Linear pulse structure and signalling in a film flow on an inclined plane

The film flow down an inclined plane has several features that make it an i nteresting prototype for studying transition in a shear flow: the basic par allel state is an exact explicit solution of the Navier-Stokes equations; t he experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets-associ ated with the full Navier-Stokes equations with viscous free-surface bounda ry conditions-are analysed by using the formalism of absolute and convectiv e instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, omega) = 0, where k is a wavenumber, omega is a frequency a nd D(k, omega) is the dispersion relation function. The main results of this paper are threefold. First, we work with the full Navier-Stokes equations with viscous free-surface boundary conditions, rath er than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instabilit y-as predicted by some model equations-is possible. Secondly, our numerical results find only convective instability, in complete agreement with exper iments. Thirdly, we find a curious saddle-point bifurcation which affects d ramatically the interpretation of the convective instability. This is the f irst finding of this type of bifurcation in a fluids problem and it may hav e implications for the analysis of wavepackets in other flows, in particula r for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the avai lable experimental data, confirming the importance of convective instabilit y for this problem. The numerical results on the position of a dominant sad dle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuat ion procedure for tracking convective instability that until now was consid ered as reliable. While for two-dimensional instabilities a numerical imple mentation of the collision criterion is readily available, the only existin g numerical procedure for studying three-dimensional wavepackets is based o n the tracking technique. For the present flow, the comparison shows a fail ure of the tracking treatment to recover a subinterval of the interval of u nstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of t he saddle point, where V is a bifurcation parameter. We argue that this bif urcation of unstable ray velocities should be observable in experiments bec ause of the abrupt increase by a factor of about 5.3 of the wavelength acro ss the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial ampli fication rate are also discussed.