3.1 Problem formulation

Table 1.

Simulation data. Each case is simulated independently in an LB domain of size $128\times 128\times 80$ at specified $Pe$ , $Ca_{G}$ and $\unicode[STIX]{x1D719}$ . NP long-time diffusivities, $\unicode[STIX]{x1D60B}_{xx}^{\infty }$ , $\unicode[STIX]{x1D60B}_{yy}^{\infty }$ , $\unicode[STIX]{x1D60B}_{zz}^{\infty }$ and $\unicode[STIX]{x1D60B}_{xy}^{\infty }$ , normalized by the theoretical Brownian diffusivity, $\unicode[STIX]{x1D60B}^{B}$ , are listed for all cases. The unlisted off-diagonal diffusivities, $\unicode[STIX]{x1D60B}_{xz}^{\infty }$ and $\unicode[STIX]{x1D60B}_{yz}^{\infty }$ , are found to be negligible. The measured diffusivities have a standard deviation less than 5 $\,\%$ . RBCs have an effective radius of $a_{2}=2.9~\unicode[STIX]{x03BC}\text{m}$ . Brownian diffusivity is calculated by $\unicode[STIX]{x1D60B}^{B}=k_{B}T/6\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}a_{1}$ at temperature $T=310$  K.

The apparent diffusivity of NPs in unbounded blood flow under simple shear is determined by NP radius $a_{1}$ , shear rate $\dot{\unicode[STIX]{x1D6FE}}$ and haematocrit $\unicode[STIX]{x1D719}$ , with NP concentration in the dilute regime. The relevant dimensionless parameters (besides $\unicode[STIX]{x1D719}$ ) primarily include the NP Péclet number,

(3.1) $$\begin{eqnarray}Pe=\frac{\dot{\unicode[STIX]{x1D6FE}}a_{1}^{2}}{\unicode[STIX]{x1D60B}^{B}},\end{eqnarray}$$

expressing the ratio of shear-induced diffusion to Brownian diffusion, and the RBC capillary number,

(3.2) $$\begin{eqnarray}Ca_{G}=\frac{\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FE}}a_{2}}{G},\end{eqnarray}$$

quantifying the competition between the fluid viscous stress and the membrane elastic stress. Here, $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the suspending plasma, $a_{2}$ is the effective radius of the RBC and $G$ is the elastic shear modulus of the RBC membrane; $\unicode[STIX]{x1D60B}^{B}$ is the Brownian diffusivity, which is determined by the Stokes–Einstein relation, $\unicode[STIX]{x1D60B}^{B}=k_{B}T/6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a_{1}$ , where $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature. The NP Péclet number quantifies the severity of the NP Brownian effect, while the RBC capillary number determines the deformability of the RBC capsule.

Figure 2.

(a) The RBC–NP complex configuration at $t\dot{\unicode[STIX]{x1D6FE}}=0$ . NPs and RBCs are randomly and uniformly mixed in an unbounded shear flow. RBCs are shown partially in the domain to visualize the NP phase. The athermal system is determined by shear rate $\dot{\unicode[STIX]{x1D6FE}}$ , haematocrit $\unicode[STIX]{x1D719}$ and NP radius $a_{1}$ . (b–f) The RBC–NP complex configuration in the long-time scale regime under a shear rate of (b)  $\dot{\unicode[STIX]{x1D6FE}}=10$ , (c) 100, (d) 1000, (e) 2000 and (f)  $10\,000~\text{s}^{-1}$ with $\unicode[STIX]{x1D719}=0.4$ .

To obtain appropriate scaling relations, we performed a large number of independent 3D simulations. Table 1 lists all the simulation parameters and the measured NP diffusivities. Wide ranges of shear rates ( $10\leqslant \dot{\unicode[STIX]{x1D6FE}}\leqslant 10\,000~\text{s}^{-1}$ ) and haematocrits ( $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.4$ ) with physiological relevance (Lipowsky Reference Lipowsky2005; Popel & Johnson Reference Popel and Johnson2005) are covered. Cases with $\dot{\unicode[STIX]{x1D6FE}}=10\,000~\text{s}^{-1}$ are to match certain vascular pathological conditions, e.g. the high shear induced thrombosis (Casa & Ku Reference Casa and Ku2017). For discussions in this section, NP size is set to $2a_{1}=100~\text{nm}$ . RBCs are assumed to be in a healthy state with an effective radius $a_{2}=2.9~\unicode[STIX]{x03BC}\text{m}$ and a membrane shear modulus $G=0.0063~\text{dyne}~\text{cm}^{-1}$ . The absolute temperature is set to $T=37\,^{\circ }\text{C}$ , at which the plasma has a viscosity $\unicode[STIX]{x1D707}=1.2~\text{cP}$ and a density $\unicode[STIX]{x1D70C}=1.0~\text{g}~\text{cm}^{-3}$ . The viscosity ratio of RBC cytoplasm to plasma is set to the physiological value $\unicode[STIX]{x1D706}=5.0$ . The density of cytoplasm is set to that of the plasma. The corresponding $Pe$ and $Ca_{G}$ lie in the ranges of $0.0066\leqslant Pe\leqslant 6.60$ and $0.0055\leqslant Ca_{G}\leqslant 5.52$ , respectively. All simulations are formulated by matching the dimensionless group, i.e. $\unicode[STIX]{x1D719}$ , $Pe$ and $Ca_{G}$ .

Simulations are initiated by imposing steady shear flow on the uniformly, randomly mixed NPs and RBCs at specific shear rate in an LEbc computational domain, as shown in figure 2(a). The domain has a dimension of $128\times 128\times 80$ ( $42.7~\unicode[STIX]{x03BC}\text{m}\times 42.7~\unicode[STIX]{x03BC}\text{m}\times 26.7~\unicode[STIX]{x03BC}\text{m}$ ) in longitudinal ( $x$ ), velocity-gradient ( $y$ ) and vorticity ( $z$ ) directions, respectively. This LB domain size matches the highest resolution applied for the rheological characterization of cellular blood flow under shear by Reasor et al. (Reference Reasor, Clausen and Aidun2013a ). The selected LB grid resolution ( $300~\text{nm}$ per lattice unit) and the equilibrium RBC mesh size (1.5 lattice units per link length) have previously proven to be fine enough to capture both the single RBC dynamics (MacMeccan et al. Reference MacMeccan, Clausen, Neitzel and Aidun2009; Reasor et al. Reference Reasor, Clausen and Aidun2012) and the rheological properties of concentrated cellular blood flow (MacMeccan et al. Reference MacMeccan, Clausen, Neitzel and Aidun2009; Reasor et al. Reference Reasor, Clausen and Aidun2013a ).

To obtain converged long-time diffusivity, sufficient strains ( $t\dot{\unicode[STIX]{x1D6FE}}\sim 1000$ ) and a large number of particles (5000 NPs and up to 208 RBCs) are employed for each simulation. The resolution of these simulations in terms of strain units and number of particles is at the high end compared to other numerical studies on particle diffusion in colloidal/non-colloidal suspensions (Foss & Brady Reference Foss and Brady1999, Reference Foss and Brady2000b ; Sierou & Brady Reference Sierou and Brady2004; Yeo & Maxey Reference Yeo and Maxey2010; Clausen et al. Reference Clausen, Reasor and Aidun2011; Gross et al. Reference Gross, Kruger and Varnik2014; Mountrakis, Lorenz & Hoekstra Reference Mountrakis, Lorenz and Hoekstra2016). As will be discussed in § 3.3.1, the selected resolution produces good agreement between the simulation results and the available experimental data.

Simulations are performed on the Intel Xeon Skylake nodes of the TACC (Texas Advanced Computing Center) Stampede-2 system where each node features 48 cores and a 2.1 GHz clock rate. For the case at $\unicode[STIX]{x1D719}=0.4$ with 208 RBCs and 5000 NPs, each run takes ${\sim}168$  h on 32 cores ( ${\sim}5376$ core hours) to accomplish 1000 strains ( $t\dot{\unicode[STIX]{x1D6FE}}$ ). The total computational cost for the 35 independent cases listed in table 1 is approximately 140 000 core hours.

3.2 Temporal growth of NP transient mobility

To understand the short-time dispersive characteristics of the NP phase, we examine the transient behaviour of NP mobility by tracking the temporal growth of NP MSDs. Cross-stream MSDs, $\langle y(t)y(t)\rangle /a_{1}^{2}$ and $\langle z(t)z(t)\rangle /a_{1}^{2}$ , are sampled from the initial configuration ( $t\dot{\unicode[STIX]{x1D6FE}}=0$ ) where NPs and RBCs are randomly and uniformly mixed, as shown in figure 2(a). Longitudinal and off-diagonal MSDs, $\langle x(t)x(t)\rangle /a_{1}^{2}$ and $\langle x(t)y(t)\rangle /a_{1}^{2}$ , are sampled starting from a long-time configuration ( $t\dot{\unicode[STIX]{x1D6FE}}=400$ ) to avoid any extra convective effect caused by the transient elongation and reorientation of RBCs. Such transient effects are found to introduce an extra affine displacement leading to ${\sim}t^{3}$ growth of MSDs and hence jeopardize the measurement of the long-time diffusivity in the $x$ -relevant directions. The necessity of eliminating the affine effect for calculating streamwise diffusivities is elaborated by Foss & Brady (Reference Foss and Brady2000b ) in the context of sheared colloidal suspensions.

Figure 3.

Normalized MSDs plotted against relative shear strains in logarithmic scale at various haematocrits for different shear rates. Panels (a)–(d) are concerned with evolutions of $\langle y(t)y(t)\rangle /a_{1}^{2}$ ; (e) and (f) depict evolutions of $\langle x(t)x(t)\rangle /a_{1}^{2}$ and $|\langle x(t)y(t)\rangle |/a_{1}^{2}$ , respectively; (g) and (h) depict evolutions of $\langle z(t)z(t)\rangle /a_{1}^{2}$ . The imposed shear rate is denoted on each plot. Inset of (b) shows the RBC–NP distribution at $t\dot{\unicode[STIX]{x1D6FE}}\sim 1$ under $\dot{\unicode[STIX]{x1D6FE}}=100~\text{s}^{-1}$ . Insets of (c) show the RBC–NP distribution at $t\dot{\unicode[STIX]{x1D6FE}}\sim 1$ (upper) and $t\dot{\unicode[STIX]{x1D6FE}}\sim 100$ (lower) under $\dot{\unicode[STIX]{x1D6FE}}=2000~\text{s}^{-1}$ and $\unicode[STIX]{x1D719}=0.4$ . An NP radius of $a_{1}=50$  nm is used for normalization. Cross-stream MSDs (a–d) and (g,h) are sampled starting from $t\dot{\unicode[STIX]{x1D6FE}}=0$ to capture the transient NP dispersive behaviour; streamwise and off-diagonal MSDs are sampled starting from $t\dot{\unicode[STIX]{x1D6FE}}=400$ to avoid the initial extra affine displacement induced by RBC elongation and orientation.

Figure 3 plots the particle normalized MSD growth with respect to the relative strains sampled. At $\unicode[STIX]{x1D719}=0$ , all diagonal MSDs grow according to the Stokes–Einstein (SE) prediction with no dependence on shear rate, while the off-diagonal MSDs produce zero value (not shown). This is consistent with the isotropic nature of NP Brownian motion in a dilute and unbounded solution, and also serves as a verification of the MSD calculations. At $\unicode[STIX]{x1D719}>0$ , a deviation from the SE relation occurs in all diagonal MSDs as a result of NP–RBC interactions. At $\dot{\unicode[STIX]{x1D6FE}}=10~\text{s}^{-1}$ , only a slight deviation from the SE relation is observed given the dominance of Brownian diffusion; see figure 3(a).

The transient deviation of NP mobility from the SE relation can be further interrogated through the evolution of cross-stream MSDs, as depicted in figure 3(a–d, g–h). In the short-time regime ( $t\dot{\unicode[STIX]{x1D6FE}}\ll 1$ ), a linear growth of MSD ( ${\sim}t$ ) is observed particularly under low-shear-rate (e.g. $\dot{\unicode[STIX]{x1D6FE}}\leqslant 100~\text{s}^{-1}$ ) or high-shear-rate and high-hematocrit (e.g. $\dot{\unicode[STIX]{x1D6FE}}\geqslant 2000~\text{s}^{-1}$ and $\unicode[STIX]{x1D719}=0.4$ ) conditions, as shown in figure 3(b) and (c), respectively. The short-time linear growth of MSD suggests a short-time diffusive mechanism. Since the short-time linear growth of MSD always occurs before the ballistic regime ( ${\sim}t^{2}$ ) where RBC–NP collisions start to occur, such a short-time diffusive driver should logically be the Brownian effect. Therefore, the initial linear MSD growth at low shear rates (with $\unicode[STIX]{x1D719}=0\sim 0.4$ ) or high shear rates (with $\unicode[STIX]{x1D719}=0.4$ ) can be explained by the Brownian diffusive time scale being much shorter than the RBC–NP collision time scale under such haemorheological conditions. However, under high shear rate (e.g. $\dot{\unicode[STIX]{x1D6FE}}\geqslant 2000~\text{s}^{-1}$ ), the diffusive behaviour ( ${\sim}t$ ) turns to ballistic ( ${\sim}t^{2}$ ) as $\unicode[STIX]{x1D719}$ increases to 0.4; see figure 3(c,d,g,h). This is probably due to the high convective (high-shear-rate) and low inertial (low-haematocrit) effects that reduce the RBC–NP collision time scale to be comparable to the Brownian diffusion time scale. As a result, the Brownian diffusion of NP is overwhelmed by the ballistic behaviour caused by insufficient RBC–NP collisions.

Figure 4.

RBC–NP partial pair distribution function, $g_{21}(\boldsymbol{r})$ , projected on the $xy$ plane under various haemorheological conditions and averaged in the strain range of $t\dot{\unicode[STIX]{x1D6FE}}=1{-}3$ .

In the intermediate-time regime, $t\dot{\unicode[STIX]{x1D6FE}}=O(1){-}O(10)$ , anomalous dispersive behaviour is observed as RBCs start to elongate and rotate to be aligned with the streamwise direction due to shear. For cases at high shear rates ( $\dot{\unicode[STIX]{x1D6FE}}\geqslant 2000~\text{s}^{-1}$ ) and high haematocrit ( $\unicode[STIX]{x1D719}=0.4$ ), we observe a sublinear growth of MSD, representing a temporary hindrance of the NP mobility. This hindrance effect is caused by a string-ordered microstructure of NP in the shearing plane accompanied by the elongation and alignment of RBCs; see the contrast between the upper inset of figure 3(c) and the inset of figure 3(b) (see also supplementary movie 2, available at https://doi.org/10.1017/jfm.2019.320). Such string-ordered NP distribution can be better visualized by plotting the RBC–NP partial pair distribution function (PPDF), $g_{21}(\boldsymbol{r})$ , projected on the $xy$ plane (the computing procedure of PPDF in the short-time regime is discussed in appendix B). As shown in figure 4, $g_{21}(\boldsymbol{r})$ in the $xy$ plane exhibits streaks showing an intensified distribution particularly near the RBC surface under high shear rate and high haematocrit, while in contrast no significant string structure is found in low shear rate ( $\dot{\unicode[STIX]{x1D6FE}}\leqslant 100~\text{s}^{-1}$ ) (see supplementary movie 1) or/and low haematocrit ( $\unicode[STIX]{x1D719}=0.1$ ) cases, as shown in figure 4(a–c). Note that a suspension string-like structure often occurs in sheared monodisperse colloidal suspensions at equilibrium as a consequence of the absence of interparticle lubrication interactions at high concentration (Xue & Grest Reference Xue and Grest1990; Foss & Brady Reference Foss and Brady2000b ) or the presence of long-range repulsive forces at low concentration (Kumar & Higdon Reference Kumar and Higdon2010). However, in the current case, ordering of NPs occurs at a non-equilibrium state that involves the change of the RBC suspension structure from a uniformly distributed and randomly oriented configuration to a streamwise-aligned and elongated configuration under high shear and high haematocrit. The sublinear growth of MSD is followed by a super-ballistic behaviour ( ${\sim}t^{3}$ ) at high shear rate ( $\dot{\unicode[STIX]{x1D6FE}}=10000~\text{s}^{-1}$ ), which might be associated with the large jump of NPs between strings when the NP phase gradually evolves from the string-like structure towards the more uniform structure featuring in the long-time diffusive regime.

In the long-time regime ( $t\dot{\unicode[STIX]{x1D6FE}}>100$ ), MSD reaches the second linear growth stage ( ${\sim}t$ ), where RESID becomes the dominate diffusive mechanism leading to a uniform distribution of NP (see lower inset of figure 3 c). Such a three-stage (diffusive/super-ballistic/diffusive) anomalous dispersion behaviour has previously been reported in the dispersion of Brownian particles subjected to external forces (Siegle, Goychuk & Hanggi Reference Siegle, Goychuk and Hanggi2010). Here, we show that such dispersive anomalies also occur in a sheared NP–RBC bidisperse suspension, where the anomalous NP dispersion is, however, driven by an internal shear-induced mechanism, i.e. the elongation and the alignment of the concentrated RBCs along the streamwise direction under high shear. Such transient shear-induced morphological adaptation of RBCs contributes to extra mobility of the NP phase, seemingly playing a role of external forces exerted on the NP phase.

Figures 3(e,f) depict the longitudinal and off-diagonal MSD evolution at $\dot{\unicode[STIX]{x1D6FE}}=10\,000~\text{s}^{-1}$ . Results for lower shear rate show similar MSD behaviours and are not presented for discussion. Because the initial transient regime is neglected, the longitudinal and off-diagonal MSDs yield classical ballistic–diffusive transitions similar to the dispersive behaviour of rigid-particle suspensions under shear (Foss & Brady Reference Foss and Brady1999; Clausen et al. Reference Clausen, Reasor and Aidun2011). The off-diagonal MSD under high shear starts with positive values during the ballistic regime, exhibits a cross-over transition involving a change of sign, and eventually maintains negative values in the diffusive regime. Absolute values are shown for the off-diagonal MSDs with the sign at certain temporal stage denoted in figure 3(f). The change of sign from positive ( $+$ ) to negative ( $-$ ) is a hallmark of the dominant NP migration direction shifted from along the extensional axes (first and third quadrants) to along the compressive axes (second and fourth quadrants) of the flow, as also observed in sheared colloidal suspensions (Foss & Brady Reference Foss and Brady1999).

Figure 5.

Normalized particle long-time RESID in (a) $xx$ , (b) $yy$ , (c) $zz$ and (d) $xy$ directions against normalized shear rate ( $Pe$ or $Ca_{G}$ ) in a log–log scale. Each curve at specific $\unicode[STIX]{x1D719}$ forms by connecting $\hat{\unicode[STIX]{x1D60B}}_{ij}^{R}$ data at shear rate $\dot{\unicode[STIX]{x1D6FE}}=10$ , 30, 100, 300, 1000, 2000 and $10\,000~\text{s}^{-1}$ , from left to right; $\hat{\unicode[STIX]{x1D60B}}_{xy}^{R}$ at $\dot{\unicode[STIX]{x1D6FE}}=10~\text{s}^{-1}$ is not shown for scaling purpose due to its small magnitude. Experimental results for RBC enhanced solute (Diller, Mikic & Drinker Reference Diller, Mikic and Drinker1980; Wang & Keller Reference Wang and Keller1985) and platelet (Grabowski, Friedman & Leonard Reference Grabowski, Friedman and Leonard1972; Antonini et al. Reference Antonini, Guiffant, Quemada and Dosne1978) diffusivity in the velocity-gradient ( $yy$ ) direction are plotted for comparison. NPs of size $2a_{1}=100~\text{nm}$ yield $\unicode[STIX]{x1D60B}^{B}=3.78~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ .

3.3 Haemorheological dependence of NP long-time diffusivity

In this section, we focus on examining the NP long-time diffusive behaviour. The NP long-time diffusivities are evaluated in the long-time regime (after 100 strain units) and listed in table 1. Experimental statistics from various sources (Grabowski et al. Reference Grabowski, Friedman and Leonard1972; Antonini et al. Reference Antonini, Guiffant, Quemada and Dosne1978; Diller et al. Reference Diller, Mikic and Drinker1980; Wang & Keller Reference Wang and Keller1985) are selected for comparison to gain credibility of the simulation results. Given the distinct time scales associated with the Brownian ( $\unicode[STIX]{x1D70F}_{B}\sim 10^{-4}~\text{s}$ ) and the long-time RBC-enhanced diffusion ( $\unicode[STIX]{x1D70F}_{R}\sim 100/\dot{\unicode[STIX]{x1D6FE}}$ ) phenomenon, the coupling of BD and RESID follows a simple superposition, $\unicode[STIX]{x1D60B}_{ij}^{\infty }=\unicode[STIX]{x1D60B}_{ij}^{R}+\unicode[STIX]{x1D60B}^{B}\unicode[STIX]{x1D6FF}_{ij}$ , as confirmed in previous studies (e.g. Liu et al. Reference Liu, Zhu, Rao, Clausen and Aidun2018), where $\unicode[STIX]{x1D60B}_{ij}^{R}$ denotes the RESID. Since $\unicode[STIX]{x1D60B}^{B}$ at infinite dilution follows the SE relation with negligible dependence on flow conditions, the haemorheological response of $\unicode[STIX]{x1D60B}_{ij}^{\infty }$ is essentially determined by $\unicode[STIX]{x1D60B}_{ij}^{R}$ .

3.3.2 Haematocrit dependence

Figure 6.

Normalized particle long-time RESID in (a) $xx$ , (b) $yy$ , (c)  $zz$ and (d) $xy$ directions against haematocrit under different shear rate in logarithm scale. Lines are the best power-law fits to the data points at specific $\dot{\unicode[STIX]{x1D6FE}}$ . The $\hat{\unicode[STIX]{x1D60B}}_{xy}^{R}$ value at $\dot{\unicode[STIX]{x1D6FE}}=10~\text{s}^{-1}$ (not shown) is positive with negligible magnitude compared to other diffusion terms. NPs of size $2a_{1}=100~\text{nm}$ yield $\unicode[STIX]{x1D60B}^{B}=3.78~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ .

Figure 6 displays the same data as figure 5 but plotted against $\unicode[STIX]{x1D719}$ to show the haematocrit dependence of RESID at different shear rate. At low shear rates ( $\dot{\unicode[STIX]{x1D6FE}}\leqslant 100$ ), all diffusion terms manifest a linear $\unicode[STIX]{x1D719}$ scaling. Note that a linear concentration dependence of particle self-diffusivity has been observed in sheared monodisperse suspensions in the presence of surface roughness (Da Cunha & Hinch Reference Da Cunha and Hinch1996) or residual Brownian motion (Brady & Morris Reference Brady and Morris1997), which causes the two-body interaction to be irreversible and hence give rise to a diffusive behaviour. Since the shear-induced diffusion of NP is driven by NP–RBC interaction, it is likely that the $\unicode[STIX]{x1D719}$ scaling of the NP diffusion tensor results from the irreversible two-body interaction between NP and RBC induced by the RBC membrane roughness/flexibility and the NP Brownian effect. The off-diagonal diffusivity $\hat{\unicode[STIX]{x1D60B}}_{xy}^{R}$ is found to be positive ( $+$ ) at very low shear rate ( $\dot{\unicode[STIX]{x1D6FE}}\leqslant 10~\text{s}^{-1}$ ) and otherwise negative ( $-$ ). The change-of-sign behaviour of the off-diagonal diffusivity designates the dominant displacement direction of NP changing from along the extensional axes to along the compressive axes of the flow as shear rate increases. A similar observation has been reported in sheared monodisperse colloidal suspensions (Foss & Brady Reference Foss and Brady1999).

As shear rate grows above $100~\text{s}^{-1}$ , various $\unicode[STIX]{x1D719}$ scaling arises in different RESID components. The $\hat{\unicode[STIX]{x1D60B}}_{xx}^{R}$ component exhibits a quadratic $\unicode[STIX]{x1D719}$ scaling ( ${\sim}\unicode[STIX]{x1D719}^{2}$ ), which suggests that in the $x$ direction additional effects exist to drive the NP diffusive motion besides the irreversible pairwise interactions. The $\hat{\unicode[STIX]{x1D60B}}_{yy}^{R}$ and $\hat{\unicode[STIX]{x1D60B}}_{xy}^{R}$ components scale linearly with $\unicode[STIX]{x1D719}$ , suggesting that the irreversible pairwise interaction remains the dominant diffusive mechanism. The $\hat{\unicode[STIX]{x1D60B}}_{zz}^{R}$ component exhibits a transition from linear to sublinear $\unicode[STIX]{x1D719}$ scale ( ${\sim}\unicode[STIX]{x1D719}^{0.6}$ ). The mostly linear $\unicode[STIX]{x1D719}$ scale in $\hat{\unicode[STIX]{x1D60B}}_{yy}^{R}$ observed in the current numerical study is consistent with the experimental observation in Wang & Keller (Reference Wang and Keller1985), where they show that the solute diffusivity in the velocity-gradient ( $y$ ) direction is augmented by approximately threefold when the particle concentration increases from 0.1 to 0.31. Since suspension viscosity typically increases with particle concentration (Foss & Brady Reference Foss and Brady2000b ), the enhancement of solute (e.g. NP) diffusivity at increased $\unicode[STIX]{x1D719}$ indicates that the RESID is due to the RBC–NP interaction rather than the secondary flow effect (Wang & Keller Reference Wang and Keller1985).

Figure 7.

RBC–NP partial pair distribution function, $g_{21}(\boldsymbol{r})$ , projected onto the $xy$ , $xz$ and $yz$ planes in the long-time regime under various haemorheological conditions, where the PPDF contour has the horizontal edge aligned with the first axis (e.g. $xy$ PPDF contour has the horizontal edge aligned with the $x$ axis). The edge length of the sampling box for computing $g_{21}(\boldsymbol{r})$ is three times the maximum diameter of the undeformed RBC.

3.4 Microstructure

The rheological properties of particle suspensions are often determined by the mechanistic phenomenon occurring on the particle length scales. To elucidate the physical mechanisms that govern the haemorheological scaling behaviours of the NP diffusion tensor observed in § 3.3, the RBC–NP PPDF, $g_{21}(\boldsymbol{r})$ , in the long-time regime is plotted to visualize the configurational microstructure of the RBC–NP bidisperse suspension under various haematocrits and shear rates. The techniques used to compute and project $g_{21}(\boldsymbol{r})$ are discussed in detail in appendix B.

Figure 7 presents a matrix of $g_{21}(\boldsymbol{r})$ projections onto the velocity–velocity gradient ( $xy$ ) plane, the velocity–vorticity ( $xz$ ) plane and the velocity gradient–vorticity ( $yz$ ) plane under two shear rates ( $\dot{\unicode[STIX]{x1D6FE}}=100$ or $2000~\text{s}^{-1}$ ) and two haematocrits ( $\unicode[STIX]{x1D719}=0.1$ or 0.4). In general, all PPDFs feature a large-scale rhombus shape as opposed to the typical circular shape commonly observed in rigid-sphere-particle suspensions (Foss & Brady Reference Foss and Brady2000a ; Wang & Brady Reference Wang and Brady2016; Pednekar, Chun & Morris Reference Pednekar, Chun and Morris2018). This can be attributed to the disk shape of RBC that causes geometry-specific anisotropy of the microstructure. The average RBC shapes under various haemorheological conditions, as shown in the central low-intensity region of the PPDF contours, are nicely captured through the PPDF sampling procedure. All PPDFs considered in the $xy$ plane show more intensified distribution (i.e. higher probability of RBC–NP interaction) near the RBC disk surfaces, of which the surface normal directions are more aligned with the compressive axes (in second and fourth quadrants) in accordance with the negative values of $\hat{\unicode[STIX]{x1D60B}}_{xy}^{R}$ . In the $xz$ and $yz$ plane, the PPDF distribution tends to be symmetric about the principal axes ( $x$ , $y$ or $z$ ), which explains the zero values of the $xz$ and $yz$ diffusivities.

As the haemorheological condition changes, the detailed configuration of the PPDF within the rhombus structure also varies. At low haematocrit ( $\unicode[STIX]{x1D719}=0.1$ ), the $xy$ PPDF shows fore–aft low intensity similar to rigid-particle suspensions (Kumar & Higdon Reference Kumar and Higdon2011). However, the break of the fore–aft symmetry, being different from the rigid-particle suspensions, seems to be related to the RBC orientation aligned with the extensional flow axes that is further caused by the tank-treading motion of the RBC membrane (Reasor et al. Reference Reasor, Clausen and Aidun2013a ). As the haematocrit increases to $\unicode[STIX]{x1D719}=0.4$ , the overall PPDF around the RBC becomes more intensified, meaning the local NP concentration near the RBC surface increases. More interestingly, the fore–aft low PPDF region observed under low haematocrit gets intensified substantially. The change of the NP microstructure with increased haematocrit can be explained by smaller inter-cell separation and hence NP getting squeezed in a smaller inter-cell region. At low shear rate ( $\dot{\unicode[STIX]{x1D6FE}}=100~\text{s}^{-1}$ ), RBC shows a close-to-equilibrium biconcave shape. Increasing the shear rate to $\dot{\unicode[STIX]{x1D6FE}}=2000~\text{s}^{-1}$ results in a significant elongation of the average RBC shape in the flow ( $x$ ) direction accompanied by certain contraction in the velocity-gradient ( $y$ ) and vorticity ( $z$ ) directions.

The above configurational changes of $g_{21}(\boldsymbol{r})$ under various shear rates and haematocrits provide possible mechanistic explanations for the nonlinear $\unicode[STIX]{x1D719}$ dependence observed at high shear rates, as discussed in § 3.3.2. The quadratic dependence of $\hat{\unicode[STIX]{x1D60B}}_{xx}^{R}$ ( ${\sim}\unicode[STIX]{x1D719}^{2}$ ), shown in figure 6(a), is probably due to the occurrence of three-body RBC–NP interactions in the longitudinal ( $x$ ) direction caused by the compound effect of the elevated NP concentration at the RBC fore–aft surface and the elongation of the RBC in the $x$ direction. The reduced $\unicode[STIX]{x1D719}$ dependence in $\hat{\unicode[STIX]{x1D60B}}_{zz}^{R}$ at high shear rate, as shown in figure 6(c), can be attributed to the relatively large contraction of the RBC in the vorticity direction, which reduces the effective $\unicode[STIX]{x1D719}$ in the $z$ direction. The lesser reduction of the $\unicode[STIX]{x1D719}$ dependence in the $y$ direction, as shown in figure 6(c), is due to the $y$ direction contraction of RBC under shear being less significant than that in the $z$ direction, as clearly indicated in the PPDF contours.

3.5 Role of RBC deformability

The above PPDF analysis shows prominent RBC morphological change with elevated shear rate, which suggests that RBC deformability may play an important role in causing the nonlinear $\dot{\unicode[STIX]{x1D6FE}}$ dependence of the NP diffusion tensor. In this section, we perform numerical experiments to further quantitatively explain the nonlinear $\dot{\unicode[STIX]{x1D6FE}}$ scaling of RESID in the intermediate-shear-rate regime ( $100\leqslant \dot{\unicode[STIX]{x1D6FE}}\leqslant 2000~\text{s}^{-1}$ ), as observed in § 3.3.1. Changing shear rate alters both the fluid inertia and RBC deformability, quantified by $Pe$ and $Ca_{G}$ , respectively. To interrogate the isolated effect of RBC deformability ( $Ca_{G}$ ), we fix $Ca_{G}$ by scaling up $G$ while increasing $Pe$ (through increasing $\dot{\unicode[STIX]{x1D6FE}}$ ). Two capillary numbers, $Ca_{G}=0.055$ and $0.55$ , are considered corresponding to the $Ca_{G}$ regime where the nonlinear shear-rate dependence of RESID occurs. For these simulations, we select a fixed haematocrit of $\unicode[STIX]{x1D719}=0.4$ and an NP size of $2a_{1}=100~\text{nm}$ . Table 2 lists all parameters and the NP diffusivity values for the cases tested.

Figure 8.

The $Pe$ dependence of the (a) diagonal and (b) off-diagonal RESID terms at fixed capillary number (by rescaling $G$ ) with $\unicode[STIX]{x1D719}=0.4$ . Two RBC capillary numbers, $Ca_{G}=0.55$ and $0.055$ , are considered to see the isolated effect of changing RBC deformability ( $Ca_{G}$ ) or adjusting the inertial effect ( $Pe$ ).

Table 2.

Simulation data for numerical experiments concerning the isolated RBC deformation effect in NP–RBC suspensions. The RBC membrane shear modulus, $G$ , is rescaled to numerically match specific $Ca_{G}$ while changing shear rate. Here, the NP size is fixed to $2a_{1}=100~\text{nm}$ ; RBC has an effective radius of $a_{2}=2.9~\unicode[STIX]{x03BC}\text{m}$ . Brownian diffusivity is calculated by $\unicode[STIX]{x1D60B}^{B}=k_{B}T/6\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}a_{1}$ at a temperature of $T=310$  K.

As presented in figure 8(a,b), instead of exhibiting nonlinear $Pe$ scaling as observed in § 3.3.1, all RESID components show a linear dependence on $Pe$ at fixed RBC capillary number of $Ca_{G}=0.55$ or $0.055$ across the wide range of shear rates. This recovers the linear $Pe$ dependence of solute/self-diffusivity in sheared rigid-particle suspensions (Zydney & Colton Reference Zydney and Colton1988; Foss & Brady Reference Foss and Brady1999; Sierou & Brady Reference Sierou and Brady2004) since we intentionally fix the RBC deformation ( $Ca_{G}$ ) while adjusting $Pe$ .

Moreover, an increase of $Ca_{G}$ (from 0.055 to 0.55) leads to an increase of $\hat{\unicode[STIX]{x1D60B}}_{xx}^{R}$ but a decrease of $\hat{\unicode[STIX]{x1D60B}}_{yy}^{R}$ and $\hat{\unicode[STIX]{x1D60B}}_{zz}^{R}$ , as shown in figure 8(a). These results are direct evidence indicating that the nonlinear shear-rate scaling of the diagonal RESID terms observed in § 3.3.1 is due to the variation of RBC deformability ( $Ca_{G}$ ) induced by changing shear rate at fixed $G$ . More specifically, the increase of $Ca_{G}$ with increasing shear rate at $100<\dot{\unicode[STIX]{x1D6FE}}<2000~\text{s}^{-1}$ under fixed $G$ leads to the superlinear shear-rate dependence of longitudinal diffusivity and meanwhile the sublinear dependence of cross-stream diffusivities. In figure 8(b), the off-diagonal diffusivity shows insignificant change at different $Ca_{G}$ , consistent with its mostly linear shear-rate dependence shown in figure 5(d).

3.6 Relevant length scale

Although the shear adaptation of the RBC deformability is shown to be responsible for the nonlinear shear-rate dependence of the RESID, it is still unclear in what way it alters the RESID. In this section, we further identify the characteristic length scale relevant to the RESID to gain a more in-depth understanding of this nonlinear phenomenon.

In a series of experiments measuring the particle diffusivity in a concentrated non-colloidal suspension, Breedveld et al. (Reference Breedveld, Van den Ende, Tripathi and Acrivos1998) show that the ratio of the diffusivity of fluid tracers to the self-diffusivity of non-colloidal particles is close to unity. Through DNS of the cellular blood flow in a microchannel, Zhao et al. (Reference Zhao, Shaqfeh and Narsimhan2012) show that the cross-stream diffusivity of platelets ( ${\sim}2~\unicode[STIX]{x03BC}\text{m}$ ) in blood exhibits similar magnitude to that of passive tracers. These findings imply that the shear-induced diffusivity is insensitive to the size of the scarce (in terms of volume fraction) small particles. Following these results, we hypothesize that RESID is insensitive to the size of NPs. Moreover, the characteristic length scale associated with the RESID should be the size of RBCs. To confirm this hypothesis, we evaluate the RESID at fixed haemorheological condition with various NP size. The NP size is kept in submicrometre scale such that the NP–RBC size ratio satisfies $a_{1}/a_{2}\ll 1$ . The volume fraction of the NP phase is kept below 0.32 % to satisfy the dilute condition. The set-up and measured diffusivities are listed in table 3.

In figure 9(a), we plot the RESID versus $Pe$ at $\dot{\unicode[STIX]{x1D6FE}}=1000~\text{s}^{-1}$ and $\unicode[STIX]{x1D719}=0.2$ . Here, $Pe$ is adjusted by NP size in the range of $50\leqslant 2a_{1}\leqslant 800~\text{nm}$ . A sublinear $\hat{\unicode[STIX]{x1D60B}}_{ij}^{R}$ – $Pe$ relationship is observed for all diffusion coefficients. Figure 9(b) further plots the same data as figure 9(a), but with $Pe$ rescaled by the NP–RBC size ratio as

(3.3) $$\begin{eqnarray}\widetilde{Pe}=Pe\left(\frac{a_{2}}{a_{1}}\right)^{2}=\frac{\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2}}{\unicode[STIX]{x1D60B}^{B}}.\end{eqnarray}$$

This simple rescaling leads to a strong linear relationship between all RESID terms and the rescaled Péclet number, $\widetilde{Pe}$ . Moreover, the observed linear relationship, $(\unicode[STIX]{x1D60B}_{ij}^{\infty }/\unicode[STIX]{x1D60B}^{B}-\unicode[STIX]{x1D6FF}_{ij})\sim O(\widetilde{Pe})$ , can be deduced to show that $\unicode[STIX]{x1D60B}_{ij}^{R}\sim O(\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2})$ , which indicates that the dimensional RESID is indeed insensitive to the NP size within the range of NP sizes considered in the current study. This also suggests that NPs at submicrometre size play a secondary role in affecting the NP apparent diffusivity primarily through altering BD.

Figure 9.

RESID plotted against (a) Péclet number $Pe$ and (b) the rescaled Péclet number $\widetilde{Pe}$ , at $\dot{\unicode[STIX]{x1D6FE}}=1000~\text{s}^{-1}$ and $\unicode[STIX]{x1D719}=0.2$ in logarithmic scales. Both $Pe$ and $\widetilde{Pe}$ are adjusted with NP size in the range of $2a_{1}=50{-}800~\text{nm}$ . (c) Linear dependence of RESID on $\widetilde{Pe}$ at fixed $Ca_{G}=0.55$ and $\unicode[STIX]{x1D719}=0.2$ ; $\widetilde{Pe}$ is adjusted by shear rate (with $2a_{1}=100~\text{nm}$ ) or NP size (with $\dot{\unicode[STIX]{x1D6FE}}=1000~\text{s}^{-1}$ ). The diffusion tensor in monodisperse colloidal suspensions (Foss & Brady Reference Foss and Brady1999) is also plotted for comparison, where $\widetilde{Pe}$ degrades to $Pe$ .

Table 3.

Simulation data for numerical experiments concerning the relevant length scale in NP–RBC suspensions. NP size is varied in the range of $2a_{1}=50{-}800~\text{nm}$ . RBC has an effective radius of $a_{2}=2.9~\unicode[STIX]{x03BC}\text{m}$ . Brownian diffusivity is calculated by $\unicode[STIX]{x1D60B}^{B}=k_{B}T/6\unicode[STIX]{x1D707}\unicode[STIX]{x03C0}a_{1}$ at temperature $T=310$  K. For the $800~\text{nm}$ case, the number of NPs is set to 500 to satisfy the dilute condition for the NP volume fraction. The last three cases are performed with adjusted $G$ to fix $Ca_{G}$ .

To further justify the universality of the rescaled Péclet number, $\widetilde{Pe}$ , i.e. whether the RBC size is the reasonable length scale associated with RBC-enhanced diffusion or not, we plot RESID against $\widetilde{Pe}$ at $Ca_{G}=0.55$ and $\unicode[STIX]{x1D719}=0.2$ , with $\widetilde{Pe}$ adjusted by either shear rate or NP size. As shown in figure 9(c), the data points for a particular RESID coefficient are found to be well aligned on the same linear $\widetilde{Pe}$ scaling line. This directly confirms that the RBC size is the characteristic length scale governing the RBC-enhanced diffusion, and $\widetilde{Pe}$ is a more general non-dimensional term quantifying the RESID.

Identifying RBC size as the RESID length scale helps in further understanding the nonlinear shear-rate dependence of the RESID. In figure 10, we present snapshots of RBC and NP distribution and morphology in the mid-cross-stream cross-sectional plane. As shown in figure 10, the $x$ direction elongation of RBCs under high shear causes a contraction of RBCs in the $yz$ plane, which causes a reduction of the average effective RBC size (i.e. the length scale $\bar{a}_{2,yz}$ ) and hence a decrease of the effective $\widetilde{Pe}$ in cross-stream directions. This explains the sublinear shear-rate scaling of the cross-stream RESID when increasing $Ca_{G}$ . Likewise, the superlinear shear-rate scaling of the longitudinal RESID can be attributed to the elongation of RBC in the streamwise direction, i.e. the increase of the RESID length scale in the longitudinal direction. The change of the RESID length scales in different principal directions due to shear is the root cause of the nonlinear shear-rate dependence of the NP diffusion tensor.

Figure 10.

RBC and NP distributions and morphology in the cross-stream plane ( $y$ – $z$ ) with (a) $Ca_{G}=0.055$ , (b) $Ca_{G}=0.55$ and (c) $Ca_{G}=1.1$ at $Pe=0.066$ and $\unicode[STIX]{x1D719}=0.4$ . Large non-circular disks are RBC cross-sectional contours. Small circles are NP cross-sectional contours. The $\bar{a}_{2,yz}$ value can be estimated from the PPDF analysis.

3.7 Empirical correlations

In this section, we construct empirical correlations for the long-time NP diffusion tensor in sheared blood based on the scaling observations in the previous sections. Since all RESID coefficients scale linearly with $\widetilde{Pe}$ at fixed $Ca_{G}$ , and the nonlinear shear-rate dependence of RESID is primarily due to the variation of $Ca_{G}$ , as demonstrated in §§ 3.5 and 3.6, there hence exists a scaling relation, $\hat{\unicode[STIX]{x1D60B}}_{ij}^{R}\sim O(\unicode[STIX]{x1D719}^{p_{1}}\widetilde{Pe}Ca_{G}^{p_{2}})$ , such that the exponents $p_{1}$ and $p_{2}$ can be estimated through matching the nonlinear $\unicode[STIX]{x1D719}$ and $\dot{\unicode[STIX]{x1D6FE}}$ scalings, respectively, as observed in § 3.3. Therefore, empirical correlations of NP long-time diffusivities can be constructed as functions of $\unicode[STIX]{x1D719}$ , $\widetilde{Pe}$ and $Ca_{G}$ in the haemorheological range of $10\leqslant \dot{\unicode[STIX]{x1D6FE}}\leqslant 2000~\text{s}^{-1}$ and $0.1\leqslant \unicode[STIX]{x1D719}\leqslant 0.4$ as follows:

(3.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D60B}_{xx}^{\infty }}{\unicode[STIX]{x1D60B}^{B}}=\left\{\begin{array}{@{}ll@{}}1+{\mathcal{C}}_{xx}^{l}\unicode[STIX]{x1D719}\widetilde{Pe},\quad & 0.0055\leqslant Ca_{G}\leqslant 0.055,\\ 1+{\mathcal{C}}_{xx}^{h}\unicode[STIX]{x1D719}^{Ca_{G}+0.945}\widetilde{Pe}Ca_{G}^{0.8},\quad & 0.055<Ca_{G}\leqslant 1.10,\end{array}\right. & \displaystyle\end{eqnarray}$$

(3.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D60B}_{yy}^{\infty }}{\unicode[STIX]{x1D60B}^{B}}=\left\{\begin{array}{@{}ll@{}}1+{\mathcal{C}}_{yy}^{l}\unicode[STIX]{x1D719}\widetilde{Pe},\quad & 0.0055\leqslant Ca_{G}\leqslant 0.055,\\ 1+{\mathcal{C}}_{yy}^{h}\unicode[STIX]{x1D719}\widetilde{Pe}Ca_{G}^{-0.3},\quad & 0.055<Ca_{G}\leqslant 1.10,\end{array}\right. & \displaystyle\end{eqnarray}$$

(3.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D60B}_{zz}^{\infty }}{\unicode[STIX]{x1D60B}^{B}}=\left\{\begin{array}{@{}ll@{}}1+{\mathcal{C}}_{zz}^{l}\unicode[STIX]{x1D719}\widetilde{Pe},\quad & 0.0055\leqslant Ca_{G}\leqslant 0.055,\\ 1+{\mathcal{C}}_{zz}^{h}\unicode[STIX]{x1D719}^{1.022-0.4Ca_{G}}\widetilde{Pe}Ca_{G}^{-0.2},\quad & 0.055<Ca_{G}\leqslant 1.10,\end{array}\right. & \displaystyle\end{eqnarray}$$

(3.4d ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D60B}_{xy}^{\infty }}{\unicode[STIX]{x1D60B}^{B}}=\frac{\unicode[STIX]{x1D60B}_{yx}^{\infty }}{\unicode[STIX]{x1D60B}^{B}}=\left\{\begin{array}{@{}ll@{}}{\mathcal{C}}_{xy}^{l},\quad & 0.0055\leqslant Ca_{G}\leqslant 0.055,\\ {\mathcal{C}}_{xy}^{h}\unicode[STIX]{x1D719}\widetilde{Pe},\quad & 0.055<Ca_{G}\leqslant 1.10,\end{array}\right. & \displaystyle\end{eqnarray}$$

${\mathcal{C}}_{ij}^{q}$

$q\in \{l,h\}$

$O(\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2})$

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D63F}^{\infty }=\unicode[STIX]{x1D60B}^{B}\unicode[STIX]{x1D644}+\unicode[STIX]{x1D719}\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2}\unicode[STIX]{x1D648}^{q},\end{eqnarray}$$

where the anisotropic tensor, $\unicode[STIX]{x1D648}^{q}$ , according to the severity of the RBC deformability ( $Ca_{G}=\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FE}}a_{2}/G$ ) at different level of shear rates, yields piecewise expressions:

(3.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D648}^{l}=\left(\begin{array}{@{}ccc@{}}{\mathcal{C}}_{xx}^{l} & {\mathcal{C}}_{xy}^{l} & 0\\ {\mathcal{C}}_{yx}^{l} & {\mathcal{C}}_{yy}^{l} & 0\\ 0 & 0 & {\mathcal{C}}_{zz}^{l}\end{array}\right),\quad 10\leqslant \dot{\unicode[STIX]{x1D6FE}}\leqslant 100~\text{s}^{-1}, & \displaystyle\end{eqnarray}$$

(3.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D648}^{h}=\left(\begin{array}{@{}ccc@{}}{\mathcal{C}}_{xx}^{h}\unicode[STIX]{x1D719}^{Ca_{G}-0.055}Ca_{G}^{0.8} & {\mathcal{C}}_{xy}^{h} & 0\\ {\mathcal{C}}_{yx}^{h} & {\mathcal{C}}_{yy}^{h}/Ca_{G}^{0.3} & 0\\ 0 & 0 & {\mathcal{C}}_{zz}^{h}/(\unicode[STIX]{x1D719}^{2Ca_{G}+0.11}Ca_{G})^{0.2}\end{array}\right), & \displaystyle \nonumber\\ \displaystyle & \displaystyle 100<\dot{\unicode[STIX]{x1D6FE}}\leqslant 2000~\text{s}^{-1}, & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D648}^{q}$

$O(\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2})$

${\mathcal{C}}_{ij}^{q}$

$\unicode[STIX]{x1D60B}_{ij}^{(R,m)}$

$\unicode[STIX]{x1D60B}_{ij}^{(R,th)}/{\mathcal{C}}_{ij}^{q}$

$i$

$j$

$q$

$\dot{\unicode[STIX]{x1D6FE}}$

$\unicode[STIX]{x1D719}$

$\unicode[STIX]{x1D60B}_{ij}^{(R,m)}=\unicode[STIX]{x1D60B}_{ij}^{\infty }-\unicode[STIX]{x1D60B}^{B}\unicode[STIX]{x1D6FF}_{ij}$

$\unicode[STIX]{x1D60B}_{ij}^{\infty }$

$\unicode[STIX]{x1D60B}_{ij}^{(R,th)}=\unicode[STIX]{x1D719}\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2}\unicode[STIX]{x1D614}_{ij}^{q}$

${\mathcal{C}}_{xy}^{l}$

${\mathcal{C}}_{xy}^{h}$

$\dot{\unicode[STIX]{x1D6FE}}<10~\text{s}^{-1}$

$\unicode[STIX]{x1D719}<0.1$

$\unicode[STIX]{x1D63F}^{\infty }$

Figure 11.

Measured NP RESID, $\unicode[STIX]{x1D60B}_{ij}^{(R,m)}=\unicode[STIX]{x1D60B}_{ij}^{\infty }-\unicode[STIX]{x1D60B}^{B}\unicode[STIX]{x1D6FF}_{ij}$ , as a function of the theoretical RESID, $\unicode[STIX]{x1D60B}_{ij}^{(R,th)}=\unicode[STIX]{x1D719}\dot{\unicode[STIX]{x1D6FE}}a_{2}^{2}\unicode[STIX]{x1D614}_{ij}^{q}$ , based on the empirical correlations presented in equations (3.6a ) and (3.6b ). For cases with $\dot{\unicode[STIX]{x1D6FE}}\leqslant 100~\text{s}^{-1}$ (filled symbols), $\unicode[STIX]{x1D614}_{ij}^{l}$ is applied to calculate $\unicode[STIX]{x1D60B}_{ij}^{(R,th)}$ . For cases with $\dot{\unicode[STIX]{x1D6FE}}>100~\text{s}^{-1}$ (open symbols), $\unicode[STIX]{x1D614}_{ij}^{h}$ is applied to calculate $\unicode[STIX]{x1D60B}_{ij}^{(R,th)}$ .

Table 4.

Correlation constants for the empirical correlations of NP diffusion tensor in sheared cellular blood flow.

Figure 11 plots the measured NP diffusivities (subtracted by BD) versus the theoretical estimation based on the empirical correlations. The good collapse of the diffusivity measurements on the $\unicode[STIX]{x1D60B}_{ij}^{(R,m)}=\unicode[STIX]{x1D60B}_{ij}^{(R,th)}$ line demonstrates the empirical correlations can well reproduce the numerically measured diffusivity tensor of NP in sheared blood.