Kinetic theory of disperse systems

A kinetic equation for the motion of solid particles in a liquid or gas is derived on the basis of the Fokker-Planck-Kolmogorov diffusion equation for the N particle distribution function. It is shown that, under appropriate assumptions, Bogolyubov's method can also be applied to equations of diffusion type. The obtained kinetic equation is a generalization of the one proposed earlier in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–132, January–February, 1980.I thank V. P. Myasnikov for suggesting the problem and for helpful discussions.     

Self-gravitating stellar systems and non-extensive thermostatistics

After introducing the fundamental properties of self-gravitating systems, we present an application of Tsallis generalized entropy to the analysis of their thermodynamic nature. By extremizing the Tsallis entropy, we obtain an equation of state known as the stellar polytrope. For a self-gravitating stellar system confined within a perfectly reflecting wall, we discuss the thermodynamic instability caused by its negative specific heat. The role of the extremum as a quasi-equilibrium is also demonstrated from the results of N-body simulations.Received: 10 July 2003, Accepted: 27 October 2003, Published online: 3 February 2004PACS: 98.10. + z, 05.70.Ln, 05.20.-yCorrespondence to: M. Sakagami     

Decay of confined,two-dimensional,spatially periodic arrays of vortices: A numerical investigation

The disarrangement of a perturbed lattice of vortices was studied numerically. The basic state is an exponentially decaying, exact solution of the Navier-Stokes equations. Square arrays of vortices with even numbers of vortex cells along each side were perturbed and their evolution was investigated. Whether the energy in the perturbation grows somewhat before it decays or decays monotonically depends on the initial strength of the vortices of the basic state, the extent of lateral confinement and the structure of the perturbation. The critical condition for temporally local instability, i.e. the critical amplitude of the basic state that must be exceeded to allow energy transfer from the basic state to the perturbation, is discussed. In the strongly confined case of a square lattice of four vortices the appearance of enchancement of global rotation is the result of energy transfer from the basic state to a temporally local unstable mode. Energy is transferred from the basic state to larger-scaled structures (inverse cascade) only if the scales of the larger structures are inherently contained in the initial structure of the perturbation. The initial structure of the double array of vortices is not maintained except for a very special form of perturbation. The facts that large scales decay more slowly than small scales and that, when non-linearities are sufficiently strong, energy is transferred from one scale to another explain the differences in the disarrangement process for different initial strengths of the vortices of the basic state. The stronger vortices, i.e. the vortices perturbed in a manner that increases their strength, tend to dominate the weaker vortices. The pairing and subsequent merging (or capture) of vortices of like sense into larger-scale vortices are described in terms of peaks in the evolution of the square root of the palinstrophy divided by the enstrophy.     

Kinetic theory of colloidal suspensions: morphology, rheology, and migration

Smoluchowski kinetic equation governing the time evolution of the pair correlation function of rigid sphericalparticles suspended in a Newtonian fluid is extended to include particle migration. The extended kinetic equation takes into account three types of forces acting on the suspended particles: a direct force generated by an interparticle potential, hydrodynamic force mediated by the host fluid, and the Faxén-type forces bringing about the across-the-streamline particle migration. For suspensions subjected to externally imposed flows, the kinetic equation is solved numerically by the proper generalized decomposition method. The imposed flow investigated inthe numerical illustrations is the Poiseuille flow. Numerical solutions provide the morphology (the pair correlation function), the rheology (the stress tensor), and the particle migration.     

A two-dimensional,higher-order,elasticity-based micromechanics model

A new, two-dimensional (2D) homogenization theory is proposed. The theory utilizes a higher-order, elasticity-based cell model (ECM) analysis. The material microstructure is modeled as a 2D periodic array of unit cells where each unit cell is discretized into four subregions (or subcells). The analysis utilizes a (truncated) eigenfunction expansion of up to fifth order for the displacement field in each subcell. The governing equations for the theory are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through an order consistent with the order of the subcell displacement field. The formulation is carried out independently of any specified constitutive models for the behavior of the individual phases (in the sense that the general governing equations hold for any constitutive model). The fifth order theory is subsequently specialized to a third order theory. Additionally, the higher order analyzes reduce to a theory equivalent to the original 2D method of cells (MOC) theory when all higher order terms are eliminated. The proposed 2D theory is the companion theory to an equivalent 3D theory [T.O. Williams, A three-dimensional, higher-order, elasticity-based micromechanics model, Int. J. Solids Struc., in press].Comparison of the predicted bulk and local responses with published results indicates that the theory accurately predicts both types of responses. The high degree of agreement between the current theory results and published results is due to the correct incorporation of the coupling effects between the local fields.The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, two-dimensional microstructures.     

On the downstream development and breakup of systems of trailing-line vortices

The downstream evolution of two configurations of trailing-line vortex systems is considered and analysed from the point of view of bi-global instability. The results lead to predictions for the breakup of the systems considered, based primarily on stability analysis. Throughout, the analysis is entirely rational from the theoretical (asymptotic, large Reynolds number) point of view, with a relatively long developmental lengthscale in the streamwise direction for the base flow, and a relatively short streamwise wavelength for the bi-global stability analysis. As such, the stability modes are inviscid in nature, and therefore likely to be one of the dominant instability mechanisms. The issue of adverse streamwise freestream pressure gradients is also addressed, and it is suggested that these can lead to a rapid breakdown/up of a vortex system, analogous to the axisymmetric case discussed by Hall (Ann Rev Fluid Mech 4:195?C218, 1972).     

Discrete vortices in the wake of various profiles in subsonic and supersonic two-dimensional gas flow

The Mach number dependence of the Strouhal number, the frequency of discrete vortices, the vortex velocity, and other parameters are determined in the wake of wedges and flat plates for low angles of attack. The studies were made using high-speed motion-picture photography through a Schlieren system and with photomultipliers. The results are presented in tabular and graphical form.Notation h transverse distance - l longitudinal distance between vortices - V freestream velocity in m/sec - n_v vortex frequency for one row of vortex street in sec - M freestream Mach number - S₁ Strouhal number based on projection of the model onto the plane perpendicular to the freestream direction - S₂ Strouhal number calculated from the wake neck width d₂ for M>1 - R Reynolds number calculated from d - R_* critical Reynolds number - model apex angle - angle of attack - L length in flow direction in mm The author wishes to thank G. I. Petrov for his interest in the study and his advice.     

Kinetic theory and rheology of dilute polymer solutions

Summary A summary is given of some recent attempts to relate the results of the kinetic theory of rigid and flexible macromolecules to continuum mechanics results.With 1 table     

Rotation,collapse, and scattering of point vortices

Relative equilibria, collapse, and scattering of point vortices in the plane are studied. Vortex systems with arbitrary choice of circulations are considered. An ordinary differential equation satisfied by polynomials with roots at the vortex positions is found. Explicit expressions for vortex double-ring configurations in the cases of two regular polygons, three regular polygons, and a quadrangle with a digon are obtained.     

Kinetic model of bubbly flow

A kinetic approach based on the approximate calculation of the fluid flow potential and formulation of Hamilton’s equations for generalized coordinates and momenta of bubbles is employed to describe processes of collective interaction of gas bubbles moving in an inviscid incompressible fluid. Kinetic equations governing the evolution of the distribution function of bubbles are derived. These equations are similar to Vlasov equations. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 130–138, September–October, 2000.     

Periodic wave patterns of two-dimensional Boussinesq systems

We prove the existence of a large family of two-dimensional travelling wave patterns for a Boussinesq system which describes three-dimensional water waves. This model equation results from full Euler equations in assuming that the depth of the fluid layer is small with respect to the horizontal wave length, and that the flow is potential, with a free surface without surface tension. Our proof uses the Lyapunov–Schmidt method which may be managed here, contrary to the case of gravity waves with full Euler equations. Our results are in a good qualitative agreement with experimental results.     

Shakedown of coupled two-dimensional discrete frictional systems

Recent results have established that Melan's theorem can be applied to discrete elastic systems governed by the Coulomb friction law only when the normal contact reactions are uncoupled from the tangential (slip) displacements. For coupled systems, periodic loading scenarios can be devised which lead to either shakedown or cyclic slip depending on the initial condition. Here we explore this issue in the simplest coupled system involving two contact nodes. The evolution of the system ‘memory’ is conveniently represented graphically by tracking the instantaneous condition in slip-displacement space. The frictional inequalities define directional straight line constraints in this space that tend to ‘sweep’ the operating point towards the safe shakedown condition if one exists. However, if the safe shakedown region is defined by a triangle in which two adjacent sides correspond to the extremal positions of the two frictional constraints for the same node, initial conditions can be found leading to cyclic slip. The critical value of a loading parameter at which this occurs can be determined by requiring that three of the four constraint lines intersect in a point. Below this value, shakedown occurs for all initial conditions. Similar concepts can be extended to multi-node systems.     

The stability and development of tip and root vortices behind a model wind turbine

An inclined turbulent jet discharging a passive scalar into a turbulent crossflow is investigated under conditions of favorable, zero and adverse streamwise pressure gradient. Experiments are conducted in water by means of magnetic resonance velocimetry and magnetic resonance concentration measurements. The velocity ratio and density ratio are equal to one for all cases. The flow configuration is relevant to film cooling technology, the molecular properties of the fluid being immaterial in the fully turbulent regime. Under favorable pressure gradient (FPG), the streamwise acceleration tilts the jet trajectory toward the wall, which would be beneficial for the film cooling performance. However, the counter-rotating vortex pair is strengthened in the accelerating flow by streamwise stretching. Also, the crossflow boundary layer is significantly thickened by increasingly adverse pressure gradient, which affects the mass transfer from the jet. Overall, the more intense counter-rotating vortices and the thinner boundary layer associated with increasingly FPG enhance the scalar dispersion into the main flow, hampering the film cooling performance in terms of surface effectiveness.     

Kinetic theory of low-temperature microscopic crack nucleation in crystals

Microscopic crack nucleation in a strained crystal is regarded as kinetic structural transformation associated with a change in the short-range order in atomic distribution and with excess-volume localization under deformation. The short-range order in the atomic distribution found in a strained medium is described by two order parameters. In the local approximation of nonequilibrium thermodynamics, equations describing the evolution of the order parameters are derived. Solutions for the decomposition of the homogeneous state with the formation of pores are proposed and microcracking conditions are considered.