Convective instabilities and directional solidification in binary mixtures
- Autores: Esteban Meca Álvarez
- Directores de la Tesis: Laureano Ramirez de la Piscina Millán (dir. tes.), Maria Isabel Mercader Calvo (codir. tes.)
- Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2010
- Idioma: español
- Tribunal Calificador de la Tesis: Francisco Marquès Truyol (presid.), Ricard Gonzalez Cinca (secret.), Mathis Plapp (voc.), Kenneth Andrew Cliffe (voc.), Alain Bergeon (voc.)
- Materias:
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- Resumen
In this thesis, the solidification and the convective instabilities of a binary mixture are discussed. By binary mixture we understand either a mixture of two different species with thermodiffusion (Soret effect) or the liquid phase of an alloy in a directional solidification experiment.
On the one hand, the double-diffusive instabilities of a laterally-heated binary mixture have been addressed, in the case in which the Lewis and the Prandtl numbers are small. The behavior of this system has been investigated by using different numerical techniques, based on a spatial discretization as given by a pseudo-spectral method (collocation), to find the temporal evolution and the steady states and discuss their stability.
Firstly, steady solutions and the time evolution have been studied for a separation ratio S=-1, including the stability of the purely diffusive solution that is present in this particular case. The stability of the different solution branches that appear as the Rayleigh number is increased has been discussed, taking into account the symmetry of these solutions, which is linked with a central symmetry of the equations, equivalent to a reflection.
In a large Rayleigh number interval, the only stable solution is a non-symmetric long- period orbit, which appears in a global bifurcation and evolves as Ra is increased in such a way that another frequency, much faster, appears superposed, with the orbit remaining perfectly closed. With increasing Ra, the long period diverges and the orbit, after becoming a chaotic attractor, disappears in a 'Blue sky catastrophe', with only a symmetric short-period orbit remaining.
The genericity of the previous results has been tested for larger values of the separation ratio and it has been found that, effectively, the same scenario is found for a slightly larger S, but the nature of the global bifurcation at which the long-period orbit is born as well as the 'Blue sky catastrophe' change in a complex way as S is further increased.
In the low-Ra regime, the origin of the long-period orbit changes from a global bifurcation to a Hopf bifurcation, following an intricate series of codimension-two bifurcations. In the High-Ra regime, the chaotic attractor becomes a torus that disappears in a homoclinic bifurcation as Ra is increased. This global bifurcation becomes also local and finally the torus as well as the non-symmetric solutions disappear altogether as S is increased, again after several codimension-two bifurcations.
In this thesis the influence of convection due to solute buoyancy in the interfacial instability of an alloy in a directional solidification setup has also been investigated. This has been done in the steady and in the transient case, given the importance of the latter in a history-dependent process.
The problem has been studied with a semi-analytical (asymptotic) computation and a purely numerical solution of the dispersion relation. From here, the stability of the flat solidifying interface has been obtained as a function of time, as well as the time to the first instability and the wavelength of the first instability.
As a summary, for the steady state it has been confirmed the known result that both instabilities (convective and morphological) tend to compensate each other, and this effect is more evident in the convective instability. In the transient regime, it has been found that the convective instability anticipates the morphological instability in parameter-space regions that in principle would be convectively-stable in the steady state. Moreover, purely transient instabilities have also been found, the presence of which is more evident for larger values of the surface energy. Finally, it has been found that in the coupled system the instability can develop as oscillations, and the parameter range where this is the first instability to develop is given.
