Correspondence between de Saint-Venant and Boussinesq. 1: Birth of the Shallow–Water Equations

The Shallow–Water Equations (SWEs), also referred to as the de Saint-Venant equations, constitute the current governing mathematical tool for free-surface water flows. These include, e.g., flood flows in rivers and in urban zones, flows across hydraulic structures as dams or wastewater facilities, flows in the environmental fields, glaciology, or meteorology. Despite this attractiveness, the system of two partial differential equations has an exact mathematical solution only for a limited number of problems of practical relevance.This historical work on the SWEs is based on a correspondence between two 19th-century scientists, de Saint-Venant and Boussinesq. Their well-known papers are thus commented from the point of development of their theory; the input of both scientists is evidenced by their writings, and comments of both to each other that led to what is commonly known as the SWEs. Given the age difference of the two of 45 years, the experienced engineer de Saint-Venant, and the mathematician Boussinesq, two eminent researchers, met to discuss not only problems in hydraulics, but in physics generally. In addition, their correspondence embraced also questions in ethics, religion, history of sciences, and personal news.The results of the SWEs cease to hold if streamline curvature effects dominate; this includes breaking waves, solitary and cnoidal waves, or non-linear waves in general. In most other cases, however, the SWEs perfectly apply to typical flows in engineering practice; they are considered the fundamental system of equations describing open channel flows. This work thus provides a background to its birth, including lots of comments as to its improvement, physical meanings, methods of solution, and a discussion of the results. This paper also deals with the steady flow equations, gives a short account on the main persons mentioned in the Correspondence, and provides a summary of further developments of the SWEs until 1920.     

Dividing a complex reaction involving a hypervalent iodine reagent into three limiting mechanisms by ab initio molecular dynamics

The electrophilic N‐trifluoromethylation of MeCN with a hypervalent iodine reagent to form a nitrilium ion, that is rapidly trapped by an azole nucleophile, is thought to occur via reductive elimination (RE). A recent study showed that, depending on the solvent representation, the S_N2 is favoured to a different extent over the RE. However, there is a discriminative solvent effect present, which calls for a statistical mechanics approach to fully account for the entropic contributions. In this study, we perform metadynamic simulations for two trifluoromethylation reactions (with N‐ and S‐nucleophiles), showing that the RE mechanism is always favoured in MeCN solution. These computations also indicate that a radical mechanism (single electron transfer) may play an important role. The computational protocol based on accelerated molecular dynamics for the exploration of the free energy surface is transferable and will be applied to similar reactions to investigate other electrophiles on the reagent. Based on the activation parameters determined, this approach also gives insight into the mechanistic details of the trifluoromethylation and shows that these commonly known mechanisms mark the limits within which the reaction proceeds. © 2015 Wiley Periodicals, Inc.     

On flows of granular materials

This article reviews the behavior of materials made up of a large assemblage of solid particles under rapid and quasi static deformations. The focus is on flows at relatively high concentrations and for conditions when the interstitial fluid plays an insignificant role. The momentum and energy exchange processes are then primarily governed by interparticle collisions and Coulomb-type frictional contact. We first discuss some physical behavior —dilatancy, internal friction, fluidization and particle segregation — that are typical to the understanding of granular flows. Bagnold's seminal Couette flow experiments and his simple stress analysis are then used to motivate the first constitutive theories that use a microstructural variable — the fluctuation energy or granular temperature — governing the subscale fluctuating motion. The kinetic theories formalize the derivation of the field equations of bulk mass, momentum and energy, and permit derivation of constitutive relations for stress, flux of fluctuation energy and its dissipation rate for simple particle assemblages and when frictional rubbing contact can be ignored. These statistical considerations also show that formulation of boundary conditions needs special attention. The frictional-collisional constitutive behavior in which both Coulomb-type rubbing contact and collisional encounters are significant are discussed. There is as yet no rigorous formulation. We finally present a phenomenological approach that describes rapid flows of granular materials under simultaneous transport of heat and close with a summary of stability analyses of the basic flow down an inclined plane.Dedicated to Professor Dr.-Ing. Franz Gustav Kollmann on the occasion of his sixtieth brithday     

A variational approach to a micro-structured theory of solid-fluid mixtures

Summary In this paper, we present a micro-structured model for describing global deformations of heterogeneous mixtures. In particular, for a saturated solid-fluid mixture, we regard the solid volume fraction as a microstructural parameter so as to enlarge the space of admissible deformations with respect to the classical theory of mixtures. According to the variational approach, the governing equations are obtained as the stationarity of a suitable action functional. The micro-structured model is then forced to establish a second-gradient mixture theory, by introducing among the considered state parameters a suitable internal constraint. Finally, we determine under which (integrability) conditions the additional balance laws, typically employed to close the theory of porous media endowed with the volume fraction, can fit the variational framework. The authors wish to thank Prof. Francesco dell'Isola from University of Rome La Sapienza for his constructive criticism about the variational approach to continuum mechanics and the interpretation of the volume-fraction balance law.     

Hall effects on the unsteady hydromagnetic oscillatory flow of a second-grade fluid

An exact solution of an oscillatory flow is constructed in a rotating fluid under the influence of an uniform transverse magnetic field. The fluid is considered as second-grade (non-Newtonian). The influence of Hall currents and material parameters of the second-grade fluid is investigated. The hydromagnetic flow is generated in the uniformly rotating fluid bounded between two rigid non-conducting parallel plates by small amplitude oscillations of the upper plate. The exact solutions of the steady and unsteady velocity fields are constructed. It is found that the steady solution depends on the Hall parameter but is independent of the material parameter of the fluid. The unsteady part of the solution depends upon both (Hall and material) parameters. Attention is focused upon the physical nature of the solution, and the structure of the various kinds of boundary layers is examined. Several results of physical interest have been deduced in limiting cases.     

A Variational Principle for the Revised Goodman–Cowin Theory with Internal Length

In the present study a variational principle is proposed for the revised Goodman–Cowin theory with internal length for cohesionless granular materials (Fang et al. in Continuum Mech Thermodyn in press). The balance equations of the internal variables employed in the theory in equilibrium states, the equilibrium expressions of the constitutive variables and the corresponding natural boundary conditions are derived by use of the proposed variational principle for both cases of compressible and incompressible grains. It is demonstrated that the derived results coincide with those obtained by use of the thermodynamic analysis. The current work serves as a supplementary variational verification of the constitutive theory proposed in Fang et al. (in Continuum Mech Thermodyn in press).     

ToF-SIMS measurements with topographic information in combined images

In 2D and 3D time-of-flight secondary ion mass spectrometric (ToF-SIMS) analysis, accentuated structures on the sample surface induce distorted element distributions in the measurement. The origin of this effect is the 45° incidence angle of the analysis beam, recording planar images with distortion of the sample surface. For the generation of correct element distributions, these artifacts associated with the sample surface need to be eliminated by measuring the sample surface topography and applying suitable algorithms. For this purpose, the next generation of ToF-SIMS instruments will feature a scanning probe microscope directly implemented in the sample chamber which allows the performance of topography measurements in situ. This work presents the combination of 2D and 3D ToF-SIMS analysis with topographic measurements by ex situ techniques such as atomic force microscopy (AFM), confocal microscopy (CM), and digital holographic microscopy (DHM). The concept of the combination of topographic and ToF-SIMS measurements in a single representation was applied to organic and inorganic samples featuring surface structures in the nanometer and micrometer ranges. The correct representation of planar and distorted ToF-SIMS images was achieved by the combination of topographic data with images of 2D as well as 3D ToF-SIMS measurements, using either AFM, CM, or DHM for the recording of topographic data.     

On the thermodynamic consistency of the equivalence principle in continuum damage mechanics

We consider theories of continuum damage mechanics involving damage effect variables of different tensorial ranks. It turns out that orthotropic damage together with the use of Lemaitre's equivalence principle for the elastic part does not allow thermodynamic potentials such as the free enthalpy to exist. As the existence of these potentials is, however, a strict thermodynamic requirement, a theory employing orthotropic damage in this way is inconsistent. We show that the use of a rank-4 damage effect variable allows a consistent use of the equivalence principle.