Stability of two-dimensional strip casting processes

The solidification of molten materials is of great significance in modern metallurgical engineering. We study disturbances of a process that is characterized by three disparate lengths: the solidification length L, the wavelength Lambda, and the depth of the slab delta(proportional to). The present analysis is motivated by the relation delta(infinity) < Lambda much less than L, which is always true in practical strip casting processes. This justifies the use of an asymptotic expansion based on shallow water equations for long waves to describe the linear stability of disturbances. The leading-order equations govern a quasiparallel flow. In this limit we found two different types of disturbances: a weakly damped stable mode that runs downstream and a strongly damped perturbation traveling upstream. We focus on the downstream moving mode and show that this disturbance is strongly frequency dependent. Although the velocity disturbances are damped, there is a regime of parameters where the perturbations of the displacement grow in the horizontal direction. In our analyses we found a region of preferred frequencies. The displacement grows at these frequencies faster than for neighboring frequencies. The wavelengths of disturbances oscillating at these preferred frequencies are in qualitative agreement with the experimental observation of the wavelengths of harmonically varying grooves in the completely solidified material. (C) 2000 American Institute of Physics. [S1070-6631(00)00105-7].