Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Particle image velocimetry studies of an incipient spot in the Blasius boundary layer

Spatial evolution of a small amplitude localized disturbance introduced into the laminar boundary layer of a flat plate has been studied experimentally using the particle image velocimetry (PIV) technique. PIV data have been acquired in the spanwise and wall normal planes. Long and well defined high and low speed streaks are seen in the spanwise plane. The number of streaks are found to increase in the downstream direction. Breathing mode type oscillation of the boundary layer is observed. Associated with the streaks and akin to the bypass transition, ‘backward’ and ‘forward’ jet like structures of the fluctuating velocity components are observed.AS Banerjee: summer trainee, IIT Kharagpur, India     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: I Homogenization Analysis

The macroscopic model governing coupled electro-chemo-mechanical phenomena in expansive clays is revisited within a rigorous homogenization procedure applied to the microscopic governing equations which describe the local interaction between charged clay particles and a binary monovalent aqueous electrolyte solution. The up-scaling of the microscopic electro-hydro-dynamics leads to a two-scale approach wherein the macroscopic model appears governed by a fully coupled form of Onsager’s reciprocity relations, mass conservation equations and a modified Terzaghi’s effective stress principle. In addition, the two-scale approach provides microscopic representations for the effective coefficients which are exploited herein to obtain further insight in the constitutive behavior of the electrochemical parameters and the swelling pressure. Among other effects, we show that these microscopic closure relations are mainly dictated by the spatial variability of a microscale electric potential which satisfies a local version of the Poisson–Boltzmann problem in a periodic unit cell, The proposed framework allows to address various relevant still open issues regarding the constitutive behavior of swelling systems, Among them we give particular emphasis on the analysis of the influence of the fluctuation and distortion of the electrical double layer upon the magnitude of the electrochemical coefficients and the precise local conditions for the validity of the symmetry of Onsager’s relations.     

Asymptotic expansions by Γ-convergence

Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.      

Stability of a 2-dimensional Mathieu-type system with quasiperiodic coefficients

In the following we consider a 2-dimensional system of ODEs containing quasiperiodic terms. The system is proposed as an extension of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The 2-d system has two ‘natural’ frequencies when the time-dependent terms are switched off, and it is internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first using numerical simulations and Floquet theory. While some instability regions are easy to anticipate, there are some surprises: within instability zones, small islands of stability develop, and unusual ‘arcs’ of instability arise also. The transition curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the ‘resonance curves’ from which bands of instability emanate. In addition, the method of multiple scales is used to examine the islands of stability near the 1:1 resonance.     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: II Computational Validation

In Part I Moyne and Murad [Transport in Porous Media 62, (2006), 333–380] a two-scale model of coupled electro-chemo-mechanical phenomena in swelling porous media was derived by a formal asymptotic homogenization analysis. The microscopic portrait of the model consists of a two-phase system composed of an electrolyte solution and colloidal clay particles. The movement of the liquid at the microscale is ruled by the modified Stokes problem; the advection, diffusion and electro-migration of monovalent ions Na⁺ and Cl⁻ are governed by the Nernst–Planck equations and the local electric potential distribution is dictated by the Poisson problem. The microscopic governing equations in the fluid domain are coupled with the elasticity problem for the clay particles through boundary conditions on the solid–fluid interface. The up-scaling procedure led to a macroscopic model based on Onsager’s reciprocity relations coupled with a modified form of Terzaghi’s effective stress principle including an additional swelling stress component. A notable consequence of the two-scale framework are the new closure problems derived for the macroscopic electro-chemo-mechanical parameters. Such local representation bridge the gap between the macroscopic Thermodynamics of Irreversible Processes and microscopic Electro-Hydrodynamics by establishing a direct correlation between the magnitude of the effective properties and the electrical double layer potential, whose local distribution is governed by a microscale Poisson–Boltzmann equation. The purpose of this paper is to validate computationally the two-scale model and to introduce new concepts inherent to the problem considering a particular form of microstructure wherein the clay fabric is composed of parallel particles of face-to-face contact. By discretizing the local Poisson–Boltzmann equation and solving numerically the closure problems, the constitutive behavior of the diffusion coefficients of cations and anions, chemico-osmotic and electro-osmotic conductivities in Darcy’s law, Onsager’s parameters, swelling pressure, electro-chemical compressibility, surface tension, primary/secondary electroviscous effects and the reflection coefficient are computed for a range particle distances and sat concentrations.     

Computational and experimental study of supersonic flow over axisymmetric cavities

Laminar and turbulent computations are presented for annular rectangular-section cavities, on a body of revolution, in a Mach 2.2 flow. Unsteady ‘open cavity flows’ result for all laminar computations for all cavity length-to-depth ratios, L/D (1.33, 10.33, 11.33 and 12.33). The turbulent computations produce ‘closed cavity flows’ for L/D of 11.33 and 12.33. Surface pressure fluctuations at the front corner of the L/D = 1.33 cavity are periodic in some cases depending on the cavity length and depth, the boundary layer at the cavity front lip and the cavity scale. The turbulent computations are supported by experimental schlieren images, obtained using a spark light source, and time-averaged surface pressure data.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

An Existence Theorem for Nonlinearly Elastic ‘Flexural’ Shells

The two-dimensional equations of a nonlinearly elastic ‘flexural’ shell have been recently identified and justified by V. Lods and B. Miara, by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. These equations can be recast as a minimization problem for a ‘two-dimensional energy’ over a manifold of ‘admissible deformations’. The stored energy function is a quadratic expression in terms of the exact difference between the curvature tensor of the deformed middle surface and that of the undeformed one; the admissible deformations are those that preserve the metric of the undeformed middle surface and satisfy boundary conditions of clamping or of simple support. We establish here that this minimization problem has at least one solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Natural vibration of a sandwich beam on an elastic foundation

The natural vibration of an elastic sandwich beam on an elastic foundation is studied. Bernoulli’s hypotheses are used to describe the kinematics of the face layers. The core layer is assumed to be stiff and compressible. The foundation reaction is described by Winkler’s model. The system of equilibrium equations is derived, and its exact solution for displacements is found. Numerical results are presented for a sandwich beam on an elastic foundation of low, medium, or high stiffness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 57–63, May 2006.     

Prediction of the effective elastic properties of spheroplastics by the generalized self-consistent method

The problem of predicting the effective elastic properties of composites with prescribed random location and radius variation in spherical inclusions is solved using the generalized self-consistent method. The problem is reduced to the solution of the averaged boundary-value problem of the theory of elasticity for a single inclusion with an inhomogeneous transition layer in a medium with desired effective elastic properties. A numerical analysis of the effective properties of a composite with rigid spherical inclusions and a composite with spherical pores is carried out. The results are compared with the known solution for the periodic structure and with the solutions obtained by the standard self-consistent methods. Perm’ State Technical University, Perm’ 614600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 186–190, May–June, 1999.     

Convective inertia effects in wall-bounded thin film flows

We apply the boundary layer equations to inertial flow in wall bounded films that might be characterized as ‘thin’, say ɛ ≤ 0.1 where ɛ is the ratio of the characteristic lengths, yet to which the lubrication approximation of Reynolds no longer applies. Two particular flow geometries are investigated, nominally parallel plates and nominally inclined plates, both with and without spatially periodic perturbation of the stationary plate. A Galerkin-B spline formulation of the governing equations is employed, and we rely on parametric continuation to obtain solutions at higher values of the Reynolds number. In particular, we are able to demonstrate that the boundary layer equations yield accurate results for a wide range of Reynolds numbers when the aspect ratio is less than 1/10. We also find that in both nominally parallel and nominally inclined geometries the sign of the inertial force correction is determined by the film contour in the neighborhood of the exit, this result might have implications in the design of MEMS devices.     

A new look at an old technique for the measurement of fatigue crack-growth rates

An automated technique is described by which a sequence of ‘beachmarks’ (clam-shell markings) can be produced in a specimen during laboratory tests on the propagation of fatigue cracks. The subsequent use of the beachmarks for the measurement of the lengths of the cracks is described, and the intrinsic accuracy of the method is estimated. It is found that, although fewer data points are obtained by this method, they are more accurate and better characterized than those obtained by the more commonly used ‘indirect’ methods. In addition, valuable supplementary information is obtained from which it is possible to assess both the confidence intervals for the results, and the degree of crack front curvature.     

A multiscale model for the ductile fracture of crystalline materials

In this paper, a multiscale model that combines both macroscopic and microscopic analyses is presented for describing the ductile fracture process of crystalline materials. In the macroscopic fracture analysis, the recently developed strain gradient plasticity theory is used to describe the fracture toughness, the shielding effects of plastic deformation on the crack growth, and the crack tip field through the use of an elastic core model. The crack tip field resulting from the macroscopic analysis using the strain gradient plasticity theory displayes the 1/2 singularity of stress within the strain gradient dominated region. In the microscopic fracture analysis, the discrete dislocation theory is used to describe the shielding effects of discrete dislocations on the crack growth. The result of the macroscopic analysis near the crack tip, i.e. a new K-field, is taken as the boundary condition for the microscopic fracture analysis. The equilibrium locations of the discrete dislocations around the crack and the shielding effects of the discrete dislocations on the crack growth at the microscale are calculated. The macroscopic fracture analysis and the microscopic fracture analysis are connected based on the elastic core model. Through a comparison of the shielding effects from plastic deformation and the discrete dislocations, the elastic core size is determined.     

Simultaneous planar measurement of droplet velocity and size with gas phase velocities in a spray by combined ILIDS and PIV techniques

A new approach for simultaneous planar measurement of droplet velocity and size with gas phase velocities is reported, which combines the out-of-focus imaging technique ‘Interferometric Laser Imaging Droplet Sizing’ (ILIDS) for planar simultaneous droplet size and velocity measurements with the in-focus technique ‘Particle Image Velocimetry’ (PIV) for gas velocity measurements in the vicinity of individual droplets. Discrimination between the gas phase seeding and the droplets is achieved in the PIV images by removing the glare points of focused droplet images, using the droplet position obtained through ILIDS processing. Combination of the two optical arrangements can result in a discrepancy in the location of the centre of a droplet, when imaging through ILIDS and PIV techniques, of up to about 1 mm, which may lead to erroneous identification of the glare points from droplets on the PIV images. The magnitude of the discrepancy is a function of position of the droplet’s image on the CCD array and the degree of defocus, but almost independent of droplet size. Specifically, it varies approximately linearly across the image along the direction corresponding to the direction of propagation of the laser sheet for a given defocus setting in ILIDS. The experimental finding is supported by a theoretical analysis, which was based on geometrical optics for a simple optical configuration that replicates the essential features of the optical system. The discrepancy in the location was measured using a monodisperse droplet generator, and this was subtracted from the droplet centres identified in the ILIDS images of a polydisperse spray without ‘seeding’ particles. This reduced the discrepancy between PIV and ILIDS droplet centres from about 1 mm to about 0.1 mm and hence increased the probability of finding the corresponding fringe patterns on the ILIDS image and glare points on the PIV image. In conclusion, it is shown that the proposed combined method can discriminate between droplets and ‘seeding’ particles and is capable of two-phase measurements in polydisperse sprays.     

Scale-dependent Macroscopic Balance Equations Governing Transport Through Porous Media: Theory and Observations

A theory is developed providing a rational framework for spatial scale- dependent fluid’s flow and heat transfer, and mass of a component migrating with it through porous media. Introducing the assumption of a non-Brownian type motion and referring to asymptotic expansion in powers of a small defined parameter, we develop a novel approach associated with macroscopic balance equations obtained by averaging over a Representative Elementary Volume (REV). We prove that these equations can be decomposed into a primary part that refers to the REV length scale and a secondary part valid at a length scale smaller than that of the corresponding REV length. Further to our previous development, we obtain two general forms of the primary and secondary macroscopic balance equations. One is based on the assumption that the advective flux of the extensive quantity is dominant over that of the dispersive flux, whereas the other disregards this assumption. Moreover we also introduce the primary and secondary macroscopic forms for the fluid heat- transfer equation. Considering a Newtonian fluid, the resulting primary Navier–Stokes equation can vary from a nonlinear wave equation to a drag-dominant equation at the fluid–solid interface (Darcy’s law). The secondary momentum balance equation describes a wave equation governing the concurrent propagation of the intensive momentum and the dispersive momentum flux, deviating from their corresponding average terms. The primary macroscopic fluid heat-transfer equation accounts for advective and dispersive heat fluxes and the secondary macroscopic heat-transfer equation involves the simultaneous advection of heat deviating from its corresponding intensive average quantity. The primary macroscopic solute mass balance equation accounts for advection and hydrodynamic dispersion. The secondary macroscopic component mass balance equation is in the form of pure advection governing migration of the deviation from the average component concentration. At this stage, we focus on establishing the viability of the developed theory. We do this by arguing that field observations of motion at small spatial scales are coherent with the hyperbolic characteristics of the secondary balance equations. Field observations under natural gradient flow conditions show excessive high concentration (average of 50 mg/L) of colloids under land irrigated by sewage effluents. We argue that this displacement of condensed colloidal parcels manifests the theoretical findings for the smaller spatial scale. Further evidence show the accumulation of particles moving behind the front of an emitted shockwave. We consider this as an experimental proof reinforcing the argument that colloidal migration is subject to the action of a shockwave in the fluid and pure advection transport, governed by the respective suggested hyperbolic macroscopic balance equations of fluid momentum and component mass at the smaller spatial scale.     

Acoustic receptivity of Mach 4.5 boundary layer with leading-edge bluntness

Boundary layer receptivity to two-dimensional slow and fast acoustic waves is investigated by solving Navier–Stokes equations for Mach 4.5 flow over a flat plate with a finite-thickness leading edge. Higher order spatial and temporal schemes are employed to obtain the solution whereby the flat-plate leading edge region is resolved by providing a sufficiently refined grid. The results show that the instability waves are generated in the leading edge region and that the boundary-layer is much more receptive to slow acoustic waves (by almost a factor of 20) as compared to the fast waves. Hence, this leading-edge receptivity mechanism is expected to be more relevant in the transition process for high Mach number flows where second mode instability is dominant. Computations are performed to investigate the effect of leading-edge thickness and it is found that bluntness tends to stabilize the boundary layer. Furthermore, the relative significance of fast acoustic waves is enhanced in the presence of bluntness. The effect of acoustic wave incidence angle is also studied and it is found that the receptivity of the boundary layer on the ‘windward’ side (with respect to the acoustic forcing) decreases by more than a factor of four when the incidence angle is increased from 0° to 45°. However, the receptivity coefficient for the ‘leeward’ side is found to vary relatively weakly with the incidence angle.      

A Matched Asymptotic Expansion Analysis of Highly Unsteady Porous Media Flows

The method of matched asymptotic expansions is used to analyse a mixture of wave and diffusive behaviours governing flow in a saturated porous medium inside an elastic pipe that is suddenly subjected to a large hydraulic gradient at its entrance. At early times and near the entrance, the head is a diffusing wave that can be reduced to the linear and non-linear telegrapher equations for the laminar and partially developed turbulent flows, respectively. At later times, laminar flows are diffusive and partially developed turbulent flows follow a ‘fast diffusion’ behaviour. In the case of fully developed turbulence, flows at later times follow a fast diffusion form which is complicated by advection at extremely high gradients. A matched asymptotic expansion approach is used to match flows at early times and near the entrance, with complementary forms that are away from the entrance and which occur at later times.     

The biaxial loading response of powder aluminum at elevated temperature

Nuclear fuel can be fabricated using powder-metallurgy processes by compacting uranium-oxide powder with aluminum powder to form a cermet and then extruding the cermet to form fuel tubes. This method of production allows greater control of uranium-oxide particle size and distribution in the tube, making the production of fuel with greater concentrations of uranium oxide possible, and thus decreasing the volume of radioactive waste remaining after the fuel is spent. As the concentration of uranium oxide increases, however, there is an increase in failures during extrusion. To address this problem, an experimental procedure was developed to examine the response of powder aluminum, a material with a structure similar to that of the cermet fuel, to biaxial loadings such as those experienced during extrusion. Biaxial loadings can be varied from pure shear to simple tension or compression, or to combinations of these loadings in a numerically controlled ‘tension-torsion’ testing machine. Data obtained using this system were used to develop a model for the post-yield behavior in extruded powder aluminum which includes information derived both from the macroscopic stress-strain behavior of 1100 aluminum and extruded powder aluminum and from the observed microscopic structure of the extruded powder aluminum. This paper describes the development of the experimental system and shows the different biaxial mechanical behavior of the two materials. Test fixtures were developed and software was written to control constant strain-rate tension, compression, torsion, combined tension-torsion, and combined compression-torsion tests performed using a computer-controlled MTS biaxial testing machine. Extruded powder aluminum and 1100 aluminum specimens were tested at 427°C, the powder-aluminum extrusion temperature, under those loading conditions. Each specimen was subjected to only one loading cycle. Data were recorded during loading only. Tested specimens were also sectioned and examined microscopically.