Bidisperse suspension balance model

The suspension balance model (SBM) for viscous Stokes flow has been well studied for the case of identical, monodisperse spherical particles in a channel, however, more work remains to be done to explore the bidisperse and polydisperse SBM. We present a simple extension of the SBM that allows for modeling of suspensions of particles of bidisperse size. The comparison with available experiments and direct simulations is found to be good. Additionally, we present a range of simulations to justify the assumptions made in the polydisperse suspension balance model for non-Brownian, neutrally buoyant particles that vary in size with a bidisperse distribution. The aim is to study the effects of moderate size variation on the rheology of the suspension and the distribution of the particles across the flow, especially in the near-wall regions. It is shown that, at the size ratios considered, bidispersity in the particle size does not significantly affect rheological parameters including the particle volume fraction and relative viscosity in the bulk of a Couette flow. The particle phase stresses are distributed between small and large particles in proportion to the relative volume fractions of each species. In bidisperse suspensions, the large particles in the wall layer develop a spatial structure and form chains in the streamwise direction.

Figure 1

Values of $A$ and $B$ for two particles in a shear flow for $\lambda =0.5$ from both FCM and FD. The exact far-field solution is given by Batchelor and Green [71] until $\xi =1.75$. The near-field solution is from Pesche [72]. We see excellent agreement between the previous results and the two numerical methods considered here.

Figure 2

(a) Total volume fraction profile. (b) Solid lines: small particle volume fraction profile. Dashed lines: large particle volume fraction profile for $\lambda =0.8,0.72$, and 0.6 with $\beta =0.75$, compared to a monodisperse simulation, averaged over $\dot{\gamma }t=100\text{–}125$. The channel wall is at $y/a_L=0$, while the channel centerline is at $y/a_L=20$ (FCM simulations). (c) Large particle volume fractions scaled by the average large particle volume fraction ($\langle {\varphi }_L\rangle =0.1$) and monodisperse volume fraction scaled by the average monodisperse volume fraction ($\varphi =0.4$). (d) Small particle volume fractions scaled by the average small particle volume fraction ($\langle {\varphi }_S\rangle =0.3$) and monodisperse volume fraction scaled by the average monodisperse volume fraction.

Figure 3

Average normal stress differences for bidisperse suspensions at a range of particle size ratios and small particle volume fractions averaged over all particles for $\dot{\gamma }t=100\text{–}125$. Left: $N_1$. Right: $N_2$. Filled symbols: FCM. Empty symbols: FD. Averages are taken over the domain $5a_L\le y\le 35a_L$ to exclude the wall layer.

Figure 4

Normal stress difference profiles $N_1$ (left) and $N_2$ (right) from FCM simulations for varying $\lambda$ at $\beta =0.75$ and a monodisperse suspension averaged over $\dot{\gamma }t=100\text{–}125$. Both $N_1$ and $N_2$ are negative on average across the channel; however, $N_1$ is positive in the wall layer.

Figure 5

Stress profiles ${\sigma }_{33}$ (top row), ${\sigma }_{22}$ (middle row), ${\sigma }_{11}$ (bottom row) for $\lambda =0.6$ for $\beta =0.25$ (left column) and $\beta =0.75$ (right column) averaged over $\dot{\gamma }t=100\text{–}125$ (FCM simulations). The stress for each size particle is distributed by the relative volume fraction of each species.

Figure 6

Ratio of particle stresses for small and large particles compared to a linear fit (dashed line) for FCM with $\lambda =0.8$ (filled symbols) and $\lambda =0.6$ (empty symbols) and FD $\lambda =0.8$ (green), $\lambda =0.72$ (red), and $\lambda =0.6$ (blue) averaged over $\dot{\gamma }t=100\text{–}125$. The relative contribution to the total stress from the small and large particles is directly proportional to the relative volume fractions of the small and large particles.

Figure 7

Profile of the total particle contact pressure, averaged over strain interval $\dot{\gamma }t=100\text{–}125$ for $\beta =0.75$ and a monodisperse suspension (FCM simulations). The pressure is constant across the channel outside of the region adjacent to the wall and increases slightly with $\lambda .$

Figure 8

Volume fraction profiles for $\lambda =0.6$ and $\beta =0.75$ (a), 0.5 (b), and 0.25(c). Comparisons between SBM and FCM simulations at strain $\dot{\gamma }t=320$.

Figure 9

Volume fraction profiles for $\lambda =0.8$ and $\beta =0.75$ (a), 0.5 (b), and 0.25(c). Comparisons between SBM and FCM simulations at strain $\dot{\gamma }t=265$.

Figure 10

Volume fraction profiles: comparisons between SBM, FCM simulations and experiments (SW)[18] ($\lambda =0.5$). ${\varphi }_T=0.2$ and $\beta =0.5$ (a); ${\varphi }_T=0.35$ and $\beta =0.71$ (b). SBM results are computed for a strain $\dot{\gamma }t=1500$ and FCM results are at strain $\dot{\gamma }t=280$. The results from [18] are scaled by an additional factor of $\sigma =1.04$ so that $H/\left(a_L\sigma \right)=32$ to provide a direct comparison with the FCM and SBM results (in [18], $H=$ $50\mu \mathrm{m}$, $a_L=$ $1.5\mu \mathrm{m}$, and $a_S=$ $0.69\mu \mathrm{m}$, so $\lambda =0.460$.)

Figure 11

Example of the wall layer in a Couette flow with $\lambda =0.6$ and $\beta =0.5$ intersecting the plane $y=a_s$ at $\dot{\gamma }t=100$ (FCM simulations). The small particles are shown as empty circles, and the large particles are shown as filled circles.

Figure 12

Pair-correlation function of the relative distances between particle centers with $\lambda =0.6$ and $\beta =0.5$ (a–d) and $\beta =0.25$ (e–h) for a Couette flow (FCM simulations). We consider the interactions between small and large particles separately. The flow is sampled at the top and bottom walls on the planes $y=a_s$ and $y=H_y-a_s$ at 40 time steps equally spaced between $\dot{\gamma }t=25$ and $\dot{\gamma }t=125$. The arrow indicates the direction of flow.