D. F. Jankowski

Energy stability of thermocapillary convection in a model of the float-zone crystal-growth process

Energy stability theory has been applied to a basic state of thermocapillary convection occurring in a cylindrical half-zone of finite length to determine conditions under which the flow will be stable. Because of the finite length of the zone, the basic state must be determined numerically. Instead of obtaining stability criteria by solving the related Euler–Lagrange equations, the variational problem is attacked directly by discretization of the integrals in the energy identity using finite differences. Results of the analysis are values of the Marangoni number, Ma E , below which axisymmetric disturbances to the basic state will decay, for various values of the other parameters governing the problem.

Linear‐stability theory of thermocapillary convection in a model of the float‐zone crystal‐growth process

Linear‐stability theory has been applied to a basic state of thermocapillary convection in a model half‐zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half‐zone is of finite, O(1) aspect ratio with two‐dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities are nonseparable partial differential equations. The disturbance equations are treated by a staggered‐grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases; they complement recent calculations of the corresponding energy‐stability limits.

Energy stability of thermocapillary convection in a model of the float‐zone crystal‐growth process. II: Nonaxisymmetric disturbances

Energy‐stability theory has been applied to investigate the stability properties of thermocapillary convection in a half‐zone model of the float‐zone crystal‐growth process. An earlier axisymmetric model has been extended to permit nonaxisymmetric disturbances, thus determining sufficient conditions for stability to disturbances of arbitrary amplitude. The results for nonaxisymmetric disturbances are compared with earlier axisymmetric results, with linear‐stability results for a geometry with an infinitely long aspect ratio and with stability boundaries from recent laboratory experiments.

The influence of initial condition on the linear stability of time‐dependent circular Couette flow

The effect of the starting condition on the linear stability properties of circular Couette flow with a time‐dependent inner‐cylinder motion is investigated. In addition to the WKBJ approach employed previously by Eagles [Proc. R. Soc. London, Ser. A 355, 209 (1977)] for slowly varying flow, an initial‐value method is also used. Results are presented for different stability criteria.

Experiments on the Stability of Viscous Flow between Eccentric Rotating Cylinders

The hydrodynamic stability of viscous flow between eccentric rotating cylinders has been studied experimentally. The ratio of the radii of the cylinders was 1 :2; several eccentricities were investigated. In the concentric configuration good agreement was achieved with previous experimental and theoretical work. Results obtained with the cylinders rotating in the same direction show that the effect of eccentricity can be stabilizing or destabilizing depending upon the speed of the outer cylinder and the amount of eccentricity. In the counterrotating case, a new flow phenomenon different from the usual vortex pattern was observed for large eccentricity.

Experiments on the onset of instability in unsteady circular Couette flow

Experiments have been performed to study the hydrodynamic stability of unsteady circular Couette flow generated by monotonic time-dependent inner-cylinder motions. The onset of instability was determined by measuring the axial component of velocity using laser-Doppler velocimetry; deviations from the pure-swirl value of zero are indicative of the initiation of Taylor vortices. The measurement technique was found to have an ‘intrusive’ effect on the flow stability, which was eliminated by the design of the experimental procedure. Significant enhancement of stability was found, in qualitative, but not quantitative, agreement with earlier theoretical predictions.

A large, sparse, and indefinite generalized eigenvalue problem from fluid mechanics

A numerical method for calculating the minimum positive eigenvalue of a sparse, indefinite, Hermitian algebraic problem has been developed. The method is based on inverse iteration and is a generalization of a procedure previously employed for the simpler problem of finding the smallest eigenvalue of a positive-definite matrix. Motivation was provided by a three-dimensional research problem from hydrodynamic stability. Stability limits obtained from the application of the method to a previously studied problem are compared to independently determined results.

Iterative solution of the eigenvalue problem in Hopf bifurcation for the Boussinesq equations

Some of the most challenging eigenvalue problems arise in the stability analysis of solutions to parameter-dependent nonlinear partial differential equations. Linearized stability analysis requires the computation of a certain purely imaginary eigenvalue pair of a very large, sparse complex matrix pencil. A computational strategy, the core of which is a method of inverse iteration type with preconditioned conjugate gradients, is used to solve this problem for the stability of thermocapillary convection. This convection arises in the float-zone model of crystal growth governed by the Boussinesq equations. The results obtained complete the stability picture augmenting the energy stability results [Mittelmann, et al., SIAM J. Sci. Statist. Comput., 13 (1992), pp. 411–424] and recent experimental results. Here a real eigenvalue of a Hermitian eigenvalue problem had to be determined.

Linear-stability theory of thermocapillary convection in a model of float-zone crystal growth

Linear-stability theory has been applied to a basic state of thermocapillary convection in a model half-zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half-zone is of finite, O(1) aspect ratio with two-dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities will remain partial differential equations. The disturbance equations are treated by a staggered-grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases.

Numerical experiments on the stability of unsteady circular Couette flow with random forcing

The stability of unsteady circular Couette flow generated by monotonically increasing the angular speed of the inner cylinder is examined through a series of numerical experiments. Random disturbances are imposed continually on the basic state by the inclusion of random forcing terms in the axisymmetric Navier–Stokes equations. Results of the calculations are compared with previous theoretical and experimental results.