Fully developed mixed convection with steady–periodic velocity oscillations in a vertical circular duct

The subject of the analysis is the combined forced and free convection of a Newtonian fluid in a vertical cylindrical duct with circular cross section. Reference is made to the non–stationary steady–periodic regime, determined by sinusoidal oscillations of the wall temperature. The fully developed region is studied, and the velocity field is assumed to be parallel to the axial direction. The investigation is performed with analytical methods. In particular, the momentum and energy local balance equations, together with the boundary conditions and the constraint equations which arise from the definition of mean velocity and mean temperature, are written in a dimensionless form and expressed as real parts of differential complex–valued equations. Two independent boundary value problems are obtained, which provide the mean value and the oscillating term of the velocity and temperature distributions. The velocity and temperature distributions, as well as the Fanning friction factor, are obtained as functions of three parameters: the Prandtl number, Pr, the dimensionless frequency ­, the ratio between the Grashof number Gr and the Reynolds number Re.