Continuum dynamics of suspensions at low Reynolds number – CORRIGENDUM

We thank an annonymous reviewer for pointing out that (5.4) was not frame invariant. This error was due to an algebraic mistake and a mislabeling of the second and third indices of the Levi-Civita tensor in (3.3). Equation (3.3) should have been

(0.1) \begin{equation} 4{\widehat {F}}^1_{ij}-{\widehat {F}}^1_{ji} = -20\pi \eta _0a^3\left (\frac {\partial v_{-m,i}}{\partial x_j} - \epsilon _{ikj}\omega _{p,k}\right ), \end{equation}

and the correct version of (5.4) is

(0.2) \begin{equation} \epsilon _{ikj}\omega _{p,k} = \frac {1}{2}\left (\frac {\partial \langle v\rangle _{-m,i}}{\partial x_j} - \frac {\partial \langle v\rangle _{-m,j}}{\partial x_i}\right ), \end{equation}

which includes a factor of one-half and gives that the average local rotational rate of the particles is equal to the suspension-averaged rotational rate.

We also found that while (3.7) does lead to the correct divergence of the stress tensor, in the stress tensor itself ${\widehat {\mathbf{F}}}^{\,2}$ should be symmetric on exchange of the second and third indices:

(0.3) \begin{eqnarray} {\widehat {F}}^2_{ijk} &=& - \frac {\pi \eta _0a^5}{6}\left (5\frac {\partial ^2 v_{-m,i}}{\partial x_j\partial x_k} + \frac {\partial ^2 v_{-m,j}}{\partial x_i\partial x_k} + \frac {\partial ^2 v_{-m,k}}{\partial x_i\partial x_j} \right )\nonumber \\ && + \frac {\pi \eta _0a^5}{6}\big ({\nabla} ^2 v_{-m,i}\delta _{jk} + 3{\nabla} ^2 v_{-m,j}\delta _{ik} + 3{\nabla} ^2 v_{-m,k} \delta _{ij}\big ). \end{eqnarray}

Finally, there was a typographical error in the coefficients that we reported for the normal stress differences. The last sentence of § 7 should have read, `Fitting the high-Pe results to a cubic polynomial, we estimate that $\sigma _{xx}-\sigma _{zz} \approx -\eta _0{\dot {\gamma }} (0.066\phi ^2 + 0.35\phi ^3 )$ and $\sigma _{xx}-\sigma _{yy} \approx -\eta _0{\dot {\gamma }} (0.37\phi ^2 + 1.9\phi ^3 )$ at large Pe.’