FIG. 3.
A hydrodynamic model for upstream micro-swimmers. (a) Force diagram of a near surface swimmer subjected to shear flow. Left: Forces due to swimming. When rolling near a surface, the sperm head experiences a net resistive force (e. g. lubrication force) that creates a torque responsible for the CW trajectories observed in Fig. 2(d). Note the propulsion force (FP) and the drag (FD) due to the sperm swimming motion do not contribute to the torque. Right: Forces due to fluid flow. Sperm experiences a near wall resistive force that is opposite to the direction of the flow. Due to the front – back asymmetry of the sperm, the head experiences a larger force than its tail, which leads to a torque that orients the sperm upstream. Note that shear is required because both the fluid flow and the no-slip boundary condition are needed for the near wall resistive force to exist. (b) Diagram for a saddle-node bifurcation. No fixed point solution can be found when γ <γc. At γ = γc, a neutral fixed point solution first shows at θ = −π/2. When γ > γc, two fixed point solutions emerge symmetrically on both sides of θ = −π/2 line, one stable θ0 which falls in the range −π/2<θ0 <π/2 and one unstable −π − θ0.