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Publication# Finite Element Approximation to Infinite Prandtl Number Boussinesq Equations with Temperature-Dependent Coefficients - Thermal Convection Problems in a Spherical Shell

2003

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Spherical shell

In geometry, a spherical shell is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii. The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: where r is the radius of the inner sphere and R is the radius of the outer sphere. An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell: when t is very small compared to r ().

Temperature coefficient

A temperature coefficient describes the relative change of a physical property that is associated with a given change in temperature. For a property R that changes when the temperature changes by dT, the temperature coefficient α is defined by the following equation: Here α has the dimension of an inverse temperature and can be expressed e.g. in 1/K or K−1. If the temperature coefficient itself does not vary too much with temperature and , a linear approximation will be useful in estimating the value R of a property at a temperature T, given its value R0 at a reference temperature T0: where ΔT is the difference between T and T0.

Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).