The primary thrombus formation is a critical phenomenon both physiologically and pathologically. It has been seen that various mechanical factors are involved the regulation of primary thrombus formation through a series of physiological and biochemical processes, including blood flow and intercellular molecular bridges. However, it has not been fully understood how the existence of red blood cells contributes to the process of platelet thrombus formation. We computationally analyzed the motions of platelets in plasma layer above which red blood cells flow assuming a background simple shear flow of a shear rate of 1000 s(-1) using Stokesian dynamics. In the computation, fluid mechanical interactions between platelets and red blood cells were taken into account together with the binding forces via plasma proteins between two platelets and between a platelet and injured vessel wall. The process of the platelets aggregation was significantly dependent on whether red blood cells were present. When red blood cells were absent, the aggregate formed grew more vertically compared to the case with red blood cells. Conversely, when red blood cells were present, the aggregate spread more horizontally because the red blood cells constrained the vertical growth when the height of the aggregate reached the level of the red blood cells. Our results suggest that red blood cells mechanically play a significant role in primary thrombus formation, which accelerates the horizontal spread of the thrombus, and point out the necessity of considering the presence of red blood cells when investigating the mechanism of thrombus formation.