Effect of surfactant on elongated bubbles in capillary tubes at high Reynolds number

The effect of surfactants on the tail and film dynamics of elongated gas bubbles propagating through circular capillary tubes is investigated by means of an extensive three-dimensional numerical study using a hybrid front-tracking/level-set method. The focus is on the visco-inertial regime, which occurs when the Reynolds number of the flow is much larger than unity. Under these conditions, “clean” bubbles exhibit interface undulations in the proximity of the tail, with an amplitude that increases with the Reynolds number. We perform a systematic analysis of the impact of a wide range of surfactant properties, including elasticity, bulk surfactant concentration, solubility, and diffusivity, on the bubble and flow dynamics in the presence of inertial effects. The results show that the introduction of surfactants is effective in suppressing the tail undulations as they tend to accumulate near the bubble tail. Here large Marangoni stresses are generated, which lead to a local “rigidification” of the bubble. This effect becomes more pronounced for larger surfactant elasticities and adsorption depths. At reduced surfactant solubility, a thicker rigid film region forms at the bubble rear, where a Couette film flow is established, while undulations still appear at the trailing edge of the downstream “clean” film region. In such conditions, the bubble length becomes an influential parameter, with short bubbles becoming completely rigid.

Figure 1

(a) Initial three-dimensional bubble shape and parabolic velocity profile imposed at the tube inlet. (b) Schematic representation of the problem in a vertical centerline $(x,y)$ plane, where $D$ denotes the tube diameter, $U_b$ is the bubble-tip velocity, $h$ is the vertical distance of the liquid-gas interface from the $y=0$ axis, and $h_0$ is the uniform liquid film thickness.

Figure 2

(a) Mesh dependence study for a “clean” bubble with reference parameters $\text{Re}=443$ and $\text{Ca}=0.0089$; (b) comparison of the experimental correlation of Han and Shikazono [4] with the simulation results from this study, for a varying capillary number and fixed $\text{Re}=443$.

Figure 3

Effect of surfactant on the flow characteristics; (a) three-dimensional representation of the bubble shape for the surfactant-free (top) and surfactant-laden (bottom) cases, with the color indicating the magnitude of surfactant interfacial concentration, $\tilde{\mathrm{\Gamma }}$; (b) variation of $\tilde{\mathrm{\Gamma }}$ along $\tilde{x}$; (c) two-dimensional projection (in the $z=0$ plane) of the bubble shape for the surfactant-free (solid line), and surfactant-laden cases in the presence (dashed) and absence (dotted) of Marangoni stresses. Also shown in (c), as a red dashed line, is the $\tilde{x}$ variation of the Marangoni stresses, $\tilde{\tau }$. The capillary numbers for the surfactant-free and surfactant-laden cases are 0.0089 and 0.0094, respectively, while the rest of the parameters are $\text{Re}=443$, ${\text{Pe}}_c={\text{Pe}}_s=100$, $\text{Da}=0.1$, $k=1$, $\text{Bi}=1$, ${\beta }_s=0.5$, and $\text{Ma}=0.13$.

Figure 4

Effect of surfactant on the magnitude of the dimensionless strain rate $\tilde{S}$, (a), and vorticity $\tilde{\mathrm{\Omega }}$, (b), for the surfactant-free (top) and Marangoni-supported surfactant-laden (bottom) cases for the same parameters as in Fig. 3; here $S$ and $\mathrm{\Omega }$ are scaled on $D/U$. The streamlines are drawn in the bubble-tip reference frame. The vortical structures identified in zones “A,” “C,” and “E” and B,” “D,” and “F” rotate in a clockwise and counterclockwise direction, respectively.

Figure 5

Effect of varying $\text{Ma}$ on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape (a), the interfacial surfactant concentration and resulting surface tension (b), and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; here $\text{Ca}=0.0089$ and the rest of the parameters remain unchanged from Fig. 3.

Figure 6

Effect of varying $\text{Ca}$ on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape (a), the interfacial surfactant concentration (b), and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; the rest of the parameters remain unchanged from Fig. 3.

Figure 7

Effect of varying $\text{Re}$ on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape (a), the interfacial surfactant concentration (b), and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; here $\text{Ca}=0.0089$ and the rest of the parameters remaining unchanged from Fig. 3.

Figure 8

Effect of varying $\text{Da}$ and $k$ on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape (a), the interfacial surfactant concentration (b), and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; here $\text{Ca}=0.0089$, and the rest of the parameters remaining unchanged from Fig. 3.

Figure 9

Effect of varying $\text{Bi}$ on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape (a), the interfacial surfactant concentration (b), and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; here $\text{Ca}=0.0089$, and the rest of the parameters remaining unchanged from Fig. 3.

Figure 10

Flow profiles associated with the $\text{Bi}=0.01$ case with the rest of the parameters remaining unchanged from Fig. 9; (a) three-dimensional representation of the bubble shape colored by the magnitude of the interfacial surfactant distribution, $\tilde{\mathrm{\Gamma }}$; (b) profiles of the dimensionless streamwise velocity component, ${\tilde{u}}_x=(u_x-U_b)/U_b$, calculated in a bubble-tip frame of reference, along the cross-stream direction, $y$, for the two axial locations indicated in (a), which are in Regions 1 and 2, as described in the text; vortical structure evolution at $\tilde{t}=3$ and at steady state, shown in (c) and (d), with the magnitude of the dimensionless vorticity, $\tilde{\mathrm{\Omega }}$, and strain rate, $\tilde{S}$, depicted in the top half and bottom half of each panel, respectively. The vortices labeled “A” and “B” in panel (c) rotate in the clockwise and counterclockwise directions, respectively. In panel (d) the vortices labeled “C” and “E” and vortex “D” rotate in the counterclockwise and clockwise directions, respectively; identical structures in the top half of the panel rotate in the opposite directions. All streamlines are presented in a frame of reference moving with the bubble tip.

Figure 11

Effect of varying ${\text{Pe}}_c$ (set equal to ${\text{Pe}}_s$) on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape (a), the interfacial surfactant concentration, (b) and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; here $\text{Ca}=0.0089$, $\text{Bi}=0.01$, and the rest of the parameters remain unchanged from Fig. 3.

Figure 12

Effect of varying the initial dimensionless bubble length, ${\tilde{L}}_b$ on the steady spatial distribution of the Marangoni stresses and two-dimensional projection (in the $z=0$ plane) of the bubble shape for ${\tilde{L}}_b=2-5$ (a), the interfacial surfactant concentration (b), and the streamwise component of the interfacial velocity in the frame of reference of the bubble tip (c), where ${\tilde{u}}_t=(u_t-U_b)/U_b$; here $\text{Bi}=0.01$, $\text{Ca}=0.0089$, and the rest of the parameters remaining unchanged from Fig. 3.