  We present a new application of Lagrangian Perturbation Theory (LPT): the
stability analysis of fluid flows. As a test case that demonstrates the
framework we focus on the plane Couette flow. The incompressible Navier-Stokes
equation is recast such that the particle position is the fundamental variable,
expressed as a function of Lagrangian coordinates. The displacement due to the
steady state flow is taken to be the zeroth order solution and the position is
formally expanded in terms of a small parameter (generally, the strength of the
initial perturbation). The resulting hierarchy of equations is solved
analytically at first order. We find that we recover the standard result in the
Eulerian frame: the plane Couette flow is asymptotically stable for all
Reynolds numbers. However, it is also well established that experiments
contradict this prediction. In the Eulerian picture, one of the proposed
explanations is the phenomenon of `transient growth' which is related to the
non-normal nature of the linear stability operator. The first order solution in
the Lagrangian frame also shows this feature, albeit qualitatively. As a first
step, and for the purposes of analytic manipulation, we consider only linear
stability of 2D perturbations but the framework presented is general and can be
extended to higher orders, other flows and/or 3D perturbations.
