Influence of the elastic deformation of a foam on its mobility in channels of linearly varying width

We study foam flow in an elementary model porous medium consisting of a convergent and a divergent channel positioned side by side and possessing a fixed joint porosity. Configurations of converging or diverging channels are ubiquitous at the pore scale in porous media, as all channels linking pores possess a converging and diverging part. The resulting flow kinematics imposes asymmetric bubble deformations in the two channels, which modulate foam-wall friction and strongly impact the flux distribution. We measure, as well as quantitatively predict, the ratio of the fluxes in the two channels as a function of the channel widths by modeling pressure drops of both viscous and capillary origins. This study reveals the crucial importance of boundary-induced bubble deformation on the mobility of a flowing foam, resulting in particular in flow irreversibility.

Figure 1

(a) Sketch of the flow configuration (not to scale). The total width is 6 cm. The obstacle dimensions are $L_{\mathrm{ch}}=10$ cm, $CE$ and $BF=2$ cm, and $\beta =11.3^{\circ }$. The lengths $r_{\mathrm{o},\mathrm{d}}$ and $r_{\mathrm{i},\mathrm{c}}$ used in the text are the distances $AB$ and $CD$, respectively. Also shown are binary images of foam flowing through the system for (b) $a_{\mathrm{c}}=1.0$ cm, (c) $a_{\mathrm{c}}=0.2$ cm, and (d) $a_{\mathrm{c}}=1.8$ cm. Flow is from left to right.

Figure 2

Maps of the velocity normalized by the average inlet velocity (top) and of the deformation quantified by the tensor $\tilde{\sigma }$ (bottom), in the symmetric case $a_{\mathrm{c}}=a_{\mathrm{d}}$.

Figure 3

Shown, from top to bottom, are the velocity component $v_x$, length $\mathcal{L}$, and pressure (set to zero at $x=-19$ mm) as functions of $x$ along the convergent [$\circ$; dash-dotted line in Fig. 1] and divergent [+; dotted line in Fig. 1] channels. The origin of the $x$ axis is taken at the entrance of the two channels. Vertical lines indicate the locations of the entrance and exit of the two channels.

Figure 4

Flux ratio $Q_{\mathrm{d}}/Q_{\mathrm{c}}$ plotted versus $a_{\mathrm{d}}/a_{\mathrm{c}}$. The experimental data for two different bubble areas are $0.11\pm 0.02 \mathrm{cm}{}^2 (\square )$ and $0.45\pm 0.10 \mathrm{cm}{}^2 (\circ )$. The dashed curve is from Eq. (5) with ${\mathcal{L}}_{\mathrm{c}}^{*}/{\mathcal{L}}_{\mathrm{d}}^{*}=0.80$. The solid curve is from Eq. (7) with ${\mathcal{L}}_{\mathrm{c}}^{*}/{\mathcal{L}}_{\mathrm{d}}^{*}=0.80$ and $B=8.7$. The asterisks denote results from 3D simulations of a Stokes flow (Newtonian fluid, vanishing Reynolds number). The inset shows a zoom at the entrance of the divergent channel when it is narrow and the bubbles are big, showing the structure as a bamboo foam.

Figure 5

Map of the normalized local fluxes $\parallel \mathbf{q}(x,y)\parallel /\langle q\rangle$ for the geometry such that $a_2=9a_1$, with the obstacle and a few streamlines drawn in blue.