Laminar convection in uniformly heated vertical pipes

Abstract

An exact solution is presented in this paper for the problem of laminar convective flow under a pressure gradient along a vertical pipe, the walls of which are heated or cooled uniformly; the solution is based on the assumption that velocity and buoyancy profiles far from the pipe entrance do not change with height, and entry-lengt effects are ignored. Two different types of behaviour are found accordingly as the pressure gradient and buoyancy forces act together or in opposition near the centre of the pipe.When an upflow is heated (or a downflow cooled) the velocity near the walls is increased relatively and that near the axis decreased until, for sufficiently large Rayleigh numbers, definite velocity and thermal boundary layers are formed.In the case of cooled upflow (or heated downflow) there is an increase in the velocity across the whole profile for small Rayleigh numbers. As the Rayleigh number is increased the velocity and buoyancy increase, slowly at first and then rapidly, and the solution ‘runs away’ at a Rayleigh number of about 33. For higher Rayleigh numbers, laminar Poiseuille flow of an increasingly complicated profile is theoretically possible, but is unlikely to be found in practise.