An analysis of convection in a mushy layer with a deformable permeable interface

Abstract

We study the dynamics of a mushy layer in directional solidification for the case of a thin near-eutectic mush with a deformable and permeable mush-liquid interface. We examine the onset of convection using linear stability analysis, and the weakly nonlinear growth of liquid inclusions that signal the onset of chimneys. This analysis is compared to past analyses in which the mush-liquid interface is replaced by a rigid impermeable lid. We find qualitative agreement between the two models, but the rigid-lid approximation gives substantially different quantitative behaviour.In linear theory, the rigid-lid approximation leads to an over-estimate of the critical Rayleigh number and wavenumber of the instability. The condition for the onset of oscillatory instability is also changed by a factor of about 5 in composition number C. In the weakly nonlinear theory, the location of the onset of liquid inclusions is near the undisturbed front for the free-boundary analysis, whereas it lies at the centre of the mushy layer when the rigid-lid approximation is used. For hexagonal patterns, the boundary between regions of parameter space in which up and down hexagons are stable, shifts as a result of coupling between the liquid and mush regions.

On nonlinear evolution of convective flow in an active mushy layer

“…To obtain the adjoint system, we multiply (16), (17), and (18) byû P ,θ , andφ, respectively, and then integrate with respect to z from z = 0 to z = δ to yield…”

“…Chung and Chen [16] derived a nonlinear evolution equation analytically. Analysis of convection in a mushy layer with a deformable permeable interface was studied by Roper et al [17]. Guba and Worster [18] studied interactions between steady and oscillatory convection.…”

“…Guba and Worster [18] studied interactions between steady and oscillatory convection. The problems considered by a number of researchers [7,[16][17][18] were essentially based on the model due to Amberg and Homsy [6], which applies several restrictions including large concentration ratio, small thickness of mushy layer, high far-field temperature, and small deviation of the permeability function from a constant value.…”

“…The Landau theory has been shown experimentally to be valid in a number of convective and shear flow systems [2, p. 450]. Although the amplitude equations derived in previous studies of convection in the mushy layer [7,16,17] indicated that the Landau constant is positive, these studies were based on some assumptions as described in the previous paragraph and of the earlier analytical modeling approach due to Amberg and Homsy [6] that led to derivation of the associated amplitude equation. For the steady case considered in the present study, the relevant Landau constant for the associated amplitude equations derived in Anderson and Worster [7], Chung and Chen [16], and Roper et al [17] is found to be positive for either constant or slightly variable permeability, but larger for the nonconstant permeability case.…”

Characterization of Three-Dimensional Dendritic Structures in Nickel-Base Single Crystals for Investigation of Defect Formation

“…In the 3-D dataset presented, over 95% of the total voids identified are united with the uppermost body, satisfying a necessary condition for convective flow and solidified material transport. However, this arrangement does not necessarily establish a sufficient condition for convective flow as the limiting flow step for such an event has been proposed as cross-flow parallel to the growth direction of the secondary dendrites [40,41]. While further investigation of mean path diameter, length and volume of these channels is in progress, initial observation of the largest single body connected to the region of superheated liquid reveals an interesting feature.…”