Particle shapes leading to Newtonian dilute suspensions

It is well known that suspensions of particles in a viscous fluid can affect the rheology significantly, producing a pronounced non-Newtonian response even in dilute suspension. However, it is unclear a priori which particle shapes lead to this behavior. We present two simple symmetry conditions on the shape which are sufficient for a dilute suspension to be Newtonian for all strain sizes and one sufficient for Newtonian behavior for small strains. We also construct a class of shapes out of thin, rigid rods not found by the symmetry argument which share this property for small strains.

Figure 1

(a) Examples of particle shapes which have Newtonian dilute suspensions for small strains. Due to their symmetry, the Platonic and Archimedean solids produce no elasticity. Some wire-frame shapes share this property if the ratios of the rod lengths are chosen correctly. The underlined shapes lead to Newtonian suspensions for all strain sizes. (b) Examples of particle shapes which, generally, lead to non-Newtonian dilute suspensions.

Figure 2

(a) One of the wire-frame particles considered, with $N=4$ in-plane legs and unequal lengths of the in and out of plane legs, $L_{\parallel }\ne L_{\perp }$. (b) The “shish kebab” procedure as applied to the shape from panel (a). The wire frame is replaced by spheres with diameter $b$ placed along each leg of the shape. The unit vectors ${\mathbf{e}}_l$ associated with some of the legs are also indicated.

Figure 3

(a) Required ratio of leg lengths, $x=L_{\perp }/L_{\parallel }$, for the elasticity to vanish plotted as a function of the number of in-plane legs, $3\le N\le 12$, for aspect ratios, $a=L_{\perp }/b=10$ (triangles) and $a\to \infty$ (circles). These ratios are solutions to (14) for $x$, which in the case $a\to \infty$ are simply $x={(N/4)}^{1/3}$. The (dark) blue and (light) orange lines indicate the solutions for $N=4$ and $N=12$, respectively. Panel (b) shows the elastic modulus, $G$, as given in (13), plotted as a function of $\gamma \approx L_{\perp }^3/L_{\parallel }^3$. The blue curve corresponds to $N=4$ and the orange $N=12$.