Convection in binary fluid mixtures. II. Localized traveling waves

Abstract

Nonlinear, spatially localized structures of traveling convection rolls that are surrounded by quiescent uid in horizontal layers of binary uids heated from below are investigated in quantitative detail as a function of Rayleigh number for two di erent Soret coupling strengths (separation ratios) with Lewis and Prandtl numbers characterizing ethanol-water mixtures. A nite-di erence method was used to solve the full hydrodynamic eld equations numerically in a vertical cross-section perpendicular to the roll axes subject to realistic horizontal and laterally periodic boundary conditions with di erent periodicity lengths. Structure and dynamics of these localized traveling waves (LTW) are dominated by the concentration eld. Like in the spatially extended convective states that are investigated in an accompanying paper, the Soret-induced concentration variations strongly in uence, via density changes, the buoyancy forces that drive convection. The spatio-temporal properties of this feed-back mechanism, involving boundary layers and concentration plumes, show that LTW's are strongly nonlinear states. Light intensity distributions are determined that can be observed in side-view shadowgraphs done with horizontal light along the roll axes. Detailed analyses of all elds are made using colour-coded isoplots, among others. In the frame comoving with their drift velocity, LTW's display a nontrivial spatio-temporal symmetry consisting of time-translation by half an oscillation period combined with vertical re ection through the horizontal midplane of the layer. A time-averaged concentration current is driven by a phase di erence between the waves of concentration and vertical velocity in the bulk of the LTW state. The associated large-scale concentration redistribution stabilizes the LTW and controls its drift velocity into the quiescent uid by generating a buoyancy-reducing concentration "barrier" ahead of the leading LTW front. All considered LTW's drift very slowly into the direction of the phase velocity of the pattern. For weak Soret coupling, = 0:08, LTW's have a small selected width and exist in a narrow band of Rayleigh numbers above the stability threshold for growth of TW's. For stronger coupling, = 0:25, LTW's exist below the bifurcation threshold for extended TW's in a narrow band of Rayleigh numbers. In its lower part, LTW's have a small selected width. For somewhat higher Rayleigh numbers, there exist two LTW attractors with two di erent widths. For yet higher Rayleigh numbers, there is again only one LTW attractor, however, with a broader width. Dynamical properties and the dependence on the system length are analysed. Comparisons with experiments are presented. 47.20.-k,47.54.+r,47.15.-x,47.10.+g Present address: Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

Confined states in large-aspect-ratio thermosolutal convection

“…Such waves are in turn to be viewed as homoclinic orbits to the origin in an appropriate amplitude equation, that is, the amplitude increases from zero at x=-w to a maximum and then decreases to zero again as x--_+_ [6]. Such states have also been found in numerical simulations of the full partial differential equations with experimental parameters and are in excellent agreement with their measured properties [7].…”

Traveling wave fronts and localized traveling wave convection in binary fluid mixtures

“…Hence, the local concentration 'barrier' ahead of the −interface does not select the drift velocity of LTWs as speculated previously [26]. It is rather the +interface that is the more important one.…”

“…Provided there are no hidden symmetries this seems to be incompatible with the robust occurrence of LTWs in experiments. Furthermore, stable pulse solutions seem to exist only in the bistable regime whereas LTWs are known to persist well above the linear onset of extended convection for weakly negative ψ [5,24,26,28,30,31,32,33,35]. Furthermore, coexisting small stable and wide unstable LTWs were never seen in the CGLE but found in experiments by Kolodner [7].…”

“…LTWs consist of slowly drifting, spatially confined convective regions that are embedded in the quiescent fluid. These intriguing structures have been investigated in experiments [4,5,7,10,18,27,28,29,30,31,32,33,34] and numerical simulations [14,24,26,35]. A discussion of various theoretical models aiming at their explanation is contained in Sec.…”

“…This is the critical linear group velocity v g if the coefficients are derived from an asymptotic reduction of the full hydrodynamic field equations. But v g is too fast by a factor of about 20-40 compared with the LTW drift velocity in experiments or simulations [7,14,21,26,28,35,76,77,78]. Brand and Deissler [79] introduced asymmetry in the pulse properties by adding nonlinear gradient terms to the CGLE.…”