Localized traveling-wave convection in binary-fluid mixtures

Abstract

The structure and dynamics of both subcritical and supercritical localized traveling-wave (LTW) convective states with different extensions and uniquely selected width are determined by numerical integration of the proper hydrodynamic field equations with realistic horizontal boundary conditions. A largescale mean concentration current loop influences the LTW significantly. It generates a concentration distribution that hinders propagation of the LTW pulse, so that the group velocity is small but finite.PACS numbers: 47.25.Qv, 47.35.+i Traveling-wave (TW) phenomena appear in many linear and nonlinear systems. An example of the latter are TW patterns of convective rolls in binary-fluid mixtures heated from below. This nonequilibrium system is experimentally and theoretically very well suited to study nonlinear pattern dynamics. Recently, spatially confined states of localized traveling-wave (LTW) convective rolls have been found 1 "" 5 in this system. Looking like wave packets that result from linear superpositions of TW's, these LTW's are nonlinear phenomena. They consist of traveling patterns of straight rolls that are localized laterally, i.e., perpendicular to the roll axes, by intensity envelopes which drop to zero into the surrounding quiescent conductive state via a leading and a trailing front. There are subcritical and supercritical LTW states below and above, respectively, the bifurcation threshold for onset of extended convection. Depending on initial conditions and driving history the system either ends up in a LTW or an extended state filling the whole space. Depending on the parameters, the extended states competing with the LTW are the basic conductive state, TW convection, or stationary convection. For certain parameters a multiplicity of LTW's with different lateral widths are stable 1,4 while for others the LTW pulse and its width are uniquely selected. 2 Many features of these LTW states are not understood. Our numerical simulations reveal that the LTW fields are not just pulses of harmonic waves with a common simple envelope. Furthermore, LTW's with different and uniquely selected widths differ only in their center parts while the leading and trailing fronts and the field structure under the respective fronts are the same. Here we provide the information lacking so far on the concentration fields and currents that play a very important role. In particular, we find in all LTW's a feedback mechanism between the LTW fields and a large-scale mean lateral concentration current loop. This interplay leads to a concentration distribution ahead of the pulse which hinders its propagation and decreases the group velocity from the large value of simple Ginzburg-Landau approaches 6 to almost zero, in agreement with experi-mental observations.Our results follow from numerically solving the hydrodynamic field equations as described elsewhere 7,8 in an x-z section of the layer with rigid, isothermal, impermeable horizontal boundaries at z=0,d and periodic boundaries at x=0 and 20d, where d is the layer thic...

Amplitude measurements of Faraday waves

“…The applied algorithm is based on a popular marker-and-cell-method (MAC) [39,40] extensively used to simulate thermal convection in fluids [41]. Therein the non-dimensionalized incompressible evolution equations read as (12) with u and w being the x-and z-component of the velocity, respectively and p being the pressure.…”

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

“…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.…”

“…Inspired by simulation results of Barten et al [35,76] which showed the important role of the concentration field for a LTW Riecke [82,83] proposed an extension of the GinzburgLandau equations. Within a weakly nonlinear expansion he coupled into the standard CGLE as an additional slow variable the amplitude C of an advected mean large-scale concentration mode that influences the growth of the critical modes.…”

Over two decades of pattern formation, a personal perspective

“…[118,119] Lücke and collaborators studied pulses by the numerical integration of the Navier-Stokes equations for binary-mixture convection. [120] It has been suggested that pulses are related to the solitons of the nonlinear Schrödinger equation, modified by the dissipative contributions to the Ginzburg-Landau equation or the physical system. [121] A detailed theoretical discussion of fronts and pulses was given by van Saarloos and Hohenberg.…”