Effect of elastic walls on suspension flow

We study suspensions of rigid particles in a plane Couette flow with deformable elastic walls. We find that, in the limit of vanishing inertia, the elastic walls induce shear thinning of the suspension flow such that the effective viscosity decreases as the wall deformability increases. This shear-thinning behavior originates from the interactions between rigid particles, soft walls, and carrier fluids; an asymmetric wall deformation induces a net lift force acting on the particles which therefore migrate towards the bulk of the channel. Based on our observations, we provide a closure for the suspension viscosity which can be used to model the rheology of suspensions with arbitrary volume fraction in elastic channels.

Figure 1

Visualization of the particle suspension in a Couette flow with elastic walls, with a sketch of the relevant geometrical parameters.

Figure 2

Effective viscosity of the suspension ${\mu }_{\mathrm{e}}$ as a function of the volume fraction $\mathrm{\Phi }$ (left) and capillary number $\text{Ca}$ (right). The effective viscosity is normalized with the fluid viscosity ${\mu }_0$. The brown, blue, orange, and red colors are used for volume fractions $\mathrm{\Phi }$ equal to 0.0016, 0.1, 0.2, and 0.29, while the symbols $▵$, $⬠$, $◊$, $○$, $\square$, and $▿$ for capillary numbers $\text{Ca}$ equal to 0.05, 0.5, 1, 2, 4, and 8. All the previous symbols and colors are used for an elastic layer of thickness $h^e=R$; the light and dark green squares represent the results for $\mathrm{\Phi }=0.2$, $\text{Ca}=4$, and $h^e=0.5R$ and $1.5R$, respectively. The black line in the left panel is the Eilers formula, Eq. (1), with ${\mathrm{\Phi }}_m=0.6$ and $B=1.7$.

Figure 3

Left: Mean particle concentration $\mathrm{\Psi }$ as a function of $y$. The shaded areas represent the variation due to the capillary number $\text{Ca}$, with the black solid and dashed lines representing the lowest and highest values of $\text{Ca}$; the gray solid lines are the data for rigid walls. Right: Distance from the wall $h-y$ at which the mean particle concentration is equal to 0.01 of the nominal value as a function of the capillary number $\text{Ca}$.

Figure 4

Root mean square of the wall-normal velocity fluctuations $v^{\prime }$ at the interface between the fluid and deformable wall $y=h$ as a function of the capillary number $\text{Ca}$.

Figure 5

Sketch of the mechanism of the lift generation induced by the interaction of a moving sphere and a rigid particle.

Figure 6

Left: Mean wall deformations ${\delta }_{\pm }$ as a function of the capillary number $\text{Ca}$. The solid and open points are the negative ${\delta }_{-}$ and positive ${\delta }_{+}$ wall deformations, respectively, while the solid lines are fits to our data in the form of Eq. (2). Right: Normalized effective viscosity ${\mu }_{\mathrm{e}}$ as a function of the effective volume fraction ${\mathrm{\Phi }}_{\mathrm{e}}$ computed from Eq. (3). The black solid line is the Eilers formula reported in Eq. (1).