  We prove long time existence of regular solutions to the Navier-Stokes
equations coupled with the heat equation. We consider the system in non-axially
symmetric cylinder with the slip boundary conditions for the Navier-Stokes
equations and the Neumann condition for the heat equation. The long time
existence is possible because we assumed that derivatives with respect to the
variable along the axis of the cylinder of the initial velocity, initial
temperature and the external force in $L_2$ norms are sufficiently small. We
proved the existence of such solutions that velocity and temperature belong to
$W_\sigma^{2,1}(\Omega\times(0,T))$, where $\sigma>{5\over3}$. The existence is
proved by the Leray-Schauder fixed point theorem.
