Pulsatile- and cellular-mode interaction in rapid directional solidification

Abstract

Merchant and Davis performed a linear stability analysis on a model for the directional solidification of a dilute binary alloy valid for all speeds; in their work, the temperature field is decoupled from the analysis. The analysis revealed that, in addition to the Mullins-Sekerka cellular mode, there is an oscillatory instability whose preferred wave number is zero. In this paper, the nonlinear interaction of these two modes in the vicinity of their simultaneous onset is analyzed. A pair of coupled Landau equations is obtained that determines the amplitudes of the modes. Any combination of primary bifurcations may occur. Secondary bifurcations also occur; there exist stable mixed modes and regions of bistability. The results are discussed in terms of experimentally relevant variables.

Stability of periodic waves generated by subcritical oscillatory instability under the action of feedback control

“…Note that there is an error in the expression for D(m) presented in [8]. The correct expression for D(m) is given by (13). D(m) is positive for all 0 m 1.…”

“…In a contradistinction to a local feedback control that demands a large number of actuators in the case of a distributed system, the above-mentioned way of control is based on changing global parameters of the system that influence the linear growth rate. For instance, in the case of an oscillatory instability of a solidification front [13,14] the suppression of blowup can be achieved by changing the crystallization front velocity or the temperature gradient near the crystal-melt interface. Similarly, in the case of an oscillatory convective instability in a binary fluid [3] the control can be achieved by changing the imposed temperature gradient.…”

Delayed feedback control of rapid directional solidification

“…Consider rapid directional solidification [14][15][16][17][18][19][20][21][22][23] of a dilute binary alloy. Since for a typical laboratory system the characteristic spatial scale of morphological perturbations at the crystal-melt interface is much smaller than the system size, we consider the crystal-melt system to be infinite.…”

Oscillatory phenomena in directional solidification

“…Ideally, this is a set ordinary differential equations ͑ODE's͒, called coupled Lan-dau equations. For the problem of rapid directional solidification such an analysis has been undertaken by Huntley and Davis 8 and Braun et al 9 Cases of particular interest are those where two or more modes of different nature interact. In this way Braun et al 9 have established coupled Landau equations for the interaction of cells and pulsations and Huntley and Davis 8 for the interaction of cells and traveling waves, and cells and standing waves.…”

“…For the problem of rapid directional solidification such an analysis has been undertaken by Huntley and Davis 8 and Braun et al 9 Cases of particular interest are those where two or more modes of different nature interact. In this way Braun et al 9 have established coupled Landau equations for the interaction of cells and pulsations and Huntley and Davis 8 for the interaction of cells and traveling waves, and cells and standing waves. The common aim of these investigations was to formulate a possible scenario for the appearance of banded structures typically observed in experiments of rapid solidification ͑see Gremaud et al 10 ͒.…”