An Internet Book on Fluid Dynamics
Linear Stability Analyses
The stability of steady laminar flows of an incompressible, Newtonian fluid of kinematic viscosity, ν,is
assessed by a linear stability analysis that begins with the relevant equations of continuity and motion,
(Bhf3) and (Bhf4):
∂ui
∂xi
=0 (Bkc1)
∂ui
∂t +uj
∂ui
∂xj
=−1
ρ
∂p
∂xi
+ν∂2ui
∂xj∂xj
(Bkc2)
Each of the flow variables, q(where qrepresents uior p) is then decomposed into a steady or time-
independent component denoted by an overbar, ¯q, and a small, time-dependent perturbation, ˜q.The
latter is envisaged as a small perturbation such that ˜q¯qso that terms that are quadratic (or higher
order) in ˜qquantities can be neglected leaving only terms that are either independent of or linear in ˜q
quantities. Consequently various perturbations, ˜q, can be linearly superposed.
We will conduct what is known as a normal mode analysis by considering perturbations that are oscillatory
in time, t, and in one spatial direction, say x,sothat˜qmay be written in the form
q=¯q(xi)+˜q(xi,t)=¯q(xi)+Re ˜q∗(xi)ei(kx−ωt)(Bkc3)
where iisthesquarerootof−1, Re{}denotes the “real part of” and ˜q∗is the amplitude of the
perturbation. The wavenumber, k, and the radian frequency, ω, may both be complex so that
k=kR+ikIand ω=ωR+iωI(Bkc4)
where the subscripts Rand Idenote the real and imaginery parts. This allows two types of solution that
are of particular interest namely
•oscillatory perturbations that have an amplitude that is growing in space but not in time so that
ωI=0andω=ωRis the perturbation frequency. Then kRis the wavenumber of the wave-like
perturbation and kIis the growth or attenuation rate. For convenience we refer to this as the spatial
growth case.
•oscillatory perturbations that have an amplitude that is growing in time but not in space so that
kI=0and2π/k =2π/kRis the wavelength of the perturbation. Then ωRis the spatial wavenumber
of the wave-like perturbation and ωIis the spatial growth or attenuation rate. For convenience we
refer to this as the temporal growth case.
Here we will focus primarily on the first case and examine the rate of growth, ωI, of perturbations of
frequency, ωR,andwavenumber,kR.
When expansions of the form (Bkc3) are substituted into the governing equation (Bkc2), the terms which
are independent of tare isolated and solved to obtain the mean motion. The terms that are linear in the
perturbations ˜q∗are the linear stability equations that are to be solved to determine the stability of the
flow.
In practice, the implementation of this procedure is very difficult unless the basic mean flow is simple. Here
we shall limit the implementation to simple planar, parallel flows in which the unperturbed flow consists
