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Thermal Convection in 2D with Varying Gravity
A study of convection in a two dimensional cell is presented. A newtonian homogeneous fluid is contained between two parallel circular glass plates (10 cm diameter) separated a small distance (1mm). Gravity can be varied by changing the inclination of the cell. The system is heated and cooled at two diametrically opposed points on the edge of the circle, which are aligned with gravity. The main experimental observations are reported for different values of the relevant dimensionless parameters. The theoretical approach stems from the full Navier-Stokes equations coupled to the mass and energy conservation laws. When the cell is in a horizontal position, and gravity has no effect, an analytic solution for the temperature distribution of the quiescent fluid is obtained :
T0=T1+T2 atanh[2rCosθ/(1+r2)],
in polar coordinates, r and θ ; the constants (temperatures) depend on the imposed gradient (Fig.1a), and it is the only case where a static solution can exist. For a slight inclination of the cell, the projection of gravity in the plane of the cell can be used as a perturbation parameter ; the equations can be solved analyticaly to first order in perturbation theory, without recourse to the Boussinesq approximation (Fig.1 b,c). Furthermore, numerical simulations using a lattice-Boltzmann approach are carried out, as well as a numerical analysis of the equations. We discuss the experiments and the theoretical results.
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