Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

Solid mechanics and its applications

Studies a nonlinear heat equation in a finite interval of space subject to a white noise forcing term. The equation without the forcing term exhibits several equilibrium configurations, two of which are stable. The solution of the complete forced equation is a stochastic process in space and time that has a unique stochastic equilibrium. The authors study this process in the limit of small noise, and obtain lower and upper bounds for the probability of large fluctuations. They then apply these estimates to calculate the transition probability between the stable configurations (tunnelling). This model problem can be interpreted as a rigorous version of some recent attempts to describe Euclidean quantum systems in terms of stochastic equilibrium states of a nonlinear stochastic differential equation in infinite dimensions. However, its significance goes beyond this situation and the authors' methods may be applicable to models in other areas of natural science.

https://iopscience.iop.org/article/10.1088/0305-4470/15/10/011/pdf

Large fluctuations for a nonlinear heat equation with noise

The design of analogue electronic experiments to investigate phenomena in nonlinear dynamics, especially stochastic phenomena, is described in practical terms. The advantages and disadvantages of this approach, in comparison to more conventional digital methods, are discussed. It is pointed out that analogue simulation provides a simple, inexpensive, technique that is easily applied in any laboratory to facilitate the design and implementation of complicated and expensive experimental projects; and that there are some important problems for which analogue methods have so far provided the only experimental approach. Applications to several topical problems are reviewed. Large rare fluctuations are studied through measurements of the prehistory probability distribution, thereby testing for the first time some fundamental tenets of fluctuation theory. It has thus been shown for example that, whereas the fluctuations of equilibrium systems obey time-reversal symmetry, those under non-equilibrium conditions are temporally asymmetric. Stochastic resonance, in which the signal-to-noise ratio for a weak periodic signal in a nonlinear system can be enhanced by added noise, has been widely studied by analogue methods, and the main results are reviewed; the closely related phenomena of noise-enhanced heterodyning and noise-induced linearization are also described. Selected examples of the use of analogue methods for the study of transient phenomena in time-evolving systems are reviewed. Analogue experiments with quasimonochromatic noise, whose power spectral density is peaked at some characteristic frequency, have led to the discovery of a range of interesting and often counter-intuitive effects. These are reviewed and related to large fluctuation phenomena. Analogue studies of two examples of deterministic nonlinear effects, modulation-induced negative differential resistance (MINDR) and zero-dispersion nonlinear resonance (ZDNR) are described. Finally, some speculative remarks about possible future directions and applications of analogue experiments are discussed.

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Analogue studies of nonlinear systems

Publisher Summary This chapter focuses on heat transfer behavior of viscoelastic fluid in turbulent pipe flow Although the asymptotic values of the heat transfer and friction factor can be calculated, there exist no firm criteria for determining whether asymptotic conditions exist Predictions of the intermediate values of the friction and heat transfer are not yet possible, even if the rheology and the thermal properties of the aqueous polymer solution are known To deals with the problems, the Weissenberg or Deborah number has to be taken into account The behavior of viscoelastic fluids flowing turbulently in noncircular channels or over external surfaces represents a relatively unexplored area of fluid mechanics Open channel flow of viscoelastic fluid is another interesting field currently being investigated The chapter concludes that to approach turbulent heat transfer behavior analytically, the usual and simplest method is to solve the uncoupled energy equation using the empirically determined velocity profile Hence, it is essential to understand the fluid mechanics of non-Newtonian fluids as well as the rheology

L. I. Sedov

Papers

Nonlinear tensor functions of several tensor arguments

Turbulent diffusion and destruction of polymer molecules in a pipe and boundary layer

Turbulent diffusion and degradation of polymer molecules in a pipe and boundary layer

MIKHAIL ALEKSEEVICH LAVRENT'EV (on his 70th birthday)

On the dynamic properties of gravitational fields