Convective Instability in the Thermal Entrance Region of a Horizontal Parallel-Plate Channel Heated from Below

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Abstract
An investigation is carried out to determine the conditions marking the onset of longitudinal vortex rolls due to buoyant forces in the thermal entrance region of a horizontal parallel-plate channel where the lower plate is heated isothermally and the upper plate is cooled isothermally. Axial heat conduction is included in an analytical solution for the Graetz problem with fully developed laminar velocity profile. Linear-stability theory based on Boussinesq approximation is employed in the derivation of perturbation equations. An iterative procedure using high-order finite-difference approximation is applied to solve the perturbation equations and a comparison is made against the conventional second-order approximation. It is found that for Pr ≥ 0.7 the flow is more stable in the thermal entrance region than in the fully developed region, but the situation is just opposite for small Prandtl number, say Pr ≤ 0.2. Graphical results for the critical Rayleigh numbers and the corresponding disturbance wavenumbers are presented for the case of Pe → ∞ with Prandtl number as a parameter and the case of air (Pr = 0.7) with Peclet number as a parameter in the range of dimensionless axial distance from the entrance between x = 0.001 and 4 × 10−1.

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