Bingham’s heritage

A hundred years after the founding paper by E.C. Bingham, we briefly review the impact of the yield stress concept and current interest in it on the scientific community. We show that yield stress fluids have only emerged as a relevant fluid type, in both mechanics and physics, over the past 20 years, opening the way to a broad range of new study areas.     

whittaker’s Reduction Method for Poincare’s Dynamical Equations

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Resolution of d’Alembert’s Paradox

We propose a resolution of d’Alembert’s Paradox comparing observation of substantial drag/lift in fluids with very small viscosity such as air and water, with the mathematical prediction of zero drag/lift of stationary irrotational solutions of the incompressible inviscid Euler equations, referred to as potential flow. We present analytical and computational evidence that (i) potential flow cannot be observed because it is illposed or unstable to perturbations, (ii) computed viscosity solutions of the Euler equations with slip boundary conditions initiated as potential flow, develop into turbulent solutions which are wellposed with respect to drag/lift and which show substantial drag/lift, in accordance with observations.     

A modified Darcy’s law

An approach to describe the turbulent flow through a complex geometry (e.g., urban area) by means of an analogy to flows through porous media is presented. Therefore, a modification of the original Darcy’s law is proposed, and its application is tested in a prototype problem with an idealized complex geometry using large eddy simulations. The numerical results indicate the validity of the modified Darcy’s law for the chosen setup.     

A generalization of Prandtl’s and Spencer’s solutions on axisymmetric viscous flow

New analytical solutions for axisymmetric deformation of a viscous hollow circular cylinder on a rigid fibre are given. One of the solutions generalizes the famous Prandtl’s solution for compression of a rigid perfectly plastic layer between two rough, parallel plates and the other is a modification of Spencer’s solution for compression of an axisymmetric rigid perfectly plastic layer on a rigid fibre. All equations are satisfied exactly whereas some boundary conditions are approximated in a standard manner. Special attention is devoted to frictional interface conditions since these conditions result in additional limitations of the applicability of the solution when compared to that based on a rigid perfectly plastic models. In particular, difficulties with the convergence of numerical solutions under certain conditions can be explained with the use of results obtained. Therefore, the solutions can serve as benchmark problems for verifying numerical codes. The solutions are also adopted to predict the brittle fracture of fibres by means of an approach used in previous studies and confirmed by experiment.     

Archie’s Law in Microsystems

Since 1942 Archie??s law is used every day to estimate, from electrical measurements, the quantity of oil present in oil fields. In this article, we perform the first experimental analysis of electric conductivity in well controlled models of porous media. We used microfluidic networks (called micromodels in the oil industry jargon), incorporating thousands of pores, with controlled wettability. Different electrode and pore geometries are considered. In all cases the evolution of the conductivity with the conductive fluid fraction (??saturation??) clearly reveals the presence of percolation thresholds, signaling that as the fraction of the conductive fluid decreases below some critical value, there are no more pathways involving only channels entirely filled with the conductive fluid that connect the electrodes. This behavior is observed in all cases, for all the network/electrode geometries and wetting properties we investigated, and is consequently likely to reflect a genuine behavior for microfluidic ??2D?? networks. The existing models??based on percolation theory or on mean field approach??reproduce correctly the structure of this behavior, but generally at a semi-quantitative level. The most successful case is obtained with the effective medium theory (EMT) model, with drainage and perpendicular electrodes. This outcome suggests that, despite the complexity of these systems, very simple models can describe correctly the physics of the system. Nonetheless, more precise modeling requires case-by-case studies. Our results are consistent with the current body of knowledge accumulated for decades on three-dimensional samples. The key point is that in 3D systems, owing to topological reasons, the threshold is extremely low in terms of water saturations. Archie??s law completely neglects the threshold effect. Nonetheless the percolation threshold should not be overlooked, and modeling should take this aspect systematically into account, as it is already done by several investigators.     

Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.     

Improvement of FEM’s dynamic property

The discretization size is limited by the sampling theorem, and the limit is one half of the wavelength of the highest frequency of the problem. However, one half of the wavelength is an ideal value. In general, the discretization size that can ensure the accuracy of the simulation is much smaller than this value in the traditional finite element method. The possible reason of this phenomenon is analyzed in this paper, and an efficient method is given to improve the simulation accuracy.     

Sensitivity Aspects of Forchheimer’s Approximation

Forchheimer’s equation, considered to be a nonlinear extension of the linear Darcy’s law, applies to a broader range of velocities for flows through porous media. In this article, we examine sensitivity of the Forchheimer model to permeability κ and a nonlinear coefficient β, using both experimental and computational data for validation. In addition to the direct observations, we were able to identify the role of temperature which influences the model by means of viscosity and density of the fluid. To get a quantifiable answer, we introduce a sensitivity index. Our results reveal a significant impact of the temperature to the model behavior.     

Maxwell’s Equations with Vector Hysteresis

Electromagnetic processes in magnetic materials are described by Maxwells equations. In ferrimagnetic insulators, assuming that D = E, we have the equationIn ferromagnetic metals, neglecting displacement currents and assuming Ohms law, we instead getAlternatively, under quasi-stationary conditions, for either material we can also deal with the magnetostatic equations:(Here f_ext and J_ext are prescribed time-dependent fields.) In any of these settings, the dependence of M on H is represented by a constitutive law accounting for hysteresis: M= (H), being a vector extension of the relay model. This is characterized by a rectangular hysteresis loop in a prescribed x-dependent direction, and accounts for high anisotropy and nonhomogeneity. The discontinuity in this constitutive relation corresponds to the possible occurrence of free boundaries.Weak formulations are provided for Cauchy problems associated with the above equations; existence of a solution is proved via approximation by time-discretization, derivation of energy-type estimates, and passage to the limit. An analogous representation is given for hysteresis in the dependence of P on E in ferroelectric materials. A model accounting for coupled ferrimagnetic and ferroelectric hysteresis is considered, too.Acknowledgement This research was partly supported by the project Free boundary problems in applied sciences of Italian M.I.U.R.. I gratefully acknowledge the useful suggestions from the reviewers.     

On Lagrange’s History of Mechanics

The birth of Lagranges Mechanics constitutes a milestone in the history of Mechanics. The genius of Lagrange should have been aware of this turning point inasmuch as he devoted a substantial part of his celebrated book to an account of the previous development of Mechanics. This historical part, located before the technical exposition in the Mécanique analytique, is the subject of the present paper. By directly consulting the original writings of the scientists that contributed to the development of Mechanics, Lagrange produced a first hand historical account. It largely influenced the historians of the XIX century and also those of the beginning of XX century. We interpret Lagranges motivations for writing his history and we suggest an interpretation of the foundations of his historical account.     

Pulsatile blood flow in large arteries: comparative study of Burton’s and McDonald’s models

To get a clear picture of the pulsatile nature of blood flow and its role in the pathogenesis of atherosclerosis, a comparative study of blood flow in large arteries is carried out using the two widely used models, McDonald＇s and Burton＇s models, for the pressure gradient. For both models, the blood velocity in the lumen is obtained analytically. Elaborate investigations on the wall shear stress （WSS） and oscillatory shear index （OSI） are carried out. The results are in good agreement with the available data in the literature. The superiority of McDonald＇s model in capturing the pulsatile nature of blood flow, especially the OSI, is highlighted. The present investigation supports the hypothesis that not only WSS but also OSI are the essential features determining the pathogenesis of atherosclerosis. Finally, by reviewing the limitations of the present investigation, the possibility of improvement is explored.     

Three-dimensional Green’s functions for composite laminates

The three-dimensional Green’s functions due to a point force in composite laminates are solved by using generalized Stroh formalism and two-dimensional Fourier transforms. Each layer of the composite is generally anisotropic and linearly elastic. The interfaces between different layers are parallel to the top and bottom surfaces of the composite and are perfectly bonded. The Green’s functions of point forces applied at the free surface, interface, and in the interior of a layer are derived in the Fourier transformed domain respectively. The surfaces are imposed by a proportional spring-type boundary condition. The spring-type condition may be reduced to traction-free, displacement-fixed, and mirror-symmetric conditions. Numerical examples are given to demonstrate the validity and elegance of the present formulation of three-dimensional point-force Green’s functions for composite laminates.     

Saint-Venant’s Problem for Compound Piezoelectric Beams

The solution of the Saint-Venant’s Problem for a slender compound piezoelectric beam presented in this paper generalizes the recent solution by the authors and E. Harash (J. Appl. Mech. 11:1–10, 2007) for a homogeneous piezoelectric beam and the solution for a compound elastic beam developed by O. Rand and the first author (Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Boston, 2005). Justification for this approximation emerges from the St. Venant’s Principle. The stress, strain and (electrical) displacement components (“solution hypothesis”) are presented as a set of initially assumed expressions involving twelve tip loading parameters, six unknown weight coefficients, and three pairs of torsion/bending functions of two variables. Each pair of functions satisfies the so-called coupled non-homogeneous Neumann problem (CNNP) in the cross-sectional domain. The work develops concepts of the torsion/bending functions, the torsional rigidity and piezoelectric shear center, the tip coupling matrix, for a compound piezoelectric beam. Examples of exact and approximate solutions for rectangular laminated beams made of transtropic materials are presented.      

Intermittent neural synchronization in Parkinson’s disease

Motor symptoms of Parkinson’s disease are related to the excessive synchronized oscillatory activity in the beta frequency band (around 20 Hz) in the basal ganglia and other parts of the brain. This review explores the dynamics and potential mechanisms of these oscillations employing ideas and methods from nonlinear dynamics. We present extensive experimental documentation of the relevance of synchronized oscillations to motor behavior in Parkinson’s disease, and we discuss the intermittent character of this synchronization. The reader is introduced to novel time-series analysis techniques aimed at the detection of the fine temporal structure of intermittent phase locking observed in the brains of Parkinsonian patients. Modeling studies of brain networks are reviewed, which may describe the observed intermittent synchrony, and we discuss what these studies reveal about brain dynamics in Parkinson’s disease. The Parkinsonian brain appears to exist on the boundary between phase-locked and nonsynchronous dynamics. Such a situation may be beneficial in the healthy state, as it may allow for easy formation and dissociation of transient patterns of synchronous activity which are required for normal motor behavior. Dopaminergic degeneration in Parkinson’s disease may shift the brain networks closer to this boundary, which would still permit some motor behavior while accounting for the associated motor deficits. Understanding the mechanisms of the intermittent synchrony in Parkinson’s disease is also important for biomedical engineering since efficient control strategies for suppression of pathological synchrony through deep brain stimulation require knowledge of the dynamics of the processes subjected to control.     

On Saint-Venant’s Problem with Concentrated Loads

The classical Saint-Venant problem is to find a solution of the traction problem of elastostatics in a finite cylinder ?? loaded over its bases. We prove that the problem has a unique solution for equilibrated surface forces $\hat{ \boldsymbol { s}}\in W^{-1,q}(\partial\Omega)$ , with q??(2?? ₀,+??) for some positive ? ₀ depending on ??. Hence $\hat{ \boldsymbol { s}}$ can model force acting on ???, concentrated on sets of zero Lebesgue surface measure of ???. Moreover, if $\hat{ \boldsymbol { s}}$ is equilibrated on each basis, we give a simple proof of the Toupin estimate expressing Saint-Venant??s principle.     

Time-harmonic Green’s functions for anisotropic magnetoelectroelasticity

Two-dimensional (2-D) and three-dimensional (3-D) time-harmonic Green’s functions for linear magnetoelectroelastic solids are derived in this paper by means of Radon-transform. Displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time-harmonic point force, point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2-D case and surface integrals over a unit sphere in 3-D case. This dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms. The singular terms coincide with the Green’s functions for the static problem and may be further reduced to closed form expressions. The proposed Green’s functions can be used in the corresponding boundary element method (BEM) formulation.     

A Constitutive Explanation of Manning’s Formula

Manning’s empirical formula for evaluating the mean velocity for a steady uniform turbulent flow in pressure conduits with circular cross-section and in a wide rectangular open channel, can be theoretically justified by introducing a virtual viscosity or mixing turbulent coefficient, depending on the velocity and the position of a particle according to Boussinesq’s hypothesis for the turbulent flow in a pipe. The balance equation yields an ordinary differential equation whose integration yield the distribution of the velocity.