On the instability of convective flow in cylinder and possible secondary regimes

A new exact solution of equations of free convection is constructed in the framework of the Oberbeck-Boussinesq approximation. The solution contains an independent parameter and describes the flow of a viscous heat-conducting liquid in the vertical cylinder with large radius. Complex rheology and radiative heating are taken into account. The considered problem reduces to the operator equation with strongly nonlinear operator. The solvability of the operator problem is proved. The iterative procedure for finding the free parameter is suggested. Three different classes of solution are obtained with the help of the procedure. The linear stability of all classes of solutions is studied numerically. Critical thermal mode is isolated. Evolution of oscillatory mode depending on Prandtl number is investigated. It is shown that under small Prandtl numbers oscillatory modes decay. If Prandtl numbers are not small a new instability type appears. This instability is connected with growing thermal disturbances. Another instability mechanism is discovered in the short waves domain. In this case the crisis is attributed to growing hydrodynamical disturbances. Secondary regimes arising in the hydrodynamical mechanism of the stability loss are calculated.