Mathematical Modeling Group - NIST

Mathematical Modeling Group - NIST

Analysis of Hydrodynamic and Interfacial Instabilities During Cooperative Monotectic Growth G.B. McFadden, NIST S.R. Coriell, NIST K.F. Gurski, NIST

B.T. Murray, SUNY Binghamton J.B. Andrews, U. Alabama, Birmingham Cooperative monotectic growth Sources of flow with a fluid-fluid interface Regular solution model of the Al-In miscibility gap

Modes of instability for a growing fluid-fluid interface Compute the morphological stability of a fluid-fluid interface during directional growth NASA Physical Sciences Research Division

Modeling Flow Effects During Monotectic Growth: Difficulty: Cooperative growth is a complex process with three phases in a complicated geometry. Typical theoretical approaches involve rough order-of-magnitude estimates or full-scale numerical calculations in 2-D or 3-D.

Idea: Idealize to two phases (fluid-fluid) in a simplified geometry (planar interface) where flow effects can be assessed quantitatively by their effects on linear stability. Related Work: Directional solidification of liquid crystals;

convective stability of liquid bi-layers. Sources of convection with a liquid-liquid interface: Thermosolutal convection (Coriell et al.) Density-change convection

Thermocapillary convection (Ratke et al.) Pressure-driven convection (Hunt et al.) Al-In Phase Diagram

C.A. Coughanowr, U. Florida (1988) Equilibrium Thermodynamics Sub-regular solution model of Al-In miscibility gap

U. Kattner, NIST; C.A. Coughanowr, U. Florida (1988) Do directional transformation of L1 () phase into L2 () phase

V

Modes of instability with a fluid-fluid interface: Double-Diffusive instability [Coriell et al. (1980)] Rayleigh-Taylor instability [Sharp (1984)] Marangoni instability [Davis (1987)]

Morphological Instability [Mullins & Sekerka (1964)] Consider the flows driven by inhomogeneities generated by morphological instability at micron-sized length scales.

V = 2 m/s Morphological Stability Analysis with No Flow [Pole in dispersion relation for k < 0]

Morphological Stability Analysis with Flow Re-introduce flow terms one at a time: BVSUP Orr-Sommerfeld equations + transport

H. Kellers approach for eigenproblem Orders of Magnitude of Flow Effects Summary

The morphological instability of a fluid-fluid interface sets a micron-sized length scale [comparable to monotectic spacing widths]; other modes of instability may also be studied. Flow interactions with the morphological mode may be computed numerically.

Buoyancy, density-driven, and thermocapillary flows interact weakly at micron scales (thermocapillary has bimodal behavior at 100 micron scale). Pressure-driven flow shows large stabilizing effect at micron scales.

In progress: Interpretation of eigenfunctions; additional modes

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