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Morphological stability analysis of a planar crystallization front with convection

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Abstract

A linear theory of morphological stability of flat crystallization front is constructed with allowance for convective motions in liquid. The cases of slow and intense convection described by conductive and convective heat and mass transfer boundary conditions are considered. The dispersion relations defining the perturbation frequency as a function of wavenumber (wavelength) and other process parameters are derived. The neutral stability curve found in the case of slow convection substantially depends on extension rate at the phase interface. This curve divides the domains of morphological instability (MI) and morphological stability (MS). In both of these domains, the constitutional supercooling (CS) condition takes place. Therefore, we arrive at two various crystallization regimes (1) CS and MI, and (2) CS and MS. These cases respectively correspond to the mushy and slurry layers developing ahead of the crystallization front. In addition, when the fluid flows from the front, it is morphologically unstable for various perturbation wavelengths. When the fluid flows to the front, it is stable for large extension rates and unstable for smaller extension rates. The dispersion relation found in the case of intense convection shows that the perturbation frequency is always negative and small morphological perturbations decay with time. It means that the crystallization process with intense convection in liquid is absolutely stable.

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Acknowledgements

This work was supported by the Russian Science Foundation (project no. 21-79-10012).

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Correspondence to Eugenya V. Makoveeva.

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Structural Transformations and Non-Equilibrium Phenomena in Multicomponent Disordered Systems. Guest editors: Liubov Toropova, Irina Nizovtseva.

Appendix: Auxiliary mathematics to the linear stability theory

Appendix: Auxiliary mathematics to the linear stability theory

Let us demonstrate the derivation of relations (11)–(14) using as an example the boundary condition \(T_{\mathrm{l}}=T_{\mathrm{s}}\) (see expression (4)) at the phase transition interface. So, expanding this condition in Taylor series at the point \(z=0\) (taking the linear terms in perturbations into account), we come to

$$\begin{aligned} &T_{\mathrm{{s}}0}+\left( \frac{\mathrm{{d}}T_{\mathrm{{s}}0}}{\mathrm{{d}}z}\right) _{z=0}Z^\prime +T_{\mathrm{s}}^\prime = T_{\mathrm{{l}}0}+\left( \frac{\mathrm{{d}}T_{\mathrm{{l}}0}}{\mathrm{{d}}z}\right) _{z=0}Z^\prime +T_{\mathrm{l}}^\prime ,\\ &\quad z=0. \end{aligned}$$

Note that the perturbed boundary condition is moved to the boundary \(z=0\) due to the smallness of perturbations. Now keeping in mind the boundary condition \(T_{\mathrm{{s}}0}=T_{\mathrm{{l}}0}\) at the unperturbed front (at \(z=0\)) and expressions (9), we obtain

$$\begin{aligned} T_{\mathrm{s}}^\prime - T_{\mathrm{l}}^\prime + (G_{\mathrm{s}}-G_{\mathrm{l}}) Z^\prime =0,\ \ z=0. \end{aligned}$$

Substituting here the perturbations \(T_{\mathrm{s}}^\prime\), \(T_{\mathrm{l}}^\prime\), and \(Z^\prime\), we arrive at the boundary condition (11). Expressions (12)–(14) are derived by analogy with formula (11).

Let us now discuss how to obtain expressions (15)–(17) considering, for example, expression (15), which follows from Eq. (1). Let \(u_{x0}\), \(u_{y0}\), and \(u_{z0}\) mean the unperturbed components of fluid velocity while \(u_x^\prime\), \(u_y^\prime\), and \(u_z^\prime\) represent the corresponding perturbations. In this case, the velocity components take the form \(u_x=u_{x0}+u_x^\prime\), \(u_y=u_{y0}+u_y^\prime\), and \(u_z=u_{z0}+u_z^\prime\). Now, keeping in mind that \(T_{\mathrm{{l}}0}=T_{\mathrm{{l}}0}(z)\), we have from Eq. (1)

$$\begin{aligned} \frac{\partial T_{\mathrm{l}}^\prime }{\partial \tau }- & {} V\frac{\mathrm{{d}} T_{\mathrm{{l}}0}}{\mathrm{{d}}z}-V\frac{\partial T_{\mathrm{l}}^\prime }{\partial z} +(u_{x0}+u_x^\prime )\frac{\partial T_{\mathrm{l}}^\prime }{\partial x}\\{} & {} +(u_{y0}+u_y^\prime )\frac{\partial T_{\mathrm{l}}^\prime }{\partial y} +(u_{z0}+u_z^\prime )\left( \frac{\mathrm{{d}}T_{\mathrm{{l}}0}}{\mathrm{{d}}z} +\frac{\partial T_{\mathrm{l}}^\prime }{\partial z} \right) \\ =&\, D_{\mathrm{l}}\left( \frac{\mathrm{{d}}^2 T_{\mathrm{{l}}0}}{\mathrm{{d}}z^2} +\frac{\partial ^2 T_{\mathrm{l}}^\prime }{\partial z^2} +\frac{\partial ^2 T_{\mathrm{l}}^\prime }{\partial x^2}+\frac{\partial ^2 T_{\mathrm{l}}^\prime }{\partial y^2}\right) . \end{aligned}$$

Also, we take into account (1) only the linear terms in perturbations, (2) the unperturbed equation

$$\begin{aligned} -V\frac{\mathrm{{d}} T_{\mathrm{{l}}0}}{\mathrm{{d}}z} = D_{\mathrm{l}} \frac{\mathrm{{d}}^2 T_{\mathrm{{l}}0}}{\mathrm{{d}}z^2}, \end{aligned}$$

(3) the boundary conditions

$$\begin{aligned} u_{x0}=u_{y0}=u_{z0}=0,\ \ \textrm{at}\ \ z=0, \end{aligned}$$

and (4) the expansion

$$\begin{aligned} u_z&=u_{z0}+\frac{\partial u_{z0}}{\partial z} Z^\prime +u_z^\prime =0,\ \ \textrm{at}\ \ z=0,\\ &\quad \textrm{or}\ \ u_z^\prime =U_z Z^\prime , \textrm{at}\ \ z=0. \end{aligned}$$

Now substituting the perturbations, we come to expression (15) at \(z=0\).

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Makoveeva, E.V., Alexandrov, D.V. Morphological stability analysis of a planar crystallization front with convection. Eur. Phys. J. Spec. Top. 232, 1109–1117 (2023). https://doi.org/10.1140/epjs/s11734-023-00824-6

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