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# Temporal evolution of the mean mucus velocity V (t) for the four different computations. The value in the caption indicates the temporal average over the simulation.

Source publication

This study provides the analysis of the generalized 3D Stokes problem in a time dependent domain, modeling a solid in motion. The fluid viscosity is a non-linear function of the shear-rate and depends on a transported and diffused quantity. This is a natural model of flow at very low Reynolds numbers, typically at the microscale, involving a miscib...

## Contexts in source publication

**Context 1**

... mucociliary clearance efficiency is quantified by the mean velocity of the mucus layer (the upper part of the ASL). This mean velocity V (t) in the proximal direction (x) is computed at each time step, it is then averaged over six beating cycles (U ): On Figure 8 the evolution of the quantity V (t) is displayed with respect to t for each previous simulation. The associated quantity U is displayed in the legend. ...

**Context 2**

... both simulations with transport (respectively without transport), one can observe that the behavior of V (t) is very similar since the curves are almost overlapping. Differences can be observed at the middle of the cilia strokes (effective and recovery) when extreme values of V (t) are reached (see zoomed parts of the Fig. ...

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Cystic fibrosis (CF) airway disease is characterized by excessive and accumulative mucus in the airways. Mucociliary clearance becomes defective as mucus secretions become hyperconcentrated and viscosity increases. The CFTR-knockout (KO) rat has been previously shown to progressively develop delayed mucociliary transport, secondary to increased vis...

## Citations

... While the latter allow a gain in accuracy, the use of collocated grids easily aligns the computational points to the experimental datasets. Optimized gridbased methods may be coupled to other methods dedicated to the flow features to be investigated: transport based on particle methods (Bandara et al. (2013), Chatelin and Poncet (2013)), anisotropic diffusion for space-variable medium (Etancelin et al. (2020), Sanchez et al. (2019)), phase-field description of multi-phase flows (Soulaine et al. (2018), Lasseux et al. (1996)) and its upscaling (Lasseux et al. (2016), Raeini et al. (2014)), complex fluid and rheology (Zami-Pierre et al. (2017), Sanchez et al. (2019), etc...). ...

This article describes how much the computed absolute permeability is impacted by the slip effect at the fluid/solid interface, in the context of single-phase pore-scale flow. While this effect is well quantified in microchannels or simple geometries, the present study focuses on its average effect in real rock matrix geometries, obtained by means of high-resolution X-ray microtomography. Due to the inherently finite resolution of the technique, an uncertainty exists on the true position of the fluid/solid interface and its morphological features below the image resolution (unseen roughness). We demonstrate that both these uncertainties can be interpreted as a slip condition, and consequently we focus on how a slip length can impact the computed absolute permeability, after having provided an estimation of a meaningful bound on the slip coefficient. To that extent, two strategies are employed: the global deviation of permeability and the theoretically established linear deviation. Three high-definition 3D geometries are used as practical examples of our methodology. Results are discussed in terms of relative deviation versus specific surface area and lead to quantities of interest involving the linear deviation of permeability.
Article Highlights
Uncertainty on the permeability is quantified by means of a slip length based on micro-CT voxel size.
The linear deviation from a permeability value is established asymptotically and involves only a prescribed slip.
A ratio involving the relative linear deviation, the specific area and the permeability is a non-dimensional quantity of interest.

... The micro-continuum approach has been used to simulate the dissolution of a calcite crystal at the pore-scale and compared successfully with microfluidic experiments . In Molins et al. (2020), the approach is compared with state-of-the-art RTM at the pore-scale using various numerical techniques including Chombo-Cruch with Level-Set (Molins et al., 2017), Lattice Boltzmann Method (Prasianakis et al., 2018), dissolFoam moving grids with conformal mapping (Starchenko et al., 2016), and Vortex methods (Sanchez et al., 2019). The benchmark consists of a 0.2 mm diameter calcite crystal posted in a 1 mm long 0.5 width channel (see Fig. 3). ...

porousMedia4Foam is a package for solving flow and transport in porous media using OpenFOAM® - a popular open-source numerical toolbox. We introduce and highlight the features of a new generation open-source hydro-geochemical module implemented within porousMedia4Foam, which relies on micro-continuum concept and which makes it possible to investigate hydro-geochemical processes occurring at multiple scales i.e. at the pore-scale, reservoir-scale and at the hybrid-scale. Geochemistry is handled by a third party package (e.g. PHREEQC) that is coupled to the flow and transport solver of OpenFOAM®. We conducted benchmarks across different scales to validate the accuracy of our simulator. We further looked at the evolution of mineral dissolution/precipitation in a fractured porous system. Application fields of this new package include the investigation of hydro-bio-geochemical processes in the critical zone, the modelling of contaminant transport in aquifers, as well as and the assessment of confinement performance for geological barriers.

... The splitting between Lagrangian transport and diffusion-stretching (i.e., ∇ · (μ∇ω) + ω · ∇V) was performed in [28] in three dimensions, and the Kozeny-Carman term has been implemented in its vorticity formulation in [32]. The diffusion equation was solved by an improved Particle-Strength-Exchange method [77], especially well suited to variable diffusion, including diffusing depending on shear-rate [94]. ...

This manuscript presents a benchmark problem for the simulation of single-phase flow, reactive transport, and solid geometry evolution at the pore scale. The problem is organized in three parts that focus on specific aspects: flow and reactive transport (part I), dissolution-driven geometry evolution in two dimensions (part II), and an experimental validation of three-dimensional dissolution-driven geometry evolution (part III). Five codes are used to obtain the solution to this benchmark problem, including Chombo-Crunch, OpenFOAM-DBS, a lattice Boltzman code, Vortex, and dissolFoam. These codes cover a good portion of the wide range of approaches typically employed for solving pore-scale problems in the literature, including discretization methods, characterization of the fluid-solid interfaces, and methods to move these interfaces as a result of fluid-solid reactions. A short review of these approaches is given in relation to selected published studies. Results from the simulations performed by the five codes show remarkable agreement both quantitatively—based on upscaled parameters such as surface area, solid volume, and effective reaction rate—and qualitatively—based on comparisons of shape evolution. This outcome is especially notable given the disparity of approaches used by the codes. Therefore, these results establish a strong benchmark for the validation and testing of pore-scale codes developed for the simulation of flow and reactive transport with evolving geometries. They also underscore the significant advances seen in the last decade in tools and approaches for simulating this type of problem.

... 42 In addition, a non-Newtonian power-law model was also employed in this study to represent the more realistic shear-thinning behaviors. 4,7,17,51,52 Specifically, the non-Newtonian μ m can be defined as ...

This study provides a quantitative analysis to investigate the effects of cough intensity and initial mucus thickness on the mucus transport and clearance in a mouth-to-trachea airway geometry using an experimentally validated Volume of Fluid (VOF) based multiphase model. In addition, the accuracy of simplifying mucus as Newtonian fluid is also quantified by the comparisons of mucus transport and clearance efficiencies with the simulations using realistic shear-thinning non-Newtonian fluid viscosities as a function of shear rate. It proves that the VOF model developed in this study can capture air–mucus interface evolution and predict the mucus transport behaviors driven by the expiratory cough waveforms. Numerical results show that noticeable differences can be identified between the simulations using simplified Newtonian fluid and the realistic non-Newtonian fluid viscosity models, which indicates that an appropriate non-Newtonian fluid model should be applied when modeling mucus transport to avoid the possible inaccuracy induced by the Newtonian fluid simplification. Furthermore, the results also indicate that an intense cough can enhance the mucus clearance efficiency in chronic obstructive pulmonary disease (COPD) upper airways. Additionally, although higher mucus clearance efficiency is observed for severe COPD conditions with a thicker mucus layer, there is a possibility of mucus accumulation and obstruction in the upper airway for such a COPD condition if the cough is not strong enough, which will possibly cause further breathing difficulty. The VOF model developed in this study can be further refined and integrated with discrete phase models to predict the mucus clearance effect on inhaled particles explicitly.

... In the present case, the model is a stationary Stokes equation combined with divergence-free condition on the velocity. On the one hand, such a flow is intrinsically useful in order to understand the properties and the mechanisms occurring inside porous media, and the Stokes problem becomes a component in coupled problems that models more complex flows, such as rheology [60], heterogeneous flows [55], reactive [50,45,44,27] or multiphase flows [64]. On the other hand, a velocity field enables to compute permeability estimations [15,22], whose accuracy is crucial for upscaling techniques. ...

... The application considered in the present work is the motion of a miscible agglomerate of Xanthan polymer, a bio-polymer heavy used in food industry and also in enhanced oil recovery, featuring a constant index N = 0.386 and µ 0 (C) = µ ∞ + A µ e C/ρB µ (1 − e −C/ρR µ ), β(C) = A β e C/ρB β , where ρ = 0.842 kg L −1 is the density of the pure material, the nominal time is A β = 2.4 s, the nominal and solvent viscosity are A µ = 0.26 Pa s and µ ∞ = 10 −3 Pa s (water), the non-dimensional ratio are B µ = 6.06 10 −4 , R µ = 6.5 10 −4 and B β = 6.17 10 −4 (see [60] for instance). This leads to the rheogram, that is to say the relations between the viscosity and the shear rate, displayed on figure 10. ...

... Solvent viscosity and power-law approximation are displayed with black lines. Experimental data and their fit to the Carreau law are taken from[60]. ...

In this article, we present a numerical iterative method for the solution of internal viscous and incompressible flows in real porous three-dimensional bodies at their pore scale. We use the penalized formulation of the problem involving velocity and vorticity: an operator splitting allows to split apart the diffusion (inherited from Stokes equation) and the penalization phenomena (which takes into account the solid matrix). By means of the numerical analysis of the splitting, we exhibit the penalization coefficient which is actually effective. This method allows to deal only with fast-evaluation operators, that is to say scaling at most as O(n log n) where n is the number of underlying grid points, such as straightforward computations of finite differences schemes or FFT solver. The numerical analysis and implementation solutions are presented, and validated on various digital rock physics geometries acquired by micro-tomography, using numerical and physical diagnostics. To enforce this validation, we also present permeability estimations of several porous samples. The simulation of transport of passive and active scalars is finally investigated in order to perform the practical upscaling to 1D models of transport and diffusion at the Darcy scale.

... The micro-continuum approach has been used to simulate the dissolution of a calcite crystal at the pore-scale and compared successfully with microfluidic experiments . In Molins et al. (2020), the approach is compared with state-of-the-art RTM at the pore-scale using various numerical techniques including Chombo-Cruch with Level-Set (Molins et al., 2017), Lattice Boltzmann Method (Prasianakis et al., 2018), dissolFoam moving grids with conformal mapping (Starchenko et al., 2016), and Vortex methods (Sanchez et al., 2019). The benchmark consists of a 0.2 mm diameter calcite crystal posted in a 1 mm long 0.5 width channel (see Fig. 3). ...

... Besides, a non-Newtonian power-law model was also employed in this study to represent realistic shear-thinning behaviors. (Cone, 2009;Fahy & Dickey, 2010;Lai et al., 2009;Ren et al., 2020;Sanchez, Hume, Chatelin, & Poncet, 2018). Specifically, the non-Newtonian can be defined as ...

A large pool fire can be generated if there is a liquefied petroleum gas (LPG) leakage during transportation or at storage sites, while the underlying mechanisms of how the hazardous matter can be generated from the LPG pool fire and delivered dose into the human lung, and evaluation of the exposure risks are still ambiguous. Thus, it is necessary to systematically study the LPG pool fire rheology and the generation, transport, and deposition of the generated aerosolized toxicants from the pool fire to the human respiratory system. To partially address the above-mentioned concerns, this study has conducted novel research efforts to investigate the characteristics of large LPG pool fires and cough-driven mucus transport behaviors in upper lung airways, which can be employed to assess the health risks from LPG fires to the pulmonary system future work. Specifically, Chapter I reviewed the previous studies concerning the LPG pool fires and mucus movement behaviors in lung airways using experimental methods and numerical approaches, as well as presented the research objectives. Chapter II was to develop an experimentally validated CFD model to estimate the surface emissive power, and predict the incident radiation from large LPG pool fires to the surrounded objects and develop the reasonable minimum distances between the pool fire and objects using CFD simulations. Chapter III performed numerical simulations using an experimentally validated CFD model to simulate large LPG pool fires and predict the fire configuration characteristics, including flame height and flame tilt. The impacts of pool diameter and wind velocity on the fire configuration characteristics were investigated. Based on the CFD results and the parametric analysis, new correlations are proposed to provide more accurate estimations of flame height and tilt specifically for large LPG pool fires. Chapter IV has built an experimentally validated Volume of Fluid (VOF) model to conduct a quantitative analysis to investigate the effects of cough intensity and initial mucus thicknesses on the mucus transport and clearance in a mouth-to-trachea airway geometry. The VOF model developed in this work can be further refined and integrated with Discrete Phase Models (DPMs) to predict the mucus clearance effect on inhaled toxic particles from LPG pool fires explicitly. In addition, Chapter V summarized the essentials of the research work done and outlined the future work.

This article shows how to consistently and accurately manage the Lagrangian formulation of chemical reaction equations coupled with the superficial velocity formalism introduced in the late 80s by Quintard and Whitaker. Lagrangian methods prove very helpful in problems in which transport effects are strong or dominant, but they need to be periodically put back in a regular lattice, a process called remeshing. In the context of digital rock physics, we need to ensure positive concentrations and regularity to accurately handle stagnation point neighborhoods. These two conditions lead to the use of kernels resulting in extra-diffusion, which can be prohibitively high when the diffusion coefficient is small. This is the case especially for reactive porous media, and the phenomenon is reinforced in porous rock matrices due to Archie’s law. This article shows how to overcome this difficulty in the context of a two-scale porosity model applied in the Darcy-Brinkman-Stokes equations, and how to obtain simultaneous sign preservation, regularity and accurate diffusion, and apply it to dissolution processes at the pore scale of actual rocks.