a complex number for c is useful for practicing the growth convention, but it does not establish that any specified profile actually possesses that mode.

For the classical smooth inviscid problem, an exponentially growing mode requires the base profile to have an inflection point in the domain. One way to understand the result is to multiply Rayleigh’s equation by an appropriate complex conjugate, integrate, and examine the imaginary part. The resulting integral cannot vanish for a growing mode if the curvature retains one strict sign. This is a powerful exclusion test under its hypotheses, rather than a universal rule about every shear-driven transition.

An inflection point does not guarantee instability, and absence of an inflection point does not settle the viscous problem. Plane Poiseuille flow illustrates why retaining viscosity can qualitatively change an eigenproblem. Even negative growth rates for every eigenmode do not prove that every initial disturbance decays monotonically: nonorthogonal modes can cooperate temporarily. Finally, a finite-amplitude perturbation can reach behavior excluded by a linear approximation. A good stability report says which of these questions it has answered and which remain open.