Hydrodynamic interaction between coaxially rising bubbles in elastoviscoplastic materials: Equal bubbles

We consider the buoyancy-driven rise and interaction between two coaxially placed bubbles of equal size and constant volume that are initially stationary inside an elastoviscoplastic material. We simulate the material using the Saramito extension of the Herschel-Bulkley constitutive model, and we fit its properties to a 0.1% aqueous Carbopol solution. The interplay between plasticity, viscoelasticity, and inertia is investigated. We observe that a “bridge” of shear stresses develops, which connects the leading and the trailing bubble, decreasing the drag force on the latter and initiating their approach. The solidlike behavior of the material preserves stresses generated by the passage of the leading bubble and makes the material “softer” for the trailing bubble. At the same time the normal stresses primarily extend the bubbles, but their finite distance eventually causes the leading bubble to adopt a hydrodynamically less favorable shape that slows it down, further promoting the approach. Moreover, we examine the effect of the geometric characteristics and the material properties. Increasing the initial distance between the bubbles delays their approach, which, however, is inevitable. Increasing the radius results in a transition from the elastoplastic to a mixed elastoplastic-inertial regime and a delayed approach. Increasing elasticity or the shear and extension thinning of the material decreases the approach time. On the contrary, an increased viscosity delays their approach. Finally, varying the yield stress induces a nonmonotonic effect. Sufficiently small values of yield stress delay the approach compared to intermediate values, because it reduces the elastic response. Raising the yield stress slightly above the entrapment conditions of a single bubble, the bubbles still interact and move.

Figure 1

Schematic representation of the two coaxial bubbles of equal size inside an elasto-visco-plastic material at the initial rest state. $\mathrm{\Omega }$ denotes the volume of the EVP material, $S_{f,TB}$ and $S_{f,LB}$ the interfaces between air and material for the trailing bubble and leading bubble, respectively, and $A_s$ and $S_{\mathrm{outer}}$ the symmetry axis and the outer boundary, respectively. The coordinate system is placed at the instantaneous middle of the distance between the bubble centers.

Figure 2

Steady shear stress vs shear rate: experimental data (hollow symbols) and model predictions with the fitted parameters (red solid line).

Figure 3

Predictions of the SHB model for the viscosity in (a) simple shear and (b) biaxial stretch.

Figure 4

Evolution of (a) velocity, (b) drag force, and (c) aspect ratio for SB, TB, and LB for the base case. Latin numerals indicate dimensionless time equal to (i) 1, (ii) 2, (iii) 6, (iv) 8.5, (v) 10.5, and (vi) 14.5, at which we will show bubble shapes, yield surfaces, and stress fields in Fig. 5.

Figure 5

Snapshots of the two bubbles in a sequence of time instances for the base case showing shear stresses to the left, normal axial stresses to the right, and the yield surface via a solid black line at times (i) 1, (ii) 2, (iii) 6, (iv) 8.5, (v) 10.5, and (vi) 14.5, indicated in Fig. 4.

Figure 6

At time instant (i) from Fig. 5, (a) contours of radial velocity to the left and axial velocity to the right and yield surface with black continuous lines, and (b) axial and radial velocities versus radial coordinate along the black dashed line to the side of the TB in (a).

Figure 7

Time evolution of normalized separation distance for all values of the initial separation distance with incubation times (a) 1.9, (b) 2.8, (c) 4, (d) 5.7, (e) 7.6, and (f) 9.5.

Figure 8

Bubble configuration at the onset of interaction showing shear stresses to the left, normal axial stresses to the right and yield surface with solid black line for (a) $d_o=3$ at $\mathrm{time}=1.9$, (b) $d_o=4$ at $\mathrm{time}=2.8$, (c) $d_o=5$ (base case) at $\mathrm{time}=4$, (d) $d_o=6$ at $\mathrm{time}=5.7$, (e) $d_o=7$ at $\mathrm{time}=7.6$, and (f) $d_o=8$ at $\mathrm{time}=9.5$.

Figure 9

Dimensional separation distance in mm vs dimensional time in seconds for all radii.

Figure 10

Time evolution of cases (j) $R_{\mathrm{eff}}=6.0\mathrm{mm}$ and (k) $R_{\mathrm{eff}}=8.0\mathrm{mm}$ in the mixed elastoplastic-inertial regime.

Figure 11

Bubble configuration showing shear stresses to the left, normal axial stresses to the right and yield surface with solid black line before the end of each simulation for (g) $R_{\mathrm{eff}}^{*}=3.5\mathrm{mm}$, (h) $R_{\mathrm{eff}}^{*}=4\mathrm{mm}$, (j) $R_{\mathrm{eff}}^{*}=6\mathrm{mm}$, (k) $R_{\mathrm{eff}}^{*}=8\mathrm{mm}$, (m) $R_{\mathrm{eff}}^{*}=12\mathrm{mm}$, and (n) $R_{\mathrm{eff}}^{*}=16\mathrm{mm}.$

Figure 12

Time evolution of normalized separation distance for the different values of the elastic modulus.

Figure 13

Time evolution of bubble velocities for (a) $G^{*}=24\mathrm{Pa}$, (b) $G^{*}=40.42$, (base case), (c) $G^{*}=60\mathrm{Pa}$, (d) $G^{*}=80\mathrm{Pa}.$

Figure 14

Bubble shapes, yield surfaces indicated with a solid black line, shear stresses to the left, normal axial stresses to the right at normalized bubble distance 0.6 for (a) $G^{*}=24\mathrm{Pa}$ at $\mathrm{time}=10.5$, (b) $G^{*}=40.42$ (base case) at $\mathrm{time}=12.5$, (c) $G^{*}=60\mathrm{Pa}$ at $\mathrm{time}=22$, and (d) $G^{*}=80\mathrm{Pa}$ at $\mathrm{time}=36.5$.

Figure 15

Time evolution of normalized separation distance for the different values of the consistency index.

Figure 16

Time evolution of bubble velocities for (e) $k^{*}=0.5\mathrm{Pa}{s}^{\mathrm{n}}$, (f) $k^{*}=1.75\mathrm{Pa}{s}^{\mathrm{n}}$ (base case), (g) $k^{*}=2.5\mathrm{Pa}{s}^{\mathrm{n}}$, and (h) $k^{*}=3.5\mathrm{Pa}{s}^{\mathrm{n}}$.

Figure 17

Bubble shapes, shear stresses to the left, normal axial stresses to the right, and yield surface indicated by a solid black line at normalized separation distance 0.5 for (e) $k^{*}=0.5\mathrm{Pa}{s}^{\mathrm{n}}$ at $\mathrm{time}=12$, (f) $k^{*}=1.75\mathrm{Pa}{s}^{\mathrm{n}}$ (base case) at $\mathrm{time}=14$, (g) $k^{*}=2.5\mathrm{Pa}{s}^{\mathrm{n}}$ at $\mathrm{time}=24$, and (h) $k^{*}=3.5\mathrm{Pa}{s}^{\mathrm{n}}$ at $\mathrm{time}=49$.

Figure 18

Time evolution of normalized separation distance for all values of flow index.

Figure 19

Time evolution of bubble velocities for (j) $n=0.2$, (k) $n=0.3$, (m) $n=0.4$, (n) $n=0.47$ (base case).

Figure 20

Bubble shapes, shear stresses to the left, normal axial stresses to the right, and yield surface indicated by a solid black line at normalized separation distance 0.6 for (j) $n=0.2$ at $\mathrm{time}=8$, (k) $n=0.3$ at $\mathrm{time}=8.5$, (m) $n=0.4$ at $\mathrm{time}=10.5$, and (n) $n=0.47$ at $\mathrm{time}=13$ (base case).

Figure 21

Time evolution of normalized separation distance for the different values of yield stress.

Figure 22

Time evolution of velocities for (p) ${\tau }_y^{*}=1\mathrm{Pa}$, (q) ${\tau }_y^{*}=3\mathrm{Pa}$, (s) ${\tau }_y^{*}=4.8\mathrm{Pa}$ (base case), and (t) ${\tau }_y^{*}=6.4\mathrm{Pa}$.

Figure 23

Bubble shapes, shear stresses to the left, normal axial stresses to the right and yield surfaces indicated with solid black line at normalized separation distance 0.6 for (p) ${\tau }_y^{*}=1\mathrm{Pa}$ at $\mathrm{time}=14.2$, (q) ${\tau }_y^{*}=3\mathrm{Pa}$ at $\mathrm{time}=12.04$, (s) ${\tau }_y^{*}=4.8\mathrm{Pa}$ at $\mathrm{time}=12.7$ (base case), and (t) ${\tau }_y^{*}=6.4\mathrm{Pa}$ at $\mathrm{time}=24.1$.

Figure 24

Close-ups of a typical initial mesh (a) at a distance to show the ellipse of structured mesh, (b) close-up, around one bubble, and (c) close-up near the rear pole of the trailing bubble.

Figure 25

Mesh convergence results of the base case at $\mathrm{time}=7.2$ (a) ${\tau }_{zz}$ along the axis of symmetry, the gaps correspond to the space occupied by the bubbles, and (b) yield surface $\left(|{\tau }_{\mathit{d}}|=\mathrm{Bn}\right)$.

Figure 26

Yield surface superimposed to the mesh for (a) the initial scale of Fig. 11 and (b) close-up of the region where the line becomes corrugated.

Figure 27

Mesh of the domain for case (n): (a) snapshot containing both bubbles, (b) zoomed area of the snapshot in the middle of the film, and (c) zoomed area of the snapshot at the corner of the TB.