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.     

Important aspects in the formulation of solid–fluid debris-flow models. Part II. Constitutive modelling

This article points at some critical issues which are connected with the theoretical formulation of the thermodynamics of solid–fluid mixtures of frictional materials. It is our view that a complete thermodynamic exploitation of the second law of thermodynamics is necessary to obtain the proper parameterizations of the constitutive quantities in such theories. These issues are explained in detail in a recently published book by Schneider and Hutter (Solid–Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context, 2009), which we wish to advertize with these notes. The model is a saturated mixture of an arbitrary number of solid and fluid constituents which may be compressible or density preserving, which exhibit visco-frictional (visco-hypoplastic) behavior, but are all subject to the same temperature. Mass exchange between the constituents may account for particle size separation and phase changes due to fragmentation and abrasion. Destabilization of a saturated soil mass from the pre- and the post-critical phases of a catastrophic motion from initiation to deposition is modeled by symmetric tensorial variables which are related to the rate independent parts of the constituent stress tensors.     

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).     

On Thermodynamic Consistency of Turbulent Closures

The problem of the implementation of the second law of thermodynamics for the determination of the thermodynamic consistency of solutions determined by turbulent closures is considered for incompressible fluids. The possibility of the application of the methods of thermodynamics to constraining constitutive laws describing turbulent flow features, but not material behaviour, is discussed. It is shown that the ordinary realizability conditions requiring non-negative values of the averaged squared fluctuations are necessary and sufficient conditions determining the thermodynamic consistency of a process governed by a closure model. Because turbulent closures are not universal, using the second law of thermodynamics to constrain them can impose unnecessary restrictions on the models, when the turbulent entropy is considered as a constitutive quantity. The notion and validity of different forms of the turbulent entropy is discussed. It is found that the form of the turbulent entropy originating from the analogy between the turbulent kinetic energy and absolute temperature contradicts the principle of irreversibility. In a particular case of small temperature fluctuations, the second law yields correct constraints, if the turbulent entropy is assumed not to be a constitutive quantity, but a variable governed by an evolution equation of special form generated by the balance equation for internal energy. Received 14 October 2000 and accepted 30 May 2001     

A viscoelastic damage model for polycrystalline ice,inspired by Weibull-distributed fiber bundle models. Part II: Thermodynamics of a rank-4 damage model

We consider a viscoelastic–viscoplastic continuum damage model for polycrystalline ice. The focus lies on the thermodynamics particularities of such a constitutive model and restrictions on the constitutive theory which are implied by the entropy principle. We use Müller’s formulation of the entropy principle, together with Liu’s method of exploiting it with the aid of Lagrange multipliers.