Cytoplasmic streaming, the persistent flow of fluid inside a cell, induces intracellular transport, which plays a key role in fundamental biological processes. In meiosis II mouse oocytes (developing egg cells) awaiting fertilization, the spindle, which is the protein structure responsible for dividing genetic material in a cell, must maintain its position near the cell cortex (the thin actin network bound to the cell membrane) for many hours. However, the cytoplasmic streaming that accompanies this stable positioning would intuitively appear to destabilize the spindle position. Here, through a combination of numerical and analytical modeling, we reveal a hydrodynamic mechanism for stable spindle positioning beneath the cortical cap. We show that this stability depends critically on the spindle size and the active driving from the cortex and demonstrate that stable spindle positioning can result purely from a hydrodynamic suction force exerted on the spindle by the cytoplasmic flow. Our findings show that local fluid dynamic forces can be sufficient to stabilize the spindle, explaining robustness against perturbations not only perpendicular but also parallel to the cortex. Our results shed light on the importance of cytoplasmic streaming in mammalian meiosis. Published by the American Physical Society 2024