On the problem of slipper shapes of red blood cells in the microvasculature

Red blood cells (RBC) are known to exhibit non symmetric (slipper) shapes in the microvasculature. Vesicles have been recently used as a model for RBC and numerical simulations proved the existence of slipper shapes under Poiseuille flow (both in unconfined and confined geometry). However, in our recent numerical simulations the transition from symmetric (parachute) shape to the slipper one was found to take place upon decreasing the flow strength, while experiments on RBCs showed the contrary. In this work we show that if the viscosity contrast (ratio between the internal over external fluid viscosities) is different from unity, as is the case with RBCs, the transition from parachute to slipper shape occurs upon increasing the flow strength, in agreement with experiments. We provide the phase diagram of shapes in the microcirculation. The slipper is found to have a higher speed than the parachute (for the same parameters), that we believe to be the basic reason for its prevailing in the microvasculature. We provide a simple geometrical picture that explains the slipper flow efficiency over the parachute one. Finally, we show that there exists in parameter space regions of co-existence of slipper/parachute shapes and suggest simple experimental protocols to test these findings. The coexistence of shapes seems to be supported by experiments, though a systematic experimental study is lacking. A potential application of this work is to guide designing flow-based experiments in order to link the shape of RBCs to pathologies affecting cell deformability, such as sickle cell diseases, malaria, and those affecting blood hematocrit, as in polycythemia vera disease.