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He II two-phase stratified flow is used in the proposed TeV Superconducting Linear Accelerator (TESLA) to provide cooling for about 20000 one meter long superconducting RF cavities. To meet this end, the 33 km long cryogenic system of TESLA is divided into twelve cryogenic units, first sloping two and the following 10 level ones. He II flow in the first two units is unique since it is driven by gravity rather than by hydraulic pressure. In addition to the high energy collider, the first two units will also support a Free Electron Laser (FEL) resulting in a higher heat load. Therefore, understanding the behavior of He II two-phase flow in the units with small slope under high heat load is vital to the proper functioning of the accelerator.