Instabilities in rapid directional solidification under weak flow

We examine a rapidly solidifying binary alloy under directional solidification with nonequilibrium interfacial thermodynamics viz. the segregation coefficient and the liquidus slope are speed dependent and attachment-kinetic effects are present. Both of these effects alone give rise to (steady) cellular instabilities, mode $S$, and a pulsatile instability, mode $P$. We examine how weak imposed boundary-layer flow of magnitude $\left|\mathcal{V}\right|$ affects these instabilities. For small $\left|\mathcal{V}\right|$, mode $S$ becomes a traveling and the flow stabilizes (destabilizes) the interface for small (large) surface energies. For small $\left|\mathcal{V}\right|$, mode $P$ has a critical wave number that shifts from zero to nonzero giving spatial structure. The flow promotes this instability and the frequencies of the complex conjugate pairs each increase (decrease) with flow for large (small) wave numbers. These results are obtained by regular perturbation theory in powers of $\mathcal{V}$ far from the point where the neutral curves cross, but requires a modified expansion in powers of ${\mathcal{V}}^{1/3}$ near the crossing. A uniform composite expansion is then obtained valid for all small $\left|\mathcal{V}\right|$.

Figure 1

Schematic of the solidification of a liquid melt pool in one form of additive manufacturing.

Figure 2

The neutral stability curve under no flow. The steady branch is shown as a solid curve and the oscillatory branch is shown as a dashed curve. Parameters used: $k_E=0.2$, $\beta =0.1$, $\mathcal{U}=0.01$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$, $\mathcal{V}=0$.

Figure 3

Neutral stability curves in rescaled variables $(\mathcal{C},\mathcal{V})$. Pulling speed versus far-field solutal concentration for both the leading-order steady (black, solid curve) and oscillatory (blue, solid curve) branches and their perturbations due to weak flow (black, dashed and blue, dashed curves). Parameters used: $\mathcal{B}=0.3,\mathcal{N}=0.05,k_E=0.8,\mathcal{R}=1,\mathcal{V}=1$.

Figure 4

Values of the second-order contributions $m_2$ for the steady mode for varying disequilibrium parameters $\beta =0.01,0.02,0.05,0.1$. Parameters used: $k_E=0.5,\mathcal{U}=0.1,\mathrm{\Gamma }=1,\mathcal{R}=1$.

Figure 5

Plots of $m_0$ (solid) and $m_0+{\mathcal{V}}^2{m}_2$ (dashed) against ${\alpha }_1$ for mode $S$, where $\beta =0.01$, $k_E=0.5$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$, $\mathcal{V}=0.5$. Only the steady branch appears at leading-order.

Figure 6

Frequencies ${\omega }_0$ (solid, thick) and ${\omega }_0+\mathcal{V}{\omega }_1$ (dashed) against ${\alpha }_1$ for mode $S$ for various disequilibrium parameters $\beta =0,0.025,0.05$. Parameters used: $k_E=0.5$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$.

Figure 7

Perturbations to the neutral curves for the leading-order (a) steady (dashed, black) and (b) oscillatory (dashed, blue) branches. The unperturbed neutral curve is overlaid for the steady (solid, black) and oscillatory (solid, blue) branches. Both the steady and the oscillatory branches appear at leading-order. There are two oscillatory branches and they lead to distinct inverse morphological numbers at higher order. Parameters used: $\beta =0.1k_E=0.5$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$, $\mathcal{V}=0.5$.

Figure 8

(a) Perturbations to the frequencies for the leading-order (a) steady (dashed, black) and (b) oscillatory (dashed, blue) branches. The unperturbed frequencies ${\omega }_0$ are overlaid for the steady (solid, black) and oscillatory (solid, blue) branches. Parameters used: $\beta =0.1k_E=0.5$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$, $\mathcal{V}=0.5$.

Figure 9

Singular perturbation results (dotted curves) near the singular root for the inverse morphological number in (a) and frequency in (b) against the wave number. Zeroth-order results (no flow) are overlain as solid curves (the steady branch is shown using thin solid curves, and the oscillatory branch is shown using thick solid curves). Parameters used: $\beta =0.1,k_E=0.5$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$, $\mathcal{V}=0.5$.

Figure 10

The composite solution (dashed), uniformly valid for all wave numbers, for the inverse morphological number in (a) and frequency in (b), against the wave number. Zeroth-order results (no flow) are shown in as solid curves (the steady branch is shown using thin solid curves, and the oscillatory branch is shown using thick solid curves). Parameters used: $\beta =0.1k_E=0.5$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$, $\mathcal{V}=0.5$.

Figure 11

Frequency ${\omega }_0+\mathcal{V}{\omega }_1$ as a function of $\mathcal{V}$ for the steady (solid) and two oscillatory (dashed) branches for $\alpha =0.06$, $k_E=0.5$, $\beta =0.1$, $\mathcal{U}=0.1$, $\mathrm{\Gamma }=1$, $\mathcal{R}=1$.

Figure 12

(a) Maximal ${\mathcal{M}}_m^{-1}$ for mode $S$ and the associated (b) frequency ${\omega }_m$ and (c) wave number ${\alpha }_m$ as a function of $\mathrm{\Gamma }$ with flow (dashed) and without flow (solid) for $k_E=0.7$, $\beta =1$, $\mathcal{U}=0.2$, $\mathcal{R}=1$.

Figure 13

(a) Cutoff wave number ${\alpha }_c$ and (b) associated frequency ${\omega }_c$ for mode $P$ as a function of $\mathcal{U}$ with (dashed) and without (solid) flow for $k_E=0.7$, $\beta =1$, $\mathrm{\Gamma }=2$, $\mathcal{R}=1$.

Figure 14

Cutoff wave number ${\alpha }_c$ for the onset of instability with flow (dashed) and without flow for mode $S$ (solid) and mode $P$ (dotted). Parameters used: $k_E=0.5$, $\beta =0.1$, $\mathcal{U}=0.1$, $\mathcal{R}=1$, $\mathcal{V}=0.5$.