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This paper is on the effect of nonlinearity in the equations for propagationof disturbances on transition in the class of Spiral Poiseuille Flows. Theproblem is approached from the fundamental point of view of following thegrowth of initially infinitesimally small disturbances into their nonlinearstage when the effect of Reynolds stresses makes itself felt. To this end arational scheme of an iterative solution method for the set of GeneralisedNonlinear Orr-Sommerfeld, Squire and Continuity equations is proposed. Thepresent proposal for Spiral Poiseuille Flows closely follows that put forth in1971 by Stuart and Stewartson in their seminal papers on the influence ofnonlinear effects during transition in bench-mark flows, which are the flow inthe gap between concentric circular cylinders (Taylor instability), and in theplane walled channel flow.  The basic feature of the proposed method is the introduction of an AmplitudeParameter and of a slow/long scale variable to account for the effect ofgrowing disturbances within the framework of a rational iteration scheme. It isshown that the effect of amplified disturbances is captured, as in thebench-mark flows, by a Ginzburg-Landau type differential equation for anAmplitude Function in terms of suitably defined slow/long scale variables.However, the coefficients in this equation are numbers that depend upon theflow parameters of the Spiral Poiseuille Flow, which are the Reynolds Number,the Swirl Number, and the geometric parameter of transverse curvature of theflow geometry.

A Nonlinear Instability Theory for a Wave System inducing Transition in Spiral Poiseuille Flows