Creeping convection in a horizontally heated ellipsoid

Abstract : Natural convection in horizontally heated ellipsoidal cavities is considered in the low Grashof number limit, solving the Laplace equation for steady thermal conduction in the unbounded solid exterior, the Oberbeck-Boussinesq equations in the fluid-filled interior, and matching the temperature at the interface. In the hierarchy of equations governing the asymptotic expansion for small Grashof number, at each order a forced Stokes problem must be solved for the momentum correction. The creeping flow is known for the sphere in closed form in terms of the toroidal and poloidal potentials, spherical coordinates, and spherical harmonics. Rather than attempting to generalize this to ellipsoidal coordinates, it is re-expressed in terms of the primitive pressure-velocity variables as polynomials in the Cartesian coordinates. This form, equivalent in the sphere, suggests solutions for the pressure in an ellipsoid, which can then be found together with the velocity in closed form by the method of un-determined coefficients. Similarly, the perturbations to the temperature satisfy Poisson equations which can be solved by the same method. Polynomial formulae are given for the creeping flow and the first-order correction to the temperature. In the limit as one of the axes of the ellipsoid tends to infinity, the three-dimensional solution reduces to a two-dimensional solution for natural convection in a horizontal elliptical cylinder, tranversely horizontally heated. This exact solution is believed to be new too.
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G Mcbain. Creeping convection in a horizontally heated ellipsoid. 20th Australasian Fluid Mechanics Conference, Dec 2016, Perth, Australia. ⟨hal-02013383⟩

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