A blanket is an essential component of a tokamak fusion reactor. It absorbs the neutrons generated in the reaction zone, thus fulfilling the triple purpose of power harvesting, shielding from radiation, and breeding tritium fuel. It is also a unique hydrodynamic system. Flows of PbLi alloy occur in the presence of very strong (up to 3MW/m2) nonuniform heating and very strong (up to 12 T) constant magnetic field. We conduct computational analysis of various components of the system and find that the effect of thermal convection is invariably strong and, in many cases, takes unexpected forms. In some configurations, the convection leads to high amplitude (tens of degrees), low-frequency fluctuations of temperature, which may results in destructive thermal stresses in the blanket’s wall. In other cases, the effect of convection is beneficial for blanket’s operation and can be utilized in new concepts of its design.