Thermal convection induced simultaneously by horizontal temperature gradient and vibration in a rectangular cavity filled with molten silicon is investigated numerically and theoretically. The time averaged equations of convection are solved in the high-frequency vibration approximation. The Chebyshev spectral collocation method and a Newton-type method based on the Frechet derivative are used in the numerical solution of the streamfunction formulation of the incompressible Navier-Stokes equations. Validation by comparison with previous works has been performed. Different values of the Grashof number Gr and vibrational Grashof number Gr v and all the possible orientations of the vibrations are considered. Numerical results show that depending on the vibration direction, the flow can be amplified or damped, with even the possibility of flow inversion which can occur between critical vibration angles 1 and 2. A general theoretical expression is derived relating these critical angles and the ratio of vibrational to buoyant convection parameters, Gr v /Gr. A very good agreement between the theoretical and numerical results is obtained.

Silicon purification for photovoltaic applications is a crucial challenge to improve the energy production performance of solar panels. In the present work, we focus on the metallurgical grade silicon purification process through a directional solidification technique, namely the horizontal Bridgman technique. By means of numerical simulations, it is shown that high frequency vibrations applied to the crucible can be used to enhance the convective level in the molten silicon (thermal-vibrational effect) and improve the purification of the final silicon ingot. The direction of the vibration, however, has to be carefully chosen, as it strongly influences the flow intensity and structure in the melt and can lead to multi-rolls and even reverse flows, with a direct effect on the induced purification.

Using the Chebyshev spectral method, the effect of high frequency (HF) vibrations on a cubic cell heated from the side and containing a liquid metal is investigated. This study extends the numerical and theoretical two-dimensional results presented in the authors' past work [Phys. Fluids 31, 043605 (2019)] about the influence of HF vibration direction on the flow structure in a rectangular crucible filled with a liquid metal. The vibration direction can now be three dimensional, and not limited to the main flow plane. In practice, the study considers that the vibration vector is contained in one of the three principal planes of the cavity (xz, xy, and yz planes). Two different cases, i.e., under weightlessness and gravity conditions, are considered for each type of vibration to better understand the separate effect of both vibration and buoyancy forces and also their combined effects. Each type of vibration has its own features and affects the flow intensity and patterns differently.