Stationary flows and periodic dynamics of binary mixtures in tall laterally heated slots

The steady and oscillatory dynamics of binary fluids contained in slots heated by the side is studied by using continuation methods, and stability analysis. The bifurcation points on the branches of solutions are determined with precision by calculating their spectra for a large range of Rayleigh numbers. It will be seen that continuation and stability methods are a powerful tool to analyze the origin of the hydrodynamic instabilities leading to steady and time periodic flows, and their dynamics. The role played by the shear stresses of the steady field, and the solutal and thermal buoyancies, at the onset of the oscillations is studied by means of the energy equation of the perturbations. With the parameters used, it is found that the shear is always the main responsible for the instabilities, and that the work done by the two buoyancies can even help to stabilize the fluid. The results also show that binary mixtures of Prandtl number order one, like pure gases, present multiple stable periodic flows coexisting in the same range of parameters, since several unstable leading multipliers remain attached to the unit circle and go back into it. However, at lower Prandtl numbers only the first branch of periodic orbits bifurcating directly from the steady state is found to be stable, because some of the unstable multipliers of the other branches quickly increase their modulus and never re-enter the unit circle.Postprint (author's final draft

Algorithms for Solving Nonlinear Systems of Equations

In this paper we survey numerical methods for solving nonlinear systems of equations F (x) = 0, where F : IR n ! IR n . We are especially interested in large problems. We describe modern implementations of the main local algorithms, as well as their globally convergent counterparts. 1. INTRODUCTION Nonlinear systems of equations appear in many real - life problems. Mor'e [1989] has reported a collection of practical examples which include: Aircraft Stability problems, Inverse Elastic Rod problems, Equations of Radiative Transfer, Elliptic Boundary Value problems, etc.. We have also worked with Power Flow problems, Distribution of Water on a Pipeline, Discretization of Evolution problems using Implicit Schemes, Chemical Plant Equilibrium problems, and others. The scope of applications becomes even greater if we include the family of Nonlinear Programming problems, since the first-order optimality conditions of these problems are nonlinear systems. Given F : IR n ! IR n ; F = (..