The dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically the particular system of three stratified layers with two internal fluid-fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared to the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney-type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilisation of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearised advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All these instabilities are followed into the nonlinear regime by carrying out extensive numerical simulations using spectral methods on periodic domains. It is established that instabilities leading to coherent structures in the form of nonlinear travelling waves are possible even at zero Reynolds number, in contrast to single interface (two-layer) systems; in addition, even in parameter regimes where the flow is linearly stable, the coupling of the flux functions and their hyperbolic-elliptic transitions lead to coherent structures for initial disturbances above a threshold value. When inertia is present an additional short-wave instability enters and the systems become general coupled Kuramoto-Sivashinsky type equations. Extensive numerical experiments indicate a rich landscape of dynamical behaviour including nonlinear travelling waves, time-periodic travelling states and chaotic dynamics. It is also established that it is possible to regularise the chaotic dynamics into travelling wave pulses by enhancing the inertialess instabilities through the advective terms. ...

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Abstract.This study considers the spatially periodic initial value problem of 2×2 quasi-linear parabolic systems having quadratic polynomial flux functions. These systems arise physically in the interfacial dynamics of viscous immiscible multilayer channel flows. The equations describe the spatiotemporal evolution of phase-separating interfaces with dissipation arising from surface tension (fourth order) and/or stable stratification effects (second order). A crucial mathematical aspect of these systems is the presence of mixed hyperbolic-elliptic flux functions that provide the only source of instability. The study concentrates on scaled spatially 2π− periodic solutions as the dissipation vanishes, and in particular the behaviour of such limits when generalised dissipation operators (spanning second to fourth order) are considered. Extensive numerical computations and asymptotic analysis suggest that the existence (or not) of bounded vanishing viscosity solutions depends crucially on the structure of the flux function. In the absence of linear terms (i.e. homogeneous flux functions) the vanishing viscosity limit does not exist in the L ∞ − norm. On the other hand, if linear terms in the flux function are present the computations strongly suggest that the solutions exist and are bounded in the L ∞ − norm as the dissipation vanishes. It is found that the key mechanism that provides such boundedness centres on persistent spatiotemporal hyperbolic-elliptic transitions. Strikingly, as the dissipation decreases, the flux function becomes almost everywhere hyperbolic except on a fractal set of elliptic regions, whose dimension depends on the order of the regularized operator. Furthermore, the spatial structures of the emerging weak solutions are found to support an increasing number of discontinuities (measure-valued solutions) located in the vicinity of the fractally distributed elliptic regions. For the unscaled problem, such spatially oscillatory solutions can be realized as extensive dynamics analogous to those found in the Kuramoto-Sivashinsky equation.

The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are 2 × 2 systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.

We present a new diagnostic diagram for local ultraluminous infrared galaxies (ULIRGs) and quasars, analysing particularly the Spitzer Space Telescope’s Infrared Spectrograph (IRS) spectra of 102 local ULIRGs and 37 Palomar Green quasars. Our diagram is based on a special non-linear mapping of these data, employing the Kernel Principal Component Analysis method. The novelty of this map lies in the fact that it distributes the galaxies under study on the surface of a well-defined ellipsoid, which, in turn, links basic concepts from geometry to physical properties of the galaxies. Particularly, we have found that the equatorial direction of the ellipsoid corresponds to the evolution of the power source of ULIRGs, starting from the pre-merger phase, moving through the starburst-dominated coalescing stage towards the active galactic nucleus (AGN)-dominated phase, and finally terminating with the post-merger quasar phase. On the other hand, the meridian directions distinguish deeply obscured power sources of the galaxies from unobscured ones. These observations have also been verified by comparison with simulated ULIRGs and quasars using radiative transfer models. The diagram correctly identifies unique galaxies with extreme features that lie distinctly away from the main distribution of the galaxies. Furthermore, special two-dimensional projections of the ellipsoid recover almost monotonic variations of the two main physical properties of the galaxies, the silicate and PAH features. This suggests that our diagram naturally extends the well-known Spoon diagram and it can serve as a diagnostic tool for existing and future infrared spectroscopic data, such as those provided by the James Webb Space Telescope.