The generalized Boussinesq equations and obliquely interacting solitary waves in a stratified fluid

In this paper, on the basis of Boussinesq’s shallow water theory, we establish the basic equations governing the motion of a stratified fluid, a kind of the generalized Boussinesq equations. And then by way of them, we study the weak interaction of two pairs of obliquely colliding solitary waves, give the second-order approximate solutions for wave profiles and maximum amplitudes, as well as conclude that when the included angle between the directions of propagation of impinging solitary waves is less than 120°, the effect of oblique interaction is stronger than that of the head-on one, but when the angle concerned is greater than 120°, the former is slightly weaker than the latter.     

An example of entropy balance in natural convection,Part 2: the thermodynamic Boussinesq equations

Numerical simulations of natural convection performed with the usual Boussinesq equations result in unbalanced irreversibility budget. The thermodynamic Boussinesq equations solve this problem, especially because they simulate production of kinetic energy within the fluid through its expansion and contraction. These fluid volume changes, without which natural convection would not occur, also induce heat transfer by piston effect. The piston effect, which appears then as an intrinsic component of buoyancy-induced natural convection, introduces the non-dimensional adiabatic temperature gradient as a control parameter of natural convection. To cite this article: M. Pons, P. Le Quéré, C. R. Mecanique 333 (2005).     

An example of entropy balance in natural convection,Part 1: the usual Boussinesq equations

Numerical simulations of natural convection in cavities performed with the usual Boussinesq equations result in an unbalanced irreversibility budget. Thermodynamic analysis shows that these equations represent a system that exchanges with the surroundings, not only two heat fluxes, but also two fluxes of mechanical energy: an input, that generates the fluid motion, and an output, due to viscous friction. After this analysis, the thermodynamic discrepancies can be explained. To cite this article: M. Pons, P. Le Quéré, C. R. Mecanique 333 (2005).     

A generalization of the method of averaging for systems with two time scales

In this paper we shall consider systems of the form x = ? f(t, ?t, x, y, ?), y = g(t,?t, x, y,?), where x and y are vectors of finite dimensions, f and g are assumed to be bounded for all t, and ? is a real parameter. Sufficient conditions are obtained for the existence of certain solutions which are bounded for all t. These solutions are shown to approach special solutions of a derived simpler averaged system of equations as ? → 0. Moreover, it is shown that there exists only one such bounded solution in the neighborhood of each special solution. In the special case when y is not present, it is shown that if a special solution is stable, solutions starting in nonlocal neighborhoods of this special solution approach the bounded solutions adjacent to it as t → ∞. These results generalize most of the existing work for systems of the type discussed here. Finally, we employ our results to study some problems of physical importance.     

The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales

As we know from experimental testing, the stiffness behaviour of carbon black-filled elastomers under dynamic deformations is weakly dependent on the frequency of deformation but strongly dependent on the amplitude. Increasing strain amplitudes lead to a decrease in the dynamic stiffness, which is known as the Payne effect. In this essay, we develop a constitutive approach of finite viscoelasticity to represent the Payne effect in the context of continuum mechanics. The starting point for the constitutive model resulting from this development is the theory of finite linear viscoelasticity for incompressible materials, where the free energy is assumed to be a linear functional of the relative Piola strain tensor. Motivated by the weak frequency dependence of the dynamic stiffness of reinforced rubber, the memory kernel of the free energy functional is of the Mittag Leffler type. We demonstrate that the model is compatible with the Second Law of Thermodynamics and equal to a fractional differential equation between the overstress of the Second Piola Kirchhoff type and the Piola strain tensor. In order to represent the dependence of the dynamic stiffness on the amplitude of strain, we replace the physical time by an intrinsic time variable. The temporal evolution of the intrinsic time is driven by an internal variable, which is a measure for the current state of the material's microstructure. The material constants of the model are estimated using a stochastic identification algorithm of the Monte Carlo type. We demonstrate that the constitutive approach pursued here represents the combined frequency and amplitude dependence of filler-reinforced rubber. In comparison with the micromechanical Kraus model developed for sinusoidal strains, the theory set out in this essay allows the representation of the stress response under arbitrary loading histories.     

Numerical solutions of fully non‐linear and highly dispersive Boussinesq equations in two horizontal dimensions

This paper investigates preconditioned iterative techniques for finite difference solutions of a high‐order Boussinesq method for modelling water waves in two horizontal dimensions. The Boussinesq method solves simultaneously for all three components of velocity at an arbitrary z‐level, removing any practical limitations based on the relative water depth. High‐order finite difference approximations are shown to be more efficient than low‐order approximations (for a given accuracy), despite the additional overhead. The resultant system of equations requires that a sparse, unsymmetric, and often ill‐conditioned matrix be solved at each stage evaluation within a simulation. Various preconditioning strategies are investigated, including full factorizations of the linearized matrix, ILU factorizations, a matrix‐free (Fourier space) method, and an approximate Schur complement approach. A detailed comparison of the methods is given for both rotational and irrotational formulations, and the strengths and limitations of each are discussed. Mesh‐independent convergence is demonstrated with many of the preconditioners for solutions of the irrotational formulation, and solutions using the Fourier space and approximate Schur complement preconditioners are shown to require an overall computational effort that scales linearly with problem size (for large problems). Calculations on a variable depth problem are also compared to experimental data, highlighting the accuracy of the model. Through combined physical and mathematical insight effective preconditioned iterative solutions are achieved for the full physical application range of the model. Copyright © 2004 John Wiley & Sons, Ltd.     

The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid

Ray theory is extended to consider the case of an accelerating disturbance which is producing waves in a rotating stratified fluid. Starting from the equations of motion, dispersion relations are derived for surface gravity waves, capillary waves, Rossby waves and internal-inertial waves. The wave system is studied in each case for the problem of a body starting impulsively from rest and for a body starting from rest and moving with constant acceleration.     

Convection in rotating porous media: The planetary geostrophic equations,used in geophysical fluid dynamics,revisited

In this paper, we give some mathematical results for a system of partial differential equations arising for instance in oceanography and meteorology. This model is called the planetary geostrophic equations or thermohaline equations. Different systems exist depending on the different viscous operators used to parameterize the small-scale processes. These choices made by physicists are empirical and are subject to numerous numerical studies. These parametrizations sometimes provide some models like Bénard convection in a rotating thin porous medium. Unfortunately this observation seems to be unknown to people working on geophysical fluid dynamics. Thus, in a first section, we discuss models which are sometimes known by several names. We compare the results obtained by people working on porous media and by people working on geophysical fluid dynamics. The last section is devoted to mathematical results. More precisely, we prove that it is possible in particular domains to extend some recent mathematical results by the use of some simple relations between the vertical velocity and the temperature. Many applications exist with such domains in meteorology. This system is also used in the food processing industry or in centrifugal filtration processes. In the last part, we indicate how to extend the results to a domain with a vanishing depth on the shore in the case of the hyperviscous parametrization. This has some applications in oceanography for instance. Received March 21, 2002 / Accepted 21 November 2002 // Published online May 9, 2003 / Brian Straughan, Durham RID="a" ID="a" e-mail: didier.bresch@math.univ-bpclermont.f RID="b" ID="b" e-mail: mamadou@math.univ-bpclermont.fr     

The stability of two stratified non-newtonian liquids in couette flow

Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.     

The stability of two stratified ”power-law“ liquids in couette flow

Consideration is given to the stability of the flow of two ”power-law“ liquids between two infinite parallel planes when one of the planes moves with constant velocity in its own plane. It is found that the ratios of the power-law parameters for each layer have a dramatic effect and can be chosen to destabilize the flow.     

The identification of two distinct laminar to turbulent transition modes in pipe flows accelerated from rest

A study was undertaken to investigate transition in a pipe flow accelerated from rest. Experiments were carried out on a vertical tube under a constant head of liquid: flow was initiated by opening a solenoid valve. A wall shear stress probe used in the role of an event recorder identified two transition events, separated by the passage of a turbulent to laminar front and a period of laminar flow. Evidence suggests that the first comprises a laminar to turbulent interface arising from a natural stable/unstable front moving up the tube as local conditions become met, while the second is consequent upon the formation of a continuous turbulent structure carried down the tube from the inlet by the bulk flow. The paper provides a formal explanation of a phenomenon which is typical of that which is observed in starting pipe flows with a disturbed inlet.     

The effect of rotation on conical wave beams in a stratified fluid

Experiments are conducted to test extant theory on the effect of uniform rotation on the angle of conical beam wave propagation excited by a sphere vertically oscillating at frequency in a density stratified fluid. The near-constant Brunt–Väisälä frequency stratification N produced in situ in a rotating cylindrical tank exhibits no effect of residual motion for the range of Froude numbers investigated. Good agreement between experiment and theory is found over the range of angles 15°<<65° using the synthetic schlieren visualization technique. In particular, the cut-off for wave propagation at =2, below which waves do not propagate, is clearly observed.     

On the frame dependence of constitutive equations. I. Heat transfer through a rarefied gas between two rotating cylinders

To verify the principle of material frame indifference a numerical calculation of the heat flux field in a rotating gas has been carried out based on the kinetic equation over wide ranges of gas rarefaction and angular velocity. It has been confirmed that a radial gradient of the temperature causes a tangential heat flux. Also, it has been found that the radial heat flux is affected by the rotation.On temporary leave from Department of Physics, Urals State University, 620083 Ekaterinburg, Russia