Journal of Non-Newtonian Fluid Mechanics

A macroscopic model for unsteady incompressible isothermal non-Newtonian flow in homogeneous porous media, taking into account inertial and slip effects at solid–fluid interfaces, is derived in this work. The development is carried out considering a general Newton’s law of viscosity for the fluid phase. Using the classical volume averaging method, the seepage velocity is shown to be solenoidal. The macroscopic momentum equation is derived in the Laplace domain, employing a simplified version of the volume averaging method, which calls upon Green’s formulas and adjoint problems for Green’s function pairs for the velocity and pressure. In the Laplace domain, the macroscopic momentum equation takes the form of Darcy’s law corrected by a term that accounts for the initial flow condition. Once transformed back into the time domain, this equation provides the macroscopic velocity that depends on two terms. The first one is under the form of a time convolution between the macroscopic pressure gradient and the time derivative of an apparent permeability tensor. The second one is a memory term that accounts for the effect of the initial flow conditions. These two effective quantities are determined from the solution of a single closure problem that naturally results from the derivations. The model is consistent with the unsteady model in the Newtonian case and simplifies to the steady versions of some non-Newtonian macroscopic flow models. The macroscopic model is validated with pore-scale simulations performed in 2D model porous structures, considering a Carreau fluid. The impact of inertia and non-Newtonian effects on the dynamics of the macroscopic coefficients is highlighted

Stretching and squeezing of sessile dielectric drops by the optical radiation pressure

We study numerically the deformation of sessile dielectric drops immersed in a second fluid when submitted to the optical radiation pressure of a continuous Gaussian laser wave. Both drop stretching and drop squeezing are investigated at steady state where capillary effects balance the optical radiation pressure. A boundary integral method is implemented to solve the axisymmetric Stokes flow in the two fluids. In the stretching case, we find that the drop shape goes from prolate to near-conical for increasing optical radiation pressure whatever the drop to beam radius ratio and the refractive index contrast between the two fluids. The semi-angle of the cone at equilibrium decreases with the drop to beam radius ratio and is weakly influenced by the index contrast. Above a threshold value of the radiation pressure, these "optical cones" become unstable and a disruption is observed. Conversely, when optically squeezed, the drop shifts from an oblate to a concave shape leading to the formation of a stable "optical torus". These findings extend the electrohydrodynamics approach of drop deformation to the much less investigated "optical domain" and reveal the openings offered by laser waves to actively manipulate droplets at the micrometer scale

Laser microfluidics: fluid actuation by light

The development of microfluidic devices is still hindered by the lack of robust fundamental building blocks that constitute any fluidic system. An attractive approach is optical actuation because light field interaction is contactless and dynamically reconfigurable, and solutions have been anticipated through the use of optical forces to manipulate microparticles in flows. Following the concept of an 'optical chip' advanced from the optical actuation of suspensions, we propose in this survey new routes to extend this concept to microfluidic two-phase flows. First, we investigate the destabilization of fluid interfaces by the optical radiation pressure and the formation of liquid jets. We analyze the droplet shedding from the jet tip and the continuous transport in laser-sustained liquid channels. In the second part, we investigate a dissipative light-flow interaction mechanism consisting in heating locally two immiscible fluids to produce thermocapillary stresses along their interface. This opto-capillary coupling is implemented in adequate microchannel geometries to manipulate two-phase flows and propose a contactless optical toolbox including valves, droplet sorters and switches, droplet dividers or droplet mergers. Finally, we discuss radiation pressure and opto-capillary effects in the context of the 'optical chip' where flows, channels and operating functions would all be performed optically on the same device

Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare?

A wide variety of techniques have been developed to homogenize transport equations in multiscale and multiphase systems. This has yielded a rich and diverse field, but has also resulted in the emergence of isolated scientific communities and disconnected bodies of literature. Here, our goal is to bridge the gap between formal multiscale asymptotics and the volume averaging theory. We illustrate the methodologies via a simple example application describing a parabolic transport problem and, in so doing, compare their respective advantages/disadvantages from a practical point of view. This paper is also intended as a pedagogical guide and may be viewed as a tutorial for graduate students as we provide historical context, detail subtle points with great care, and reference many fundamental works

KG2B, a collaborative benchmarking exercise for estimating the permeability of the Grimsel granodiorite - Part 1: Measurements, pressure dependence and pore-fluid effects

Measuring the permeability of tight rocks remains a challenging task. In addition to the traditional sources of errors that affect more permeable formations (e.g. sample selection, non-representative specimens, disturbance introduced during sample acquisition and preparation), tight rocks can be particularly prone to solid–fluid interactions and thus more sensitive to the methods, procedures and techniques used to measure permeability. To address this problem, it is desirable to collect, for a single material, measurements obtained by different methods and pore-fluids. For that purpose a collaborative benchmarking exercise involving 24 laboratories was organized for measuring the permeability of a single low permeability material, the Grimsel granodiorite, at a common effective confining pressure (5 MPa). The objectives of the benchmark were: (i) to compare the results for a given method, (ii) to compare the results between different methods, (iii) to analyze the accuracy of each method, (iv) to study the influence of experimental conditions (especially the nature of pore fluid), (v) to discuss the relevance of indirect methods and models and finally (vi) to suggest good practice for low permeability measurements. In total 39 measurements were collected that allowed us to discuss the influence of (i) pore-fluid, (ii) measurement method, (iii) sample size and (iv) pressure sensitivity. Discarding some outliers from the bulk data set (4 out of 39) an average permeability of 1.11 × 10−18 m² with a standard deviation of 0.57 × 10−18 m² was obtained. The most striking result was the large difference in permeability for gas measurements compared to liquid measurements. Regardless of the method used, gas permeability was higher than liquid permeability by a factor approximately 2 (kgas = 1.28 × 10−18 m² compared to kliquid = 0.65 × 10−18 m²). Possible explanations are that (i) liquid permeability was underestimated due to fluid-rock interactions (ii) gas permeability was overestimated due to insufficient correction for gas slippage and/or (iii) gases and liquids do not probe exactly the same porous networks. The analysis of Knudsen numbers shows that the gas permeability measurements were performed in conditions for which the Klinkenberg correction is sufficient. Smaller samples had a larger scatter of permeability values, suggesting that their volume were below the Representative Elementary Volume. The pressure dependence of permeability was studied by some of the participating teams in the range 1–30 MPa and could be fitted to an exponential law k = ko.exp(–γPeff) with γ = 0.093 MPa−1. Good practice rules for measuring permeability in tight materials are also provided

Multiphase, Multicomponent Fluid Flow in Homogeneous and Heterogeneous Porous Media Écoulement de fluides multiconstituants polyphasiques dans des milieux poreux homogènes et hétérogènes

The flow of several components and several phases through a porous medium is generally described by introducing macroscopic mass-balance equations under the form of generalized dispersion equations. This model raises several questions that are discussed in this paper on the basis of results obtained from the volume averaging method, coupled with pore-scale simulations of the multiphase flow. The study is limited to a binary, two-phase system, and we assume that the momentum equations can be solved independently from the diffusion/advection equations. The assumption of local-equilibrium is discussed and several length-scale and time-scale constraints are provided. A key issue concerns the impact on the dispersion tensors of the pore-scale equilibrium condition at the interface between the different phases. Our results show that this phenomenon may lead to significant variations of the dispersion coefficients with respect to passive dispersion, i. e. , dispersion without interfacial mass fluxes. Macroscopic equations are then obtained in the general case, and several local closure problems are provided that allow one to calculate the dispersion tensors and others properties, from the pore-scale geometry, velocities, and fluid characteristics. Examples of solutions of these closure problems are given in the case of two-dimensional representative unit cells. The two-phase flow equations are solved in two different ways : a boundary element technique, or a modified lattice Boltzmann approach. Solutions of the closure problems associated with the dispersion equations are then given using a finite volume element formulation of the partial differential equations. The results show the influence of velocity and saturation on the effective parameters. They emphasize the importance of geometry on the behavior of the dispersion tensor. Extension of these results to a larger-scale including the effect of heterogeneities is proposed in a limited case corresponding to the flow of one phase, the other phase being at residual saturation. A new large-scale dispersion equation is provided, which features a large-scale dispersion tensor that can be determined from the heterogeneity characteristics through a set of closure problems. Results are extended to a more general two-phase flow problem, when the large-scale two-phase flow can be assumed to be quasi-static. Indications are given on the difficulties associated with flow under large-scale dynamic conditions, with abnormal dispersion. <br> L'écoulement polyphasique de plusieurs constituants à travers un milieu poreux est généralement décrit en introduisant des équations macroscopiques de conservation de la masse sous la forme d'équations de dispersion généralisées. Cette modélisation soulève plusieurs questions qui sont débattues dans cet article en se basant sur des résultats obtenus à partir d'une prise de moyenne volumique, couplée avec une simulation à l'échelle du pore de l'écoulement polyphasique. L'étude est limitée à un système binaire comportant deux phases et nous supposons que les équations de quantité de mouvement peuvent être résolues indépendamment des équations de diffusion/advection. L'hypothèse d'équilibre local est discutée et plusieurs contraintes d'échelles de longueur et de temps sont prises en compte. Une des questions concerne l'influence sur les tenseurs de dispersion de la condition d'équilibre à l'échelle du pore à l'interface entre les différentes phases. Nos résultats montrent que ces phénomènes peuvent conduire à des variations significatives des coefficients de dispersion en rapport avec la dispersion passive, c'est-à-dire la dispersion sans flux de masse aux interfaces. Des équations macroscopiques sont alors obtenues dans le cas général ainsi que plusieurs équations locales de fermeture permettant de calculer les tenseurs de dispersion et d'autres propriétés à partir des géométries à l'échelle du pore, des vitesses et des caractéristiques des fluides. Des exemples de solutions de ces équations de fermeture sont donnés dans le cas de cellules unitaires représentatives à deux dimensions. Les équations des écoulements biphasiques sont résolues de deux manières différentes : par une technique par éléments frontières ou une approche par réseau de Boltzmann modifié. Des solutions aux équations de fermeture associées aux équations de dispersion sont ensuite apportées au moyen d'une formulation par volumes finis des équations aux dérivées partielles. Les résultats montrent l'influence de la vitesse et de la saturation sur les paramètres effectifs. Ils mettent en évidence l'importance de la géométrie sur le comportement du tenseur de dispersion. L'extension de ces résultats à une plus grande échelle incluant l'effet des hétérogénéité est proposée dans un cas limité correspondant à l'écoulement d'une phase, l'autre phase étant à la saturation résiduelle. Une nouvelle équation de dispersion à grande échelle est obtenue, comportant un tenseur de dispersion à grande échelle que l'on peut déterminer à partir des caractéristiques d'hétérogénéité en passant par un système d'équations de fermeture. Les résultats sont étendus à un problème plus général d'écoulement biphasique, lorsque l'écoulement biphasique à grande échelle peut être supposé quasi statique. Des indications sont données concernant les difficultés associées aux écoulements soumis à des conditions fortement dynamiques et avec une dispersion anormale

Prediction and Measurement of Sealing Properties of Joints Between Wavy Metal Surfaces

The transmissivity of metal-metal sealing joints is investigated experimentally and compared to predictions obtained by modelling. The focus is laid upon a wavy surface contacting a flat rigid part, representative of a seat-to-plug contact in an internal sealing valve encountered in nuclear power plants for instance. Experimental transmissivities are obtained from water leak-rate and pressure drop measurements carried out on a model ring-shape sample seat holding a controlled wavy defect and pressed against a rigid flat plug with a controlled normal load. The sample seat surface is manufactured by face turning a tubular part under radial stress and waviness is obtained after elastic relaxation. Modelling is performed on a 3D finite element model of the assembly, composed of the plug, the sample seat and its holder. The upper sample seat surface, which topography is recorded by confocal microscopy, is reconstructed using a modal decomposition on a basis of vibrational eigen modes. Its lower surface, in contact with the holder, is considered as perfectly flat or with its own defects. The contact aperture field between the seat and the plug is computed for a given normal load and is used to solve the incompressible Reynolds equation with a boundary element method, yielding the transmissivity. Predicted transmissivities reveal to be in good agreement with experimental data at low clamping loads and are overestimated for larger ones. Defects on the lower surface of the seat are shown to have a significant impact on the seat-to plug contact transmissivity. Nomenclature A Apparent contact area: A = π(r 2 e − r 2 i) (m 2) C contour of the contact zones F clamping load (N) h(r, θ) local aperture (m) I identity matrix K stiffness matrix (N.m −1) K measured transmissivity (m 3) K rr computed transmissivity (m 3