Rheology of suspensions of viscoelastic spheres: Deformability as an effective volume fraction

We study suspensions of deformable (viscoelastic) spheres in a Newtonian solvent in plane Couette geometry, by means of direct numerical simulations. We find that in the limit of vanishing inertia, the effective viscosity $\mu$ of the suspension increases as the volume fraction occupied by the spheres $\mathrm{\Phi }$ increases and decreases as the elastic modulus of the spheres $G$ decreases; the function $\mu (\mathrm{\Phi },G)$ collapses to a universal function $\mu \left({\mathrm{\Phi }}_{\mathrm{e}}\right)$ with a reduced effective volume fraction ${\mathrm{\Phi }}_{\mathrm{e}}(\mathrm{\Phi },G)$. Remarkably, the function $\mu \left({\mathrm{\Phi }}_{\mathrm{e}}\right)$ is the well-known Eilers fit that describes the rheology of suspension of rigid spheres at all $\mathrm{\Phi }$. Our results suggest different ways to interpret the macrorheology of blood.

Figure 1

Top: The fractional increase in effective viscosity $\mu /{\mu }_{\mathrm{f}}$ as a function of the volume fraction $\mathrm{\Phi }$ for several different values of the capillary numbers $\text{Ca}=0.02$ (+), 0.1 ($\times$), 0.2 ($*$), 0.4 ($⊡$), and 2 ($△$). All the cases have $K=1$. For comparison we also plot the same data for rigid particles [8] $[\text{Ca}=0$ ($•$)]. The inset shows the same data replotted as a function of $\text{Ca}$ for different $\mathrm{\Phi }\approx 0.0016$ (green), 0.11 (blue), 0.22 (magenta), and 0.33 (red). Bottom: The same data replotted as a function of effective volume fraction ${\mathrm{\Phi }}_{\mathrm{e}}$ collapses to a universal function given by the Eilers fit, Eq. (2), with ${\mathrm{\Phi }}_{\mathrm{m}}=0.6$ and $B=1.7$. The horizontal error bars show the standard deviation of the effective volume fraction. In the figure we also show the fitted data for three more cases at $\mathrm{\Phi }=0.11,\text{Ca}=0.2$, and viscosity ratio $K=0.01$ (black), 0.1 (brown), and 10 (orange). The interested reader is referred to the Supplemental Material [16] for a discussion of the effect of $K$ on the effective viscosity.

Figure 2

Top: Sketch of the channel geometry. The top and bottom walls, located at $y=\pm h$, move with opposite velocities $\pm V_{\mathrm{w}}$ in the streamwise $x$ direction. Left: Shape of the deformed particle for lowest $\mathrm{\Phi }$ (just a single object in the computational box) for three different capillary numbers, $\text{Ca}=0.02$, 0.2, and 2. Right: Shape of the deformed particles at $\text{Ca}=0.2$ for three different volume fractions, $\mathrm{\Phi }=0.11$, 0.22, and 0.33. The intensity of color shows $B^{12}$. In all our simulations we use $\text{Re}=0.1$ with several different values of $\mathrm{\Phi }\approx 0.0016$, 0.11, 0.22, and 0.33, and $\text{Ca}=0.02$, 0.1, 0.2, 0.4, and 2. We use the viscosity ratio ${\mu }_{\mathrm{s}}/{\mu }_{\mathrm{f}}=1$ for all our simulations, except for three more cases with $\varphi =0.11$ and $\text{Ca}=0.2$, where ${\mu }_{\mathrm{s}}/{\mu }_{\mathrm{f}}=0.01$, 0.1, and 10.

Figure 3

Top: The Taylor deformation parameter $\mathcal{T}$ Eq. (4) as a function of capillary number $\text{Ca}=0.02$ (+), 0.1 ($\times$), 0.2 ($*$), 0.4 ($⊡$), and 2 ($△$) for $\mathrm{\Phi }\approx 0.0016$ (green), 0.11 (blue), 0.22 (magenta), and 0.33 (red). All the cases have $K=1$. The black solid line shows the result of the perturbative calculation of Ref. [12] expected to hold for small $\text{Ca}$ and $\mathrm{\Phi }$. The black symbols are numerical results from the literature for a single particle in a box. In particular, the triangle and rhombus are the results from Refs. [26, 27] which were calculated for $\mathrm{\Phi }\approx 0.06$, while the circle the two-dimensional simulation from Ref. [28] with $\mathrm{\Phi }\approx 0.05$. Bottom: The effective volume fraction ${\mathrm{\Phi }}_{\mathrm{e}}$ as a function of the volume fraction $\mathrm{\Phi }$. The inset shows ${\mathrm{\Phi }}_{\mathrm{e}}/\mathrm{\Phi }$ as a function of the capillary number $\text{Ca}$, with logarithmic scale for the $x$ axis.

Figure 4

Left: The effective viscosity of suspensions of RBCs for different deformabilities and viscosity ratio plotted as a function of the effective volume fraction collapses to the Eilers fit. The data are obtained from Fig. 5 of Ref. [35]. The brown points correspond to normal RBCs in saline, while the blue, red, and green ones to RBCs in dextran solutions of viscosities 3.2, 11, and 67 cP, respectively. The gray symbols are the rigid RBCs treated with acetylaldehyde. Right: The effective volume fraction ${\mathrm{\Phi }}_{\mathrm{e}}$ as a function of $\mathrm{\Phi }$ necessary to obtain the collapse in the left panel.