  We examine the Navier-Stokes equations with homogeneous slip boundary
conditions coupled with the heat equation with homogeneous Neumann conditions
in a bounded domain in $R^3$. The considered domain is a cylinder with
$x_3$-axis. The aim of this paper is to show long time estimates without
smallness of the initial velocity, the initial temperature and the external
force. To prove the estimate we need however smallness of $L_2$ norms of
derivatives with respect to $x_3$ of the initial velocity, the initial
temperature and the external force.
