Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model

Abstract

In this paper we consider the effect of a model boundary-layer flow on the hydrodynamic and morphological stability of a simple model of the solidification of a binary alloy. We conduct a linear analysis and develop asymptotic solutions for large Schmidt number and large Reynolds number. We also present numerical solutions for data appropriate to a lead–tin alloy. We show that for modes parallel to the free-stream velocity the flow is responsible for the appearance of travelling waves and, for common values of the material parameters, may stabilize the morphological stability of the interface. However the morphological stability of modes perpendicular to the free-stream velocity is unaffected by the presence of the flow. The hydrodynamic stability of the boundary layer is very weakly affected by the presence of the interface, which we attribute to the large Schmidt numbers associated with real crystal growth situations.

Flow-induced changes of the morphological stability in directional solidification: localized morphologies

“…It includes the slug fl.ow in the limit of an infinitely thin viscous layer. The obtained results are consistent with those obtained by Forth & Wheeler [10]. A comparison with a solution of [17] shows good agreement as well.…”

“…by [7] (the plane Couette fl.ow) and [11], [17] (asymptotic suction profile with uniform weak remote fl.ow far from the interface) or originally by [9] (parabolic velocity profile near the interface). The velocity profile used in the present paper is very similar to that used by Forth & Wheeler [10]. Their analytical analysis has been performed for very thick viscous layers.…”

“…The case of an asymptotic suction profile, where P is constant in ~, has been considered in the past by several authors ( see e.g. [10], [17], [ll],or [9] who used the first two terms in apower series expansion of equation (55)). The asymptotic suction profile is an exact solution of the momentum equation for a translational motion of the crystal, when the condition far from the interface becomes u =Pas z' ~ oo.…”

Asymptotic behavior of modulated Taylor-Couette flows with a crystalline inner cylinder

“…In [20], the asymptotic suction profile [16], and the buoyant instability of the parallel flow inside a vertical annulus with lateral heating [21,20]. [21,20], Rayleigh-Benard convection [17], steady…”

“…Examples include plane Couette flow [9, 10], thermosolutal convection [11,12], plane stagnation flow [13,14], rotating disk flow [15], and the asymptotic suction profile [16]. The effect of a crystal-melt interface on the hydrodynamic stability of the melt has also been considered for a variety of flows, including Rayleigh-Benard convection [17], thermosolutal convection [7, 18,19], plane Poiseuille flow [20], the asymptotic suction profile [16], thermally-driven flow in an annulus [21,20], and steady Taylor-Couette flow [22,23].…”