Shape transitions of fluid vesicles and red blood cells in capillary flows

Abstract

The dynamics of fluid vesicles and red blood cells (RBCs) in cylindrical capillary flow is studied by using a three-dimensional mesoscopic simulation approach. As flow velocity increases, a model RBC is found to transit from a nonaxisymmetric discocyteto an axisymmetric parachute shape (coaxial with the flow axis), while a fluid vesicle is found to transit from a discocyte to a prolate ellipsoid. Both shape transitions reduce the flow resistance. The critical velocities of the shape transitions are linearly dependent on the bending rigidity and on the shear modulus of the membrane. Slipper-like shapes of the RBC model are observed around the transition velocities. Our results are in good agreement with experiments on RBCs.

Fig. 1.

Snapshots of vesicles in capillary flow (with bending rigidity κ/k_BT = 20).(A) Fluid vesicle with discoidal shape at the mean fluid velocity v_mτ/R_cap = 41, both in side and top views. (B) Elastic vesicle (RBC model) with parachute shape at v_mτ/R_cap = 218 (with shear modulus μR²₀/k_BT = 110). The blue arrows represent the velocity field of the solvent. (C) Elastic vesicle with slipper-like shape at v_mτ/R_cap = 80 (with μR²₀/k_BT = 110). The inside and outside of the membrane are depicted in red and green, respectively. The upper front quarter of the vesicle in B and the front half of the vesicle in C are removed to allow a look into the interior; the black circles indicate the lines where the membrane has been cut in this procedure. Thick black lines indicate the walls the cylindrical capillary.

Fig. 2.

Dependence of the shape-transition velocity on the bending rigidity κ (A) and the shear modulus μ (B). (A) Transitions of elastic vesicles (with μR²₀/k_BT = 110) from disk to parachute (○), as well as transitions of fluid vesicles from disk to prolate (□) are shown. Solid and broken lines are linear fits to the data, v_mτ/R_cap = 10 + 4.2κ/k_BT and v_mτ/R_cap = -16 + 3.9κ/k_BT, respectively. The sliced snapshots for κ/k_BT = 20 show elastic vesicles at v_mτ/R_cap = 52 and 107 (upper row) and fluid vesicles at v_mτ/R_cap = 41 and 69 (lower row). (B) Transition velocities of elastic vesicles for κ/k_BT = 10, at the discocyte-to-parachute transition (upper curve) and at the parachute-to-discocyte transition (lower curve). The sliced snapshots for μR²₀/k_BT = 900 show slipper and parachute shapes at v_mτ/R_cap = 106 and 217, respectively.

Fig. 3.

Dependence of the curvatures of discoidal vesicles on the flow-velocity, for κ/k_BT = 20. The curvatures of the front and rear of a vesicle are calculated from a least-squares fit on the plane, which is normal to the eigenvector of the moment-of-inertia tensor with the smallest eigenvalue. The solid and dashed lines correspond to elastic vesicles (with μR²₀/k_BT = 110) and fluid vesicles, respectively.

Fig. 4.

Dependence of the shape of parachute vesicles on the bending rigidity κ, for μR²₀/k_BT = 110 and v_mτ/R_cap = 218. The average ratio of maximum length to radius, 〈l_max〉/2〈r_max〉, the average ratio of dimple depth to maximum length, 〈l_dmp〉/〈l_max〉, and the average mean curvature 〈C〉 = 〈(C₁ + C₂)/2〉 of the front and rear (dimple) of vesicles. Note that the rear (dimple) curvature is negative. The solid and dashed lines represent data for elastic and fluid vesicles, respectively. The sliced snapshots of elastic vesicles at κ/k_BT = 5 (left) and κ/k_BT = 30 (right), and the definition of the lengths are shown as Insets.

Fig. 5.

Profiles of the fluid velocity v_z along the capillary for various distances r from the capillary axis, for an elastic vesicle with κ/k_BT = 20, μR²₀/k_BT = 110, and v_mτ/R_cap = 218. The vesicle exhibits a parachute shape; its center of mass is located on z = 0. From top to bottom, r/R_cap changes from 0.03125 to 0.96875 in intervals of 0.0625.

Fig. 6.

Velocity v_m dependence of hematocrit ratio H_T/H_D = v_m/v_ves (A) and flow resistance ΔP_ves/v_m (B) per vesicle, for κ/k_BT = 20 and μR²₀/k_BT = 110. The solid and broken lines correspond to elastic and fluid vesicles, respectively. B Inset shows the v_m dependence of (r_max/R_cap)⁴. Data are shown for discoidal vesicles (□), prolate-ellipsoidal vesicles (⋄), and parachute-shaped vesicles (▵).