In a core-shell quantum-well (QW) nanorod (NR) structure, because of the non-uniform constituent atom supply for QW growth at different heights on a sidewall, the QW thickness and indium content vary with NR height. Multi-section NRs can be grown by controlling the supply duration of Ga source for decreasing the size of catalytic Ga droplet and hence tapering the NR cross-sectional size. The sidewall QWs of such a multi-section NR can emit light of a broad spectrum due to the larger variation ranges of QW thickness and indium content between sections of different cross-sectional sizes. In this paper, besides the growth processes of the aforementioned NR structures are reported, two models are built for simulating the sidewall QW growth and the tapering process of NR. Based on one of the models, the theories show the consistent results of increasing QW thickness and indium content in increasing NR height with the experimental observations. Based on another model, Ga adatoms diffuse on the slant facets from the Ga droplet on the NR top to deposit GaN on the slant facets for forming a gradient layer. In this situation, the angle of the slant facet increases from ~43 to ~62 degrees during the tapering process. The results are consistent with what observed in experiment. In this paper, besides NRs grown from patterned circular holes, the growth results of NRs from patterned elliptical holes are illustrated.