Rheological properties of the soft-disk model of two-dimensional foams

The soft-disk model previously developed and applied by Durian [D. J. Durian, Phys. Rev. Lett. 75, 4780 (1995)] is brought to bear on problems of foam rheology of longstanding and current interest, using two-dimensional systems. The questions at issue include the origin of the Herschel-Bulkley relation, normal stress effects (dilatancy), and localization in the presence of wall drag. We show that even a model that incorporates only linear viscous effects at the local level gives rise to nonlinear (power-law) dependence of the limit stress on strain rate. With wall drag, shear localization is found. Its nonexponential form and the variation of localization length with boundary velocity are well described by a continuum model in the spirit of Janiaud et al. [Phys. Rev. Lett. 97, 038302 (2006)]. Other results satisfactorily link localization to model parameters, and hence tie together continuum and local descriptions.

Figure 1

Three types of 2D foams: (a) monolayer of air bubbles sitting at liquid/air interface (Bragg raft); (b) bubbles floating in liquid under a glass plate; and (c) bubbles confined between two glass plates. There are large effects due to the drag associated with motion relative to solid boundaries in both (b) and (c).

Figure 2

Overlap ${\Delta }_{ij}$ between two contacting bubbles of radii $R_i$ and $R_j$, located at ${\mathbf{r}}_{\mathbf{i}}$ and ${\mathbf{r}}_{\mathbf{j}}$, respectively.

Figure 3

Snapshot of 2D soft-disk foam. The sample is periodic in the horizontal direction; bubbles at the upper and lower boundary are shown in gray. In the simulation the upper boundary is moved with velocity $V$.

Figure 4

(a) Stress on the moving boundary as a function of the Deborah number $\text{De}=\dot{\gamma }{c}_b/\kappa$. The solid line is a fit to Eq. (1) (reexpressed in terms of De), resulting in the Herschel-Bulkley exponent $a=0.54\pm 0.01$. In (b) we have subtracted the fitted value of the yield stress ${\dot{\sigma }}_y$ from the data to show the power-law behavior in a double logarithmic plot.

Figure 5

Pressure acting on the moving boundary as a function of the Deborah number. The data is well described by Eq. (7) with an exponent $0.4\pm 0.01$. In (b) we have subtracted the fit parameter ${\Pi }_0$ to show the power law behavior in a double-logarithmic plot.

Figure 6

Normalized velocity profiles in the foam for a range of values of the parameter $\chi =c_b/c_w$. Shear close to the moving boundary is enhanced as wall friction is increased, i.e., with decreasing $\chi$. The solid lines, which agree closely with the data, were obtained numerically from the continuum model, using the Herschel-Bulkley index $a=0.54$.

Figure 7

Localization length ${\lambda }_{1/10}$ as a function of the dimensionless ratio $\chi =c_b/c_w$. The solid line is a fit of the data for $\chi <500$ to a power law, resulting in ${\lambda }_{1/10}/R_0\propto {\chi }^{0.64\pm 0.01}$. The power-law behavior is limited by the width of the sample, which is indicated by the dashed line (in units of the average bubble radius $R_0$).

Figure 8

Normalized localization length ${\lambda }_{1/10}/R_0$ as a function of the Deborah number. The solid line is a fit of the data to a power law, resulting in ${\lambda }_{1/10}/R_0=0.87{\text{De}}^{-0.30}$, in excellent agreement with theory. (Here the sample width was $28R_0$.)