Global stability of flowing red blood cell trains

A train of red blood cells flowing in a round tube will either advect steadily or break down into a complex and irregular flow, depending upon its degree of confinement. We analyze this apparent instability, including full coupling between the viscous fluid flow and the elastic cell membranes. A linear stability analysis is constructed via a complete set of orthogonal perturbations to a boundary integral formulation of the flow equations. Both transiently $\left(t\to 0^{+}\right)$ and asymptotically $\left(t\to \infty \right)$ amplifying disturbances are identified. Those that amplify transiently have short-wavelength shape distortions that carry significant membrane strain energy. In contrast, asymptotic disturbances are primarily rigid-body-like tilts and translations. It is shown that an intermediate cell-cell spacing of about half a tube diameter suppresses long-time train instability, particularly when the vessel diameter is relatively small. Altering the viscosity ratio between the cytosol fluid within the cell and the suspending fluid is found to be asymptotically destabilizing for both higher and lower viscosity ratios. Altering the cytosol volume away from that of a nominally healthy discocyte alters the stability with complex dependence on train density and vessel diameter. Several of the observations are consistent with a switch from predominantly cell-cell interactions for dense trains and predominantly cell-wall interactions for less dense trains. Direct numerical simulations are used to verify the linear stability analysis and track the perturbation growth into a self-sustaining disordered regime.

Figure 1

An example of the empirical instability of model red blood cells flowing in a uniformly spaced train. The flow is simulated using the methods of Sec. 3. A transition to apparent disorder is observed when increasing the number of cells from (a) $N=8$ to (b) $N=12$.

Figure 2

The model system and boundary conditions.

Figure 3

Eigenvalues of $\mathbf{A}$ for a two-cell case with $D=10r_o,\varphi =0.2$, and viscosity ratio $\lambda =5$. The same data are shown in (a) and (b) with different scales.

Figure 4

Behavior and verification for a case with $N=2,D=10r_o,\varphi =0.2$, viscosity ratio $\lambda =5$, and initial disturbance magnitude $\widehat{\varepsilon }={10}^{-3}{r}_o$.

Figure 5

Example base flow configurations: (a) $D=6r_o,\varphi =0.2$, (b) $D=6r_o,\varphi =0.7$, (c) $D=10r_o,\varphi =0.2$, (d) $D=10r_o,\varphi =0.7$.

Figure 6

(a) Spectral abscissa amplification $\alpha$ for the full range of $D$ and $\varphi$. The constant $\alpha$ curves suggest a region of apparent marginal stability (see text). (b) Example ${\vec{\mathbf{v}}}_{\alpha }$ for (i) $D=6r_o,\varphi =0.2$, (ii) $D=6r_o,\varphi =0.7$, (iii) $D=10r_o,\varphi =0.2$, and (iv) $D=10r_o,\varphi =0.7$, as labeled in (a). The wrinkled appearance is due to the $\vec{\mathbf{s}}+10{\vec{\mathbf{v}}}_{\alpha }$ magnification used to visualize the small perturbation.

Figure 7

Streamlines relative to the cell-attached-frame for $D=6r_o$ for (a) an example unstable loosely packed $\varphi =0.3$ case, (b) a marginally stable $\varphi =0.5$ case, and (c) an unstable densely packed $\varphi =0.7$ case. The dashed line is the tube center line.

Figure 8

(a) Numerical abscissa $\eta$. Solid black lines are curves of constant $\eta$ for values as labeled. (b) Example ${\vec{\mathbf{v}}}_{\eta }$ for (i) $D=6r_o,\varphi =0.2$, (ii) $D=6r_o,\varphi =0.7$, (iii) $D=10r_o,\varphi =0.2$, and (iv) $D=10r_o,\varphi =0.7$, visualized as $\vec{\mathbf{s}}+10{\vec{\mathbf{v}}}_{\eta }$. Again, the exaggerated wrinkled appearance is due to the magnification used to visualize the small perturbation.

Figure 9

DNS evolution and linear predictions for ${\vec{\mathbf{v}}}_{\alpha }$ and ${\vec{\mathbf{v}}}_{\eta }$, are shown with relatively large $\widehat{\varepsilon }=0.1r_o$ initial disturbance for cases (a) $D=10r_o,\varphi =0.2$ and (b) $D=10r_o,\varphi =0.7$. Linear predictions of ${\vec{\mathbf{v}}}_{\alpha }$ remain good for long times as expected, though transient disturbances ${\vec{\mathbf{v}}}_{\eta }$ undergo rapid decay.

Figure 10

Cell-averaged deformation spectra (16) for ${\vec{\mathbf{v}}}_{\alpha }$ and ${\vec{\mathbf{v}}}_{\eta }$ for $D=10r_o$ and $\varphi =0.7$.

Figure 11

$\parallel {\vec{\mathbf{\varepsilon }}}_x\left(t\right)\parallel$ for ${\vec{\mathbf{v}}}_{\alpha }$ and an ad hoc disturbance ${\vec{\mathbf{v}}}_{\text{ad}}\text{hoc}$ (see text) with $\widehat{\varepsilon }=0.1r_o$ and corresponding DNS for $D=10r_o$ and $\varphi =0.2$ and 0.7, as indicated.

Figure 12

The time evolution and subsequent train breakup of ${\vec{\mathbf{v}}}_{\alpha }$ for $D=10r_o$ and $\varphi =0.7$. The velocities of the cell centroids relative to that of the initial mean flow, ${\widehat{\mathbf{u}}}^{\prime }\equiv {\mathbf{u}}^{\prime }-{\mathbf{u}}^{\infty }$, are shown as black vectors. Two initially adjacent cells are labeled as $\mathsf{A}$ and $\mathsf{B}$ to illustrate the relative cell motion. For $tU/r_o\le 61$ blue vectors show the surface velocity relative to the centroid, ${\widehat{\mathbf{u}}}_s\equiv \mathbf{u}-{\widehat{\mathbf{u}}}^{\prime }$. Walls are not shown, though they can be seen in the base state visualization in Fig. 5. The times were selected to illustrate the flow development.

Figure 13

(a) Numerical $\left(\eta \right)$ and (b) spectral $\left(\alpha \right)$ abscissa for $\lambda =1$ and 5 and $D=8r_o$ and $12r_o$ as labeled versus $\varphi$.

Figure 14

Amplification factor $\alpha$ and tank-treading and tumbling metrics for varying $\lambda$ and (a) $\varphi =0.2$ and (b) $\varphi =0.7$.

Figure 15

Cell trains with $D=8r_o$ for the (a) smallest $\text{Ca}=0.4$ and (b) largest $\text{Ca}=20.5$ we consider.

Figure 16

Dependence of amplification rate $\alpha$ on flow strength $\text{Ca}$ for $\lambda =1$ and (a) $\varphi =0.2$ and (b) $\varphi =0.7$. The dashed lines indicate the $\text{Ca}=0.66$ of most other cases.

Figure 17

Equilibrium shapes for reduced volume $v$. The reference healthy red blood cell shape used in previous sections has $v=0.64$.

Figure 18

Base configurations of cell trains for $D=8r_o$ and $\varphi =0.7$: (a) $v=0.40$ and (b) $v=0.86$.

Figure 19

Maximum $t\to \infty$ amplification $\alpha$ for varying $D/r_o,v$, and (a) $\varphi =0.2$ and (b) $\varphi =0.7$. All cases have $\lambda =1$. Vertical dashed lines indicate the nominal healthy red blood cell $v=0.64$ used in most other cases. In (b), the shaded region $v>0.86$ indicates states that cannot form a train without overlapping.

Figure 20

Modified growth rate $\tilde{\alpha }\left(\tilde{N}\right)$ for $D=8r_o$, (a) varying $\text{Ca}$ and $v=0.64$, and (b) varying $v$ and $\text{Ca}=0.66$. Both (i) $\varphi =0.2$ and (ii) $\varphi =0.7$ are shown.