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We perform linear stability calculations for horizontal bilayers of a two-component fluid that can undergo a phase transformation, taking into account both buoyancy effects and thermocapillary effects in the presence of a vertical temperature gradient. Critical values for the applied temperature difference across the system that is necessary to produce instability are obtained by a linear stability analysis, using both numerical computations and small wavenumber approximations. Thermophysical properties are taken from the aluminum–indium monotectic system, which includes a liquid–liquid miscibility gap. In addition to buoyant and thermocapillary modes of instability, we find an oscillatory phase-change instability due to the combined effects of solute diffusion and fluid flow that persists at small wavenumbers. This mode is sensitive to the ratio of the layer depths, and for certain layer depths can occur for heating from either above or below.

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the di erential equationwhere k is a constant; g is continuous, continuously di erentiable with respect to x; and is periodic of period P in the variable t; " … t † is continuous and periodic of period P; and when @g=@x satis®es some additional boundedness conditions. This means that there exist initial values x … 0 †ˆ¬ ¤ and _ x x … 0 †ˆ ¤ so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x … 0 †ˆ¬ and _ x x … 0 †ˆ ; a path in the phase plane can be produced, starting at … ¬; † and terminating at … ¬ ¤ ; ¤ † : Both the theoretical proof and a constructive proof are discussed and a Mathematica implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on di erential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary di erential equations, linear algebra, and numerical analysis.

Abstract. We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion equation. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this method in numerical simulations using a spectral element discretization.Key words. Convection-Diffusion, Domain Decomposition, Preconditioning, Spectral Element Method 1. Introduction. Numerical simulation of fluid flow allows for improved prediction and design of natural and engineered systems such as those involving water, oil, or blood. The interplay between inertial and viscous forces in a fluid flow dictates the length scale where energy is transferred, thus determining the resolution required to capture flow information accurately. This resolution requirement poses computational challenges in situations where the convective nature of the flow dominates diffusive effects. In such flows, convection and diffusion occur on disparate scales, causing sharp flow features that require fine numerical grid resolution. This leads to a large system of equations which is often solved using an iterative method. Exacerbating the challenge of solving a large linear system, the discrete convection-diffusion operator is non-symmetric and poorly conditioned. This leads to slow convergence of iterative solvers. In total, as convection dominates the flow the discrete fluid model becomes exceedingly challenging to solve.In recent years, the spectral element method has gained popularity as a technique for numerical simulation of fluids [9], [18]. This is due in part to the method's high-order accuracy, which produces solutions with low dissipation and low dispersion with relatively few degrees of freedom. Also important is the inherent computational efficiency gained through the use of a hierarchical grid structure based on unstructured macro-elements with fine tensor-structured interiors. This structure has enabled the development of efficient multi-level solvers and preconditioners based on Fast Diagonalization and Domain Decomposition [8], [10], [21]. Application of these techniques, however, has been restricted to symmetric systems.One way to apply such methods to non-symmetric systems is through use of timesplitting techniques, which split the system into symmetric and non-symmetric components. For convection-diffusion systems, the standard method for performing steady and unsteady flow simulations with spectral elements is operator integration factor splitting (OIFS) [12], which requires time integration even in steady flow simulations. Using this standard approach, convection and diffusion are treated separately; convection components are tackled explicitly using a sequence of small time steps that satisfy a CFL condition, and diffusive components are treated implicitly with larger t...