A model associated fluid flow and heat transfer of longitudinal flows of oil in pipes and channels are considered. The model is an extended version of the Bingham flow on the case of flow with one or two solid cores. The fluid part is a two-phase medium, wherein the disperse phase may consist of solid particles, droplets and gas bubbles. The sliding on the borders of pipes and channels, thermal conductivity and viscosity of such systems and their impact on the characteristics of the solid core are discussed. The flow with a yield point dependence on the temperature is also considered.