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

Abstract

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 non-inertial steady motion of a finite size gas bubble in both a plane 2D channel and an axi-symmetric tube filled by a Bingham fluid has been studied numerically. The Bingham number, Bn, is in the range 0 ≤ Bn ≤ 3, where 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 Ca=0.05, 0.10, and 0.25. The volume of all axi-symmetric 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 width/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 (axi-symmetric)/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 non-monotonic depending on Ca values and 2D/axi-symmetric 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 the flow pattern in the bubble frame of reference varies significantly by Ca and Bn.

Surface-tension-driven evolution of a viscoplastic liquid coating the interior of a cylindrical tube

“…Turning to viscoplastic flows with applications in airway modelling, Craster & Matar (2000) modelled surfactant-driven flow on a single-layer or two-layer film of viscoplastic or Newtonian fluids, showing that, at least in the single-layer case, yield stress decreases spreading rates and can cause the layer to become frozen in a non-trivial static shape. Modelling of propagation of viscoplastic liquid plugs in tubes and channels has shown that increasing the yield stress increases the stress applied to the wall and increases the thickness of the layer of liquid left behind as a plug propagates (Zamankhan et al 2012;Zamankhan, Takayama & Grotberg 2018). Rupture of viscoplastic liquid plugs has also been modelled both experimentally (Hu et al 2015) and numerically (Hu, Romanò & Grotberg 2020), showing that increased yield stress can inhibit plug rupture because a larger pressure drop is required across the plug to make it yield.…”

“…Modelling of propagation of viscoplastic liquid plugs in tubes and channels has shown that increasing the yield stress increases the stress applied to the wall and increases the thickness of the layer of liquid left behind as a plug propagates (Zamankhan et al. 2012; Zamankhan, Takayama & Grotberg 2018). Rupture of viscoplastic liquid plugs has also been modelled both experimentally (Hu et al.…”

Effects of gravity and surface tension on steady microbubble propagation in asymmetric bifurcating airways

“…For normal breathing conditions of tidal volume equal to 500 ml and 12 breaths/min, typical small reopening velocities are studied in the literature range from 1 mm/s to 10 mm/s. 13 Assuming the fluid viscosity and air–liquid surface tension as 47 cP and 72 dyn/cm, respectively, the corresponding Re, Ca, and Bo are then given by and However, for lung fluids, having surface tension much lower than 72 dyn/cm 12 would lead to larger Ca and Bo values. 15 To investigate all these possible Ca and Bo values that may exist in the human lungs, inertial effects were neglected, and about baseline conditions (i.e., Ca = 0.06 and Bo = 0.003), parameter independent variation studies were then performed for a range of Ca and Bo values.…”

Bursting bubble in a viscoplastic medium

“…2015; Lopez, Naccache & de Souza Mendes 2018) and bubbles moving inside tubes filled with viscoplastic fluids (Jalaal & Balmforth 2016; Laborie et al. 2017; Zamankhan, Takayama & Grotberg 2018). We will show that the introduction of non-Newtonian properties can significantly influence the bursting behaviour of bubbles on a free surface.…”