This analysis deals with advances in models concerning the floating zone technique as well as with novel results on the relative importance of various parameters in the crystal-growth process. The attention is focused in particular on microgravity fluid-dynamic aspects and on the effect of the volume of the liquid melt since the cylindrical configuration is expected to be only a very special case under microgravity conditions. The instability of Marangoni flow is investigated by direct three-dimensional and time-dependent simulation of the problem and parallel computations. Body-fitted curvilinear co-ordinates are adopted to handle the non-cylindrical shape. A novel realistic distribution is considered to model the surface heat flux generated by a ring heater positioned around the equatorial plane at a fixed distance from the axis of the liquid (full) zone. The fluid-dynamic environment that occurs inside the melt is very sensitive to the geometrical aspect ratio A F (length/diameter) of the floating zone and to the volume factor S (ratio of the volume of the liquid zone and the volume of the corresponding cylindrical configuration: convex S>1, concave S<1). The role played by the geometrical constraints and degrees of freedom of the Marangoni toroidal rolls in determining the azimuthal structure and the stability of the flow field is discussed.