Abstract: Reynolds stresses in liquid flow with bubbles are made up of two parts: a part induced by bubble displacement and a part produced by liquid turbulence. They are the main geneses of Reynolds stresses respectively in laminar flow and in turbulent flow. In this paper, approximate velocity expressions of uniform viscous liquid passing a sphere bubble are given, and a new formula of shear Reynolds stress induced by bubbles is derived using cell-ensemble average. The formula is in proportion to the square root of Reynolds number, and valid for low void fraction condition. By the method of Sato(1981), the best valid shear Reynolds stress coefficient is acquired. The new formula and that of Sato are compared. It is shown that the new formula is more reasonable.