Steady displacement of long gas bubbles in channels and tubes filled by a Bingham fluid

Bingham fluids behave like solids below a von Mises stress threshold, the yield stress, while above it they behave like Newtonian fluids. They are characterized by a dimensionless parameter, Bingham number (Bn), which is the ratio of the yield stress to a characteristic viscous stress. In this study, the noninertial steady motion of a finite-size gas bubble in both a plane two-dimensional (2D) channel and an axisymmetric tube filled by a Bingham fluid has been studied numerically. The Bingham number, Bn, is in the range $0\le \mathrm{Bn}\le 3$, where $\mathrm{Bn}=0$ is the Newtonian case, while the capillary number, which is the ratio of a characteristic viscous force to the surface tension, has values $\mathrm{Ca}=0.05,0.10$, and 0.25. The volume of all axisymmetric and 2D bubbles has been chosen to be identical for all parameter choices and large enough for the bubbles to be long compared to the channel, tube height, and diameter. The Bingham fluid constitutive equation is approximated by a regularized equation. During the motion, the bubble interface is separated from the wall by a static liquid film. The film thickness scaled by the tube radius (axisymmetric) and half of the channel height (2D) is the dimensionless film thickness, $h$. The results show that increasing Bn initially leads to an increase in $h$; however, the profile $h$ versus Bn can be monotonic or nonmonotonic depending on Ca values and 2D or axisymmetric configurations. The yield stress also alters the shape of the front and rear of the bubble and suppresses the capillary waves at the rear of the bubble. The yield stress increases the magnitude of the wall shear stress and its gradient and therefore increases the potential for epithelial cell injuries in applications to lung airway mucus plugs. The topology of the yield surfaces as well as the flow pattern in the bubble frame of reference varies significantly by Ca and Bn.

Figure 1

Schematics of (a) a bubble, (b) a liquid plug, and (c) an advancing (left) and a receding (right) gas finger.

Figure 2

A schematic from a deformed airway conduit.

Figure 3

A schematic of the domain of calculation for steady motion of a bubble in a 2D channel and an axisymmetric tube.

Figure 4

The shape of a bubble moving in a Newtonian fluid for different Ca values. (a) Axisymmetric. (b) Axisymmetric versus 2D.

Figure 5

(a)–(c) The shape of an axisymmetric bubble for different Ca and Bn. (d) Comparison between 2D and axisymmetric.

Figure 6

(a)–(c) Close-up from the rear of axisymmetric bubbles for different Bn and Ca.

Figure 7

$G$ in terms of Bn for different Ca values.

Figure 8

(a) $h$ in terms of Bn for different Ca values. (b) Area average of the fluid axial velocity upstream (downstream) of (from) the bubble in terms of Bn for different Ca values. The velocity values are given in the wall frame of reference.

Figure 9

Variation of $D_{12}$ along the wall with Bn for different Ca values. (a) $\mathrm{Ca}=0.05$. (b) $\mathrm{Ca}=0.10$. (c) $\mathrm{Ca}=0.25$. (d) Comparison between axisymmetric and 2D.

Figure 10

Variation of ${\tau }_{12}$ along the wall with Bn for different Ca values. (a) $\mathrm{Ca}=0.05$. (b) $\mathrm{Ca}=0.10$. (c) $\mathrm{Ca}=0.25$. (d) Comparison between axisymmetric and 2D for $\mathrm{Ca}=0.1$ and $\mathrm{Bn}=2.0$.

Figure 11

Distribution of ${\tau }_{12}$ around a region of the wall where $|d{\tau }_{12}/\phantom{d{\tau }_{12}dz}dz|$ is maximum. (a) $\mathrm{Ca}=0.05$. (b) $\mathrm{Ca}=0.1$. (c) Comparison between axisymmetric and 2D configurations.

Figure 12

Variation of pressure on the wall in a 2D channel (a) with Ca and (b) with Bn. The broken lines show the wall pressure for the Poiseuille and Bingham-Poiseuille equations with the same volumetric flow rate delivery.

Figure 13

von Mises stress for $\mathrm{Ca}=0.1$ and varying Bn. The unyielded regions are in white. The insets in panels (b) and (f) magnify the area around the distinct unyielded region beneath the back of the bubble.

Figure 14

Axial velocity in the unyielded strips attached to the tube centerline in terms of Bn for different Ca. The values are given for the bubble frame of reference.

Figure 15

Contours of von Mises stress for $\mathrm{Ca}=0.0725$ and $\mathrm{Bn}=3.0$. The unyielded regions, white color, in the front, back, and the middle are disconnected.

Figure 16

Fluid path lines and contours of $V_z$ for the axisymmetric bubble with $\mathrm{Ca}=0.1$ and varying Bn. Top: Fluid path lines. Bottom: Contours of $V_z$.