We investigate the long-wave Marangoni instability in a binary-liquid layer in the limit of a small Biot number B. The surface deformation and the Soret effect are both taken into account. It is shown that the problem is characterized by two distinct asymptotic limits for the disturbance wave number k, k∼B1∕4 and k∼B1∕2, which are caused by the action of two instability mechanisms, namely, the thermocapillary and solutocapillary effects. The asymptotic limit of k∼B1∕2 is novel and is not known for pure liquids. A diversity of instability modes is revealed. Specifically, a new long-wave oscillatory mode is found for sufficiently small values of the Galileo number.

We investigate bifurcations of stationary periodic solutions of a convective CahnHilliard equation, ut+Duux+(u−u 3 +uxx)xx = 0, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter D < D 0 = √ 2/3, the periodic stationary solutions are unstable. For D > D 0 , the periodic stationary solutions are stable if their wavelength belongs to a certain stability interval. It is therefore shown that in a driven phase-separating system that undergoes spinodal decomposition the coarsening can be stopped by the driving force, and formation of stable periodic structures is possible. The modes that destroy the stability at the boundaries of the stability interval are also found.

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We investigate the long-wave Marangoni instability in binary-liquid layers in the presence of the Soret effect in the case of finite Biot numbers. Linear stability theory reveals both long-wave monotonic and oscillatory modes of instability in various parameter domains. A set of nonlinear evolution equations governing the spatiotemporal dynamics of a thin binary-liquid film is derived. Based on this set of equations, weakly nonlinear analysis is carried out. Selection of stable supercritical patterns is investigated in the limit of low gravity. Various parameter domains are examined in which supercritical standing and traveling waves are found. Stability of superposed two-wave traveling solutions is also investigated.

We consider a system which consists of a layer of an incompressible binary liquid with a deformable free surface, and a thick solid substrate subjected to a differential heating across it. We investigate the long-wave thermosolutal Marangoni instability in the case of asymptotically small Lewis and Galileo numbers for finite capillary and Biot numbers with the Soret effect taken into account. We find both long-wave monotonic and oscillatory modes of instability in various parameter domains of Biot and Soret numbers. In the domain of finite wave numbers the monotonic instability is found, but the minimum of the monotonic neutral curve is shown to be located in the long-wave region. A set of nonlinear evolution equations is derived for the description of the spatiotemporal dynamics of the oscillatory instability. The weakly nonlinear analysis is carried out for the monotonic instability.