  The tumbling to tank-treading (TB-TT) transition for red blood cells (RBCs)
has been widely investigated, with a main focus on the effects of the viscosity
ratio $\lambda$ (i.e., the ratio between the viscosities of the fluids inside
and outside the membrane) and the shear rate $\dot{\gamma}$ applied to the RBC.
However, the membrane viscosity $\mu_m$ plays a major role in a realistic
description of RBC's dynamics, and only a few works have systematically focused
on its effects on the TB-TT transition. In this work, we provide a parametric
investigation on the effect of membrane viscosity $\mu_m$ on the TB-TT
transition, for a single RBC. It is found that, at fixed viscosity ratios
$\lambda$, larger values of $\mu_m$ lead to an increased range of values of
capillary number at which the TB-TT transition occurs. We systematically
quantify such an increase by means of mesoscale numerical simulations based on
the lattice Boltzmann models.
