Mixed Convective Flows Around a Slowly Rotating Isothermal Sphere

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Abstract
The mixed convective flows generated by a heated rotating sphere have been investigated theoretically. The solutions are obtained by considering the full Navier-Stokes and energy equations along with the Boussinesq approximation. The governing equations are expressed in the R-θ-φ coordinates and due to the nature of the flow field generated, all three velocity components appear in the formulation. Due to the symmetry of the problem studied, ∂ξ/∂φ = 0, where ξ is any dependent variable considered. The R and θ momentum equations are expressed in the stream function-vorticity formulation. The resulting four coupled elliptic equations (for stream function, vorticity, vφ, and temperature) are solved numerically. Results have been obtained over a large range of Grashof and Reynolds (based on the rotational velocity of the sphere surface) numbers. The study reveals interesting flow patterns for the mixed convective problems. The gravitationally induced free convection is significant for the slowly rotating sphere where the Grashof number is of the order of or more than the square of the Reynolds number. The results are compared with previously published experimental observations and theoretical predictions based on the boundary layer theory.

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