The pulsatile motion of a semi-infinite bubble in a channel: flow fields, and transport of an inactive surface-associated contaminant

Abstract

We investigate a theoretical model of the pulsatile motion of a contaminant-doped semi-infinite bubble in a rectangular channel. We examine the fluid mechanical behaviour of the pulsatile bubble, and its influence on the transport of a surface-inactive contaminant (termed surfinactant). This investigation is used to develop a preliminary understanding of surfactant responses during unsteady pulmonary airway reopening. Reopening is modelled as the pulsatile motion of a semi-infinite gas bubble in a horizontal channel of width 2$a$ filled with a Newtonian liquid of viscosity $\mu$ and constant surface tension $\gamma$. A modified Langmuir sorption model is assumed, which allows for the creation and respreading of a surface multilayer. The bubble is forced via a time-dependent volume flux $Q(t)$ with mean and oscillatory components ($Q_{M}$ and $Q_{\omega }$, respectively) at frequency $\omega $. The flow behaviour is governed by the dimensionless parameters: Ca$_{M} \,{=}\,\mu Q_{M}/(2a\gamma $), a steady-state capillary number, which represents the ratio of viscous to surface tension forces; Ca$_{\Omega } \,{=}\,\mu Q_{\omega }/(2a\gamma $), an oscillatory forcing magnitude; $\Omega \,{=}\,\omega \mu a/\gamma $, a dimensionless frequency that represents the ratio of viscous relaxation to oscillatory-forcing timescales; and $A\,{=}\,2\hbox{\it Ca}_{\Omega }/\Omega $, a dimensionless oscillation amplitude. Our simulations indicate that contaminant deposition and retention in the bubble cap region occurs at moderate frequencies if retrograde bubble motion develops during the oscillation cycle. However, if oscillations are too rapid the ensuing large forward tip velocities cause a net loss of contaminant from the bubble tip. Determination of an optimal oscillation range may be important in reducing ventilator-induced lung injury associated with infant and adult respiratory distress syndromes by increasing surfactant transport to regions of collapsed airways.

The pulsatile propagation of a finger of air within a fluid-occluded cylindrical tube

“…This behavior clearly demonstrates that the time-dependent nature of the system modifies the interfacial geometry and flow field even though inertia is not included in this model. In addition, the separatrix was not observed in prior 2-D planar investigations of pulsatile flow (Zimmer et al 2005), highlighting the importance of geometry on the flow fields.…”

“…Velocity (u) and/or stress (τ) boundary conditions are applied on two of the four degrees of freedom. Nodes adjacent to two surfaces (Figure 2) have 4 DOF for each surface, as discussed in Zimmer et al (2005 The matrices T and U are of size 2N × 2N and 2N × 3N respectively, where N is the number of nodes. The vector w contains the velocities in the form w 2j-1 = u zj , w 2j = u rj and t contains stress data t 2j-1 = τ zj and t 2j = τ rj , where and j = 1,2,...,N. U is 2N × 3N so that two stresses may defined at the corner nodes.…”

“…In the BEM region we apply the following condition: [2.18] where M = (r m (s,t), z m (s,t)) is the interfacial position vector. Details of this technique are provided by (Zimmer et al 2005). Likewise, in the lubrication region we apply [2.19] …”

“…As such, panels a-f present instantaneous streamlines every 1/6 th cycle, and identify the time-dependent locations of converging (+) and diverging (-) stagnation points as well a separatrix (X) that temporarily exists during deceleration. These stagnation points are emphasized because of their importance in interfacial transport (Stebe & Barthes-Biesel 1995;Stebe & Maldarelli 1994;Zimmer et al 2005). Note that since this is an unsteady flow, the interface is not a streamline and therefore streamlines may intersect the free surface.…”

Equation of state for a coarse-grained DPPC monolayer at the air/water interface

“…This work is part of a multi-scale modeling approach to the study of the dynamics of pulmonary surfactant during the respiration cycle. As a component of this approach, the surface tension-surface concentration relationship, thus obtained on the mesoscopic scale, will subsequently be integrated with macroscopic simulations of the system [4][5][6].…”