Hyperuniform flow fields resulting from hyperuniform configurations of circular disks

Motivated by cancer cell capture and sorting applications, we investigate the scalar velocity field of two-dimensional (2D) laminar flows through configurations of fixed congruent circular disks with a disordered hyperuniform (HU) distribution of the disk centers. Disordered HU many-particle systems suppress large-scale density fluctuations like perfect crystals and yet possess no Bragg peaks, and are endowed with many novel physical properties. Here, we generate 2D HU configurations of congruent nonoverlapping (hard) circular disks with various packing densities via Monte Carlo simulations and obtain the corresponding scalar velocity fields of laminar flows via the lattice Boltzmann method. Using numerical spectral analysis, we find that hyperuniform configurations of the disks can lead to hyperuniform flow fields for a certain range of disk packing fraction $\varphi$ and normalized flow rate $q$ over porosity $\epsilon =1-\varphi$. The degree of hyperuniformity of the flow fields, as quantified via the spectral density value at the zero-wave-number limit, reduces as $\varphi$ and $q$ increase and eventually the fields become nonhyperuniform. The transition from HU flow fields to non-HU fields as $\varphi$ and $q$ vary is captured using a “phase diagram.” For purpose of comparison, disordered nonhyperuniform configurations of circular disks are generated via the random sequential addition process and the resulting flow field is found to be nonhyperuniform. To complement our numerical analysis, we show analytically that HU configurations of circular disks, in the dilute (low-packing-density) limit, lead to HU flow field. Our findings have implications in the design and optimization of microfluidic chips for capture and differentiation of subpopulations in circulating tumor cells.

Figure 1

Representative configurations with different degrees of order analyzed in this work. From left to right: ordered triangular-lattice packing, disordered HU configuration, and disordered RSA configuration.

Figure 2

Flow field (i.e., scalar velocity field normalized with respect to the average flow velocity in the system) associated with hyperuniform configurations with varying packing densities $\varphi$ and normalized flow rates $q$. Blue color indicates low field velocity and red color indicates high flow velocity.

Figure 3

Vector-form spectral density $\widehat{\psi }\left(\mathbf{k}\right)$ associated with the HU configurations shown in Fig. 2.

Figure 4

Angularly averaged spectral density $\widehat{\psi }\left(k\right)$ associated with the HU configurations shown in Fig. 2.

Figure 5

Color map showing the spectral density $\widehat{\psi }\left(k\right)$ value at the limit $k\to 0$ associated with the HU configurations for varying packing density $\varphi$ and flow rate $q$. The transition from HU flow regime to non-HU regime can be clearly seen.

Figure 6

Spectral analysis of the flow fields associated with triangular-lattice configurations of disks. Upper panels: representative flow fields associated with $\varphi =0.15$ and $q=0.00001,0.001,0.01$, and 1 ${\mathrm{mm}}^2$/s from left to right. Middle panels: vector-form spectral density $\widehat{\psi }\left(\mathbf{k}\right)$ associated with the configurations shown in the upper panels. Lower panels: angularly averaged spectral density $\widehat{\psi }\left(\mathbf{k}\right)$ associated with the configurations shown in the upper panels.

Figure 7

Spectral analysis of the flow fields associated with RSA configurations of disks. Upper panels: representative flow fields associated with $\varphi =0.15$ and $q=0.00001,0.001,0.01$, and 1 ${\mathrm{mm}}^2$/s from left to right. Middle panels: vector-form spectral density $\widehat{\psi }\left(\mathbf{k}\right)$ associated with the configurations shown in the upper panels. Lower panels: angularly averaged spectral density $\widehat{\psi }\left(k\right)$ associated with the configurations shown in the upper panels.

Figure 8

Distributions of normalized local flow velocity $v_s/{\overline{v}}_s=q/\overline{q}$ for the flow fields associated with the hyperuniform, triangular-lattice, and RSA configurations at $\varphi =0.1$. The left panel shows the evolution of the flow velocity distribution in the HU configuration for varying flow rates $q\in (0.00001,0.01){\mathrm{mm}}^2$/s. Note that distributions for small flow rates are virtually identical and coincide with one another. The right panel shows the comparison of the flow velocity distributions in three different configurations at $q=0.01{\mathrm{mm}}^2$/s.

Figure 9

Fourier transform of the single-particle flow field $K\left(\mathbf{x}\right)$.