Solution of a rank 2 hydrodynamic submodel with a linear velocity field

For gas dynamics equations, we consider one invariant rank 2 submodel of the evolutionary type in the cylindrical coordinate system. All its solutions for the general state equation with the assumption of linear dependence of the radial velocity component on the spatial coordinate are found. Two different cases of solutions are highlighted. In the first case, the form of the equation of state is found explicitly. In the second case, a differential equation is obtained for determining the form of the state equations. And the explicit view is found only for special cases. As an example, three special cases of gas particle motion are considered. Formulas for the world lines of the motion of gas particles in the cylindrical coordinate system are written. Trajectories of particles motion for each case are constructed. The obtained solutions describe the motion of gas particles along spiral trajectories located on the surface of the cylinder, sphere, and surface formed by the rotation around the $Ox$ axis of the Neil parabola arc.