Degassing cascades in a shear-thinning viscoelastic fluid

We report the experimental study of the degassing dynamics through a thin layer of shear-thinning viscoelastic fluid when a constant air flow is imposed at its bottom. The fluid is an aqueous solution of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal). Over a large range of parameters, the air is periodically released through a series of successive bubbles, hereafter named cascades. Each cascade is followed by a continuous degassing, lasting for several seconds, corresponding to an open channel crossing the fluid layer. The periodicity between two cascades does not depend on the injected flow rate. Inside one cascade, the properties of the overpressure signal associated with the successive bubbles vary continuously. The pressure threshold above which the fluid starts flowing, fluid deformation and pressure drop due to degassing through the thin fluid layer can be simply described by a Maxwell model.

Figure 1

Experimental setup. A constant air flow is supplied at the base of a thin layer of micellar fluid by an air-flow controller F (flow rate $Q$), via a chamber of volume $V$. $h$ is of the order of a typical bubble diameter (a few millimeters to centimeters).

Figure 2

Phase diagram of the different degassing regimes depending on the parameters (V,Q) for $c=0.1$ mol L${}^{-1}$ (a) and $c=0.5$ mol L${}^{-1}$ (b) ($•$, bubbles; $▴$, intermittence; $\square$, open channel). Dashed lines are a guide to the eyes. The gray zone corresponds to the space of parameters where bubble cascades are observed ($h=5$ mm).

Figure 3

Top: Pressure signal displaying the gas discharge via bubble cascades. $\delta P_{\text{max}}$ indicates the maximum overpressure reached in the cascades. Middle: (a) The first bubbles in the cascade exhibit a linear pressure increase. The fluid starts flowing only when the bubble is about to be emitted (pressure jump). (b) The last bubbles in the cascade exhibit a curved pressure increase, characteristic of fluid deformation (flowing) and bubble growth. (c) Between two bubble cascades, an open channel connects the air nozzle to the fluid free surface. The overpressure is constant, almost equal to zero (see text). Bottom: Detail of the pressure signal for a bubble formation and emission inside a cascade: (1) linear pressure increase; the dashed gray line corresponds to the linear pressure increase in the fixed volume $V$, $\delta P=(P_0/V)Qt$, without any adjustable parameters (see text); (2) bubble formation and growth; and (3) the bubble pierces the free surface and the gas is released. Note the pressure oscillations subsequent to the bubble emission ($c=0.1$ mol L${}^{-1}$, $V=147$ mL, $Q=0.65$ mL/s).

Figure 4

Time interval $\Delta t_{\text{casc}}$ separating a sequence “bubble cascade $+$ hole,” as a function of the air flow rate $Q$. Inset: Maximum overpressure $\delta P_{\text{max}}$ reached in the cascades (see Fig. 3). [Symbol, $V$ (mL)]: ($•$, 56); ($\circ$, 71); ($▵$, 83); ($\square$, 161). Dashed lines indicate the mean value of all the data.

Figure 5

(a) Number of bubbles per cascade $n$ as a function of the air flow rate $Q$. [Symbol, $V$ (mL)]: ($▴$, 48); ($•$, 56); ($\circ$, 71); ($\blacksquare$, 105); ($\square$, 161). The gray lines correspond to the linear interpolation for each series of experiments ($V$ fixed) ($h=5$ mm). (b) $dn/dQ$ as a function of $V$. The series reported in (a) display a linear, decreasing relationship up to $V=105$ mL. For larger fluid layer height, the slope is almost constant. [Symbol, $h$ (mm)]: ($•$, 5); ($\circ$, 8); ($▵$, 18); ($\square$, 27); ($▴$, 35). Inset: $dn/dQ$ as a function of the fluid concentration ($V=147$ mL, $Q=0.65$ mL/s, $h=5$ mm). No apparent relationship is found.

Figure 6

Maxwell model for the CTAB/NaSal. This simple model consists in an ideal elastic spring (elastic modulus $G^{\prime }$) and a dashpot representing the viscous loss (viscosity $\eta$). $r$ and $r_n$ indicate the displacements generated by the rising bubble and the dashpot shift after the $n$th bubble, respectively.

Figure 7

Evolution of $\delta P_y$ as a function of the bubble number in the cascade. In the main part of the cascade, $\delta P_y$ is a linear decaying function of the bubble number $n$ [Symbol, $c$ (mol L${}^{-1}$)]: ($\circ ,0.1$); ($▴,0.07$) ($V=147$ mL, $Q=0.65$ mL/s).

Figure 8

Mean pressure drop for a bubble vs time interval for bubble emission ($V=147$ mL, $h=5$ mm, $c=0.1$ mol L${}^{-1}$, $Q=0.65$ mL/s). Inset: Variations of $\delta t_c$ with the fluid characteristic time $\tau$, determined here as $\tau =G^{\prime }/G^{\prime \prime }\omega$ (see text). From left to right: $c=0.1,0.07$, and $0.05$ mol L${}^{-1}$.

Figure 9

The flow curve, shear stress $\sigma$ vs shear rate $\dot{\gamma }$ obtained by increasing $\dot{\gamma }$ ($c=0.1$ mol L${}^{-1}$). Inset: Viscosity as a function of shear rate. The arrow indicates the shear-thinning behavior (C-VOR 150 Bohlin rheometer, plate-plate geometry, diameter 60 mm, gap 400 $\mu$m, waiting time 60 s per point).

Figure 10

Elastic ($G^{\prime }$) and viscous ($G^{\prime \prime }$) moduli as a function of the applied stress for the CTAB/NaSal mixture at $c=0.1$ mol L${}^{-1}$ (oscillation test $\omega =1$ Hz, AR1000 rheometer, plate-plate geometry).

Figure 11

Average elastic ($G^{\prime }$) and viscous ($G^{\prime \prime }$) moduli corresponding to the plateau value in Fig. 10 as a function of the fluid concentration. Lines are given as a guide to the eyes.