179 research outputs found
The effect of gravity on liquid plug propagation in a two-dimensional channel
The effect of plug propagation speed and gravity on the quasisteady motion of a liquid plug in a two-dimensional liquid-lined channel oriented at an angle αα with respect to gravity is studied. The problem is motivated by the transport of liquid plugs instilled into pulmonary airways in medical treatments such as surfactant replacement therapy, drug delivery, and liquid ventilation. The capillary number Ca is assumed to be small, while the Bond number Bo is arbitrary. Using matched asymptotic expansions and lubrication theory, expressions are obtained for the thickness of the trailing films left behind by the plug and the pressure drop across it as functions of Ca, Bo, αα and the thickness of the precursor films. When the Bond number is small it is found that the trailing film thickness and the flow contribution to the pressure drop scale as Ca2/3Ca2∕3 at leading order with coefficients that depend on Bo and αα. The first correction to the film thickness is found to occur at O(Ca)O(Ca) compared to O(Ca4/3)O(Ca4∕3) in the Bo = 0Bo=0 case. Asymmetry in the liquid distribution is quantified by calculating the ratio of liquid volumes above and below the centerline of the channel, VṘ. VR = 1VR=1 at Bo = 0Bo=0, indicating a symmetric distribution, and decreases with Bo and Ca, but increases with the plug length LpLp. The decrease of VRVR with Ca suggests that higher propagation speeds in small airways may result in less homogenous liquid distribution, which is in contrast to the expected effect in large airways. For given values of the other parameters, a maximum capillary number CacCac is identified above which the plug will eventually rupture. When the Bond number becomes equal to an orientation-dependent critical value BocBoc, it is found that the scaling of the film thickness and pressure drop change to Ca1/2Ca1∕2 and Ca1/6Ca1∕6, respectively. It is shown that this scaling is valid for small increments of the Bond number over its critical value, Bo = Boc+BCa1/6Bo=Boc+BCa1∕6, but for higher Bond numbers the asymptotic approach breaks down.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87914/2/031507_1.pd
The propagation of a surfactant laden liquid plug in a capillary tube
This paper considers the propagation of a liquid plug, forced by a driving pressure ΔP, within a rigid tube. The tube is already lined with a liquid precursor film of thickness 2.h̄2. Both the plug and the precursor film, as well as the interface, contain small amounts of surfactant whose concentrations are assumed to be near equilibrium. Since the motions are slow, we seek asymptotic solutions for small capillary number, Ca≪1,Ca≪1, and also assume that sorption kinetics control the surfactant flux to the interface compared to bulk diffusion. An additional asymptotic assumption is that the Stanton number, St, is sufficiently large such that β∝Ca1/3/St≪1,β∝Ca1/3/St≪1, which relates the importance of sorption kinetics to convection. The surfactant strength is measured by the surface elasticity, E = M/βE=M/β where M is the Marangoni number. The results of the analysis are that, for a given plug Ca, ΔP increases with increasing E but decreases with increasing 2.h̄2. The trailing film thickness, 1,h̄1, increases with ΔP, but at a slower rate when E is larger. For 1<2,h̄1<h̄2, criteria for plug rupture are established. This model is relevant to delivery of surfactants into the lung by direct instillation into the bronchial network as is done in surfactant replacement therapy and the use of surfactant solutions to carry other substances (e.g., genetic material) into the airways. © 2002 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69965/2/PHFLE6-14-2-471-1.pd
The cascade structure of linear instability in collapsible channel flows
This paper studies the unsteady behaviour and linear stability of the flow in a collapsible channel using a fluid–beam model. The solid mechanics is analysed in a plane strain configuration, in which the principal stretch is defined with a zero initial strain. Two approaches are employed: unsteady numerical simulations solving the nonlinear fully coupled fluid–structure interaction problem; and the corresponding linearized eigenvalue approach solving the Orr–Sommerfeld equations modified by the beam. The two approaches give good agreement with each other in predicting the frequencies and growth rates of the perturbation modes, close to the neutral curves. For a given Reynolds number in the range of 200–600, a cascade of instabilities is discovered as the wall stiffness (or effective tension) is reduced. Under small perturbation to steady solutions for the same Reynolds number, the system loses stability by passing through a succession of unstable zones, with mode number increasing as the wall stiffness is decreased. It is found that this cascade structure can, in principle, be extended to many modes, depending on the parameters. A puzzling ‘tongue’ shaped stable zone in the wall stiffness–Re space turns out to be the zone sandwiched by the mode-2 and mode-3 instabilities. Self-excited oscillations dominated by modes 2–4 are found near their corresponding neutral curves. These modes can also interact and form period-doubling oscillations. Extensive comparisons of the results with existing analytical models are made, and a physical explanation for the cascade structure is proposed
Liquid plug formation in an airway closure model
The closure of a human lung airway is modeled as an instability of a two-phase flow in a pipe coated internally with a Newtonian liquid. For a thick enough coating, the Plateau-Rayleigh instability creates a liquid plug which blocks the airway, halting distal gas exchange. Owing to a bifrontal plug growth, this airway closure flow induces high stress levels on the wall, which is the location of airway epithelial cells. A parametric numerical study is carried out simulating relevant conditions for human lungs, in either ordinary or pathological situations. Our simulations can represent the physical process from pre- to postcoalescence phases. Previous studies have been limited to precoalescence only. The topological change during coalescence induces a high level of stress and stress gradients on the epithelial cells, which are large enough to damage them, causing sublethal or lethal responses. We find that postcoalescence wall stresses can be in the range of 300% to 600% greater than precoalescence values and so introduce an important source of mechanical perturbation to the cells
Pulmonary Fluid Flow Challenges for Experimental and Mathematical Modeling
Modeling the flow of fluid in the lungs, even under baseline healthy conditions, presents many challenges. The complex rheology of the fluids, interaction between fluids and structures, and complicated multi-scale geometry all add to the complexity of the problem. We provide a brief overview of approaches used to model three aspects of pulmonary fluid and flow: the surfactant layer in the deep branches of the lung, the mucus layer in the upper airway branches, and closure/reopening of the airway. We discuss models of each aspect, the potential to capture biological and therapeutic information, and open questions worthy of further investigation. We hope to promote multi-disciplinary collaboration by providing insights into mathematical descriptions of fluid-mechanics in the lung and the kinds of predictions these models can make
Effects of elastoviscoplastic properties of mucus on airway closure in healthy and pathological conditions
Airway mucus is a complex material with both viscoelastic and viscoplastic properties that vary with healthy and pathological conditions of the lung. In this study, the effects of these conditions on airway closure are examined in a model problem, where an elastoviscoplastic (EVP) single liquid layer lines the inner wall of a rigid pipe and surrounds the air core. The EVP liquid layer is modelled using the Saramito-HB model. The parameters for the model are obtained for the mucus in healthy, asthma, chronic obstructive pulmonary disease (COPD), and cystic fibrosis (CF) conditions by fitting the rheological model to the experimental data. Then the liquid plug formation is studied by varying the Laplace number and undisturbed liquid film thickness. Airway closure is a surface-tension-driven phenomenon that occurs when the ratio of the pulmonary liquid layer thickness to the airway radius exceeds a certain threshold. In previous studies, it has been found that airway epithelial cells can be lethally or sublethally damaged due to the high peak of the wall stresses and stress gradients during the liquid plug formation. Here we demonstrate that these stresses are also related to the EVP features of the liquid layer. Yielded zones of the liquid layer are investigated for the different mucus conditions, and it is found that the liquid layer is in a chiefly unyielded state before the closure, which indicates that this phase is dominated by the elastic behavior and solvent viscosity. This is further confirmed by showing that the elastic coefficient is one of the most critical parameters determining whether the closure occurs. This parameter also largely affects the closure time. The wall stresses are also investigated for the pathological and healthy cases. Their peaks for COPD and CF are found to be the highest due to the viscoelastic extra stress contribution. Contrary to the Newtonian case, the wall stresses for COPD and CF do not smoothly relax after closure, as they rather remain effectively almost as high as the Newtonian peak. Moreover, the local normal wall stress gradients are smaller for the COPD and CF liquid layer due to their higher stiffness causing a smaller curvature at the capillary wave. The local tangential wall stress gradients are also shown to be smaller for these cases because of the slower accumulation of the liquid at the bulge
On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within
In this chapter, we analyze the steady-state microscale fluid--structure
interaction (FSI) between a generalized Newtonian fluid and a hyperelastic
tube. Physiological flows, especially in hemodynamics, serve as primary
examples of such FSI phenomena. The small scale of the physical system renders
the flow field, under the power-law rheological model, amenable to a
closed-form solution using the lubrication approximation. On the other hand,
negligible shear stresses on the walls of a long vessel allow the structure to
be treated as a pressure vessel. The constitutive equation for the microtube is
prescribed via the strain energy functional for an incompressible, isotropic
Mooney--Rivlin material. We employ both the thin- and thick-walled formulations
of the pressure vessel theory, and derive the static relation between the
pressure load and the deformation of the structure. We harness the latter to
determine the flow rate--pressure drop relationship for non-Newtonian flow in
thin- and thick-walled soft hyperelastic microtubes. Through illustrative
examples, we discuss how a hyperelastic tube supports the same pressure load as
a linearly elastic tube with smaller deformation, thus requiring a higher
pressure drop across itself to maintain a fixed flow rate.Comment: 19 pages, 3 figures, Springer book class; v2: minor revisions, final
form of invited contribution to the Springer volume entitled "Dynamical
Processes in Generalized Continua and Structures" (in honour of Academician
D.I. Indeitsev), eds. H. Altenbach, A. Belyaev, V. A. Eremeyev, A. Krivtsov
and A. V. Porubo
A viscoelastic deadly fluid in carnivorous pitcher plants
Background : The carnivorous plants of the genus Nepenthes, widely
distributed in the Asian tropics, rely mostly on nutrients derived from
arthropods trapped in their pitcher-shaped leaves and digested by their
enzymatic fluid. The genus exhibits a great diversity of prey and pitcher forms
and its mechanism of trapping has long intrigued scientists. The slippery inner
surfaces of the pitchers, which can be waxy or highly wettable, have so far
been considered as the key trapping devices. However, the occurrence of species
lacking such epidermal specializations but still effective at trapping insects
suggests the possible implication of other mechanisms. Methodology/Principal
Findings : Using a combination of insect bioassays, high-speed video and
rheological measurements, we show that the digestive fluid of Nepenthes
rafflesiana is highly viscoelastic and that this physical property is crucial
for the retention of insects in its traps. Trapping efficiency is shown to
remain strong even when the fluid is highly diluted by water, as long as the
elastic relaxation time of the fluid is higher than the typical time scale of
insect movements. Conclusions/Significance : This finding challenges the common
classification of Nepenthes pitchers as simple passive traps and is of great
adaptive significance for these tropical plants, which are often submitted to
high rainfalls and variations in fluid concentration. The viscoelastic trap
constitutes a cryptic but potentially widespread adaptation of Nepenthes
species and could be a homologous trait shared through common ancestry with the
sundew (Drosera) flypaper plants. Such large production of a highly
viscoelastic biopolymer fluid in permanent pools is nevertheless unique in the
plant kingdom and suggests novel applications for pest control
The Effect of Wall Inertia on High-Frequency Instabilities of Flow Through an Elastic-Walled Tube
We examine the effect of wall inertia on the onset of high-frequency self-excited oscillations in flow through an elastic-walled tube. The previous asymptotic model of Whittaker et al. (Proc. Roy. Soc. A466, 2010), for a long-wavelength high-frequency instability in a Starling-resistor set-up, neglected inertia in the tube wall. Here, we extend this model by modifying the ‘tube-law’ for the wall mechanics to include inertial effects. The resulting coupled model for the fluid and solid mechanics is solved to find the normal modes of oscillation for the system, together with their frequencies and growth rates. In the system and parameter regime considered, the addition of wall inertia reduces the oscillation frequency of each mode, however its effect on the stability of the system is not as straightforward. Increasing wall inertia lowers the mean flow rate required for the onset of instability, and is therefore destabilising. However, at higher flow rates the instability growth rate is decreased, and so wall inertia is stabilising here. Overall, the addition of wall inertia decreases the sensitivity of the system to the mean axial flow rate. The theoretical results show good qualitative and reasonable quantitative agreement with direct numerical simulations performed using the oomph-lib framework
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