This paper presents a temporal, spatial, and spatio-temporal linear stability analysis of the two-layer film flow down a plate tilted at an angle θ. It is based on a zero Reynolds number approximation to the Orr-Sommerfeld equations and a zero surface tension approximation to both surface boundary conditions. The combined effects of density and viscosity stratifications are systematically investigated. The subtle influence of density stratification is first put into light by a temporal analysis for θ=0.2; when increasing/decreasing the density ratio (upper fluid/lower fluid), the two-layer film flow becomes much more unstable/stable with respect to the finite wavelength instability. Moreover, below a critical density ratio this finite wavelength instability even disappears, whatever the viscous ratio. Concerning the long wave instability, it becomes dominant when decreasing the density ratio below 1 and is even triggered in a region which was stable for equal density layers. The spatio-temporal analysis shows that the instability is convective for incline angles that are not too small as θ=0.2. The study of the local growth rates of the spatio-temporal instability as a function of the ray velocity V shows that there is a transition between long wave and short wave instabilities which has been determined by using the Briggs-Bers collision criterion. Accordingly, there exists a jump for the local oscillatory frequency, spatial amplification rate, and spatial wave number due to this transition. Due to the existence of the absolute Rayleigh-Taylor instability for θ=0, the transition from convective to absolute instability can be detected for values of θ smaller than 0.2, and absolute/convective instability boundary curves have been obtained for varying characteristic parameters.

A detailed temporal and spatiotemporal stability analysis of two-layer falling films with density and viscosity stratification is performed by using the Chebyshev collocation method to solve the full system of linear stability equations. From the neutral curves Re(k) for the surface mode and the interface mode of instability, obtained for different density ratios gamma of the upper layer to the lower layer, it is found that smaller density ratios make the surface mode and the short-wave interface mode much more stable, and can even make the short-wave interfacial instability disappear. Moreover, through the study of the local growth rates of the spatiotemporal instability as a function of the ray velocity V , it is found that for not too small incline angles like theta=0.2, the two-layer flow is always convectively unstable, and there is a transition between long- and short-wave instabilities which is determined by the Briggs-Bers collision criterion. Due to the existence of the absolute Rayleigh-Taylor instability for gamma>0 and theta=0, a transition from convective to absolute instability can be detected at small incline angles, and the corresponding boundary curves are plotted for different Reynolds numbers, viscosity ratios, and incline angles. It is found that there exists a limit Reynolds number above which the two-layer film flow can only be convectively unstable for a fixed small incline angle. The spatial amplification properties of the convective waves are finally presented for both surface and interface modes.

Temporal and spatiotemporal instabilities of Poiseuille-Rayleigh-Bénard flows in binary fluids with Soret effect have been investigated by a Chebyshev collocation method. Both situations corresponding to the fluid layer heated from below or from above have been studied. When heating is from below and for positive separation factors, the critical thresholds strongly increase when the throughflow is applied, and the boundary curves between absolute and convective instabilities ͑AI/ CI͒ increase as well, but more steeply. For large enough positive separation factors, there exist three local minima in the neutral curves Ra͑k͒ ͑Rayleigh number against wavenumber͒ for moderate Reynolds numbers ͑Re͒, which results in the discontinuity of the critical wavenumber curve and the nonsmoothness of the critical Rayleigh number curve when the Reynolds number is varied. For negative separation factors, there exists a contact point between the critical Rayleigh number curve and the AI/ CI boundary curve at which the fluid system is directly changed from stable to absolutely unstable without crossing the convectively unstable region. This contact point has been characterized and localized for different negative separation factors. When heating is from above, the main observation is that the stationary curve obtained at Re= 0 is replaced by two critical curves, one stationary and the other oscillatory, when a throughflow is applied. An energy budget analysis for the binary fluid system is also performed. A better insight into the role played by the solutal buoyancy contribution in the different situations is thus obtained.

Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka,G,Pr,L, χ,M,B), as well as the inclination angle (β) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (χ≥0), and oscillatory for χ<0. Moreover, the P-mode which is short-wave unstable for χ=0 remains so for χ<0, but becomes long-wave unstable for χ>0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

Three-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

The spatiotemporal evolution of Poiseuille-Rayleigh-Bénard flows in binary fluids with Soret effect is investigated by carrying out fully nonlinear two-dimensional numerical simulations initiated by a pulselike disturbance. The traveling wave packets for positive as well as negative separation factors psi are obtained numerically for ethanol-water-like mixtures (Prandtl number Pr=10 , Lewis number Le=0.01) and selected combinations of Rayleigh and Reynolds numbers at psi=0.01, 0.1 and psi=-0.1. The characteristics of the wave fronts and the transitions observed between absolute and convective instabilities when changing the parameters are compared with the results previously obtained by linear spatiotemporal stability analysis. The simulations are in very good agreement with the stability results, which confirms the validity of both approaches. Finally, in order to characterize the possible interaction between the two wave packets of the so-called downstream and upstream modes for psi<0, the spatiotemporal stability analysis is used to detect a boundary curve in the (Re, Ra) parameter region beyond which the two wave packets will never completely separate. Numerical simulations illustrate the different evolutions of the wave packets on both sides of this boundary.

A linear spatio-temporal stability analysis is conducted for the ice growth under a falling water film along an inclined ice plane. The full system of linear stability equations is solved by using the Chebyshev collocation method. By plotting the boundary curve between the linear absolute and convective instabilities (AI/CI) of the ice mode in the parameter plane of the Reynolds number and incline angle, it is found that the linear absolute instability exists and occurs above a minimum Reynolds number and below a maximum inclined angle. Furthermore, by plotting the critical Reynolds number curves with respect to the inclined angle for the downstream and upstream branches, the convectively unstable region is determined and divided into three parts, one of which has both downstream and upstream convectively unstable wavepackets and the other two have only downstream or upstream convectively unstable wavepacket. Finally, the effect of the Stefan number and the thickness of the ice layer on the AI/CI boundary curve is investigated.

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