No abstract

This paper treats the buoyant convection of a liquid metal in a circular cylinder with a uniform, steady, axial magnetic field and with the random residual accelerations encountered on earth-orbiting vehicles. The objective is to model the magnetic damping of the melt motion during semiconductor crystal growth by the Bridgman process in space. For a typical process with a magnetic flux density of 0.2 T, the convective heat transfer and the nonlinear inertial effects are negligible, so that the governing equations are linear. Therefore, for residual accelerations or ‘‘g-jitters’’ whose directions are random functions of time, the buoyant convection is given by a superposition of the convections for two unidirectional accelerations: an axial acceleration which is parallel to the cylinder’s axis and a transverse acceleration which is perpendicular to this axis. Similarly, the response to accelerations whose amplitudes are random functions of time is given by a Fourier-transform superposition of the buoyant convections for sinusoidally periodic accelerations for all frequencies. Since the temperature gradient in the Bridgman process is primarily axial, the magnitude of the three-dimensional buoyant convection for the transverse acceleration is much larger than the magnitude of the axisymmetric convection for the axial acceleration. At a low frequency, only the magnetic damping limits the magnitude of the buoyant convection. As the frequency is increased, linear inertial effects augment the magnetic damping, so that the magnitude of the convection decreases, and its phase shifts to a quarter-period lag after the acceleration.

This paper treats the decay of the buoyant convection in a circular cylinder after a brief spike of large acceleration. There is a steady, uniform, axial magnetic field which is sufficiently strong that nonlinear inertial effects and convective heat transfer are negligible. Since the governing equations are linear, the solution for a spike at any angle to the cylinder’s axis is given by the superposition of two solutions for an axial spike and a transverse spike. The flows are axisymmetric and three-dimensional for the axial and transverse spikes, respectively. With the transverse spike, the axial temperature gradient drives a single cell of circulation with transverse vorticity, and the radial temperature gradient drives two opposite cells of circulation with axial vorticity on opposite sides of the diameter parallel to the acceleration. Although the axial temperature gradient is larger than the radial one in the Bridgman crystal-growth process, the damping of transverse vorticity with an axial magnetic field is much stronger than the magnetic damping of the axial vorticity, so that the two-cell axial-vorticity circulation dominates. For a typical Bridgman process with a 0.2 T magnetic field and with silicon, the buoyant convections driven by the axial and transverse spikes of acceleration decay to one percent of their magnitudes immediately after the spikes in 3 and 14 s, respectively.

No abstract

No abstract