Rayleigh Problem Research Articles

Optimum slider profile of a slider bearing lubricated with a micropolar fluid

The slider profile which gives the maximum load capacity for a slider bearing lubricated with a micropolar fluid is considered. The optimum profile is a step function with a riser location and a step height ratio that depend on the two basic dimensionless parameters of micropolar fluid dynamics. The results reduce to those of the classical Rayleigh problem when the coupling parameter tends to zero. The optimum load capacity increases with the coupling parameter.

The Rayleigh problem for a wavy wall

The problem of the flow generated in a viscous fluid by the impulsive motion of a wavy wall is treated as a perturbation about the known solution for a straight wall. It is shown that, while a unified treatment for high and low Reynolds numbers is possible in principle, the two limiting cases have to be treated separately in order to get results in closed form. It is also shown that a straightforward perturbation expansion in Reynolds number is not possible because the known solution is of exponential order in that parameter. At low Reynolds numbers the waviness of the wall quickly ceases to be of importance as the liquid is dragged along by the wall. At high Reynolds numbers on the other hand, the effects of viscosity are shown to be confined to a narrow layer close to the wall and the known potential sohtion emerges in time. The latter solution is a good illustration of the interaction between a viscous fluid field and its related inviscid field.

Unsteady-state slip of a gas near an infinite plane with diffusion-mirror reflection of molecules

On the basis of a “two-point” approximation for the distribution function, an approximate analytical solution is obtained to the Rayleigh problem describing (for an arbitrary moment of time) the unsteady-state slip of a rarefied gas near a surface with diffusion-mirror reflection of molecules. In the solution of the problem it was postulated that the characteristic value of the macroscopic velocity of the gas is small in comparison with the speed of sound. An approximate analogy is established with the propagation of free vibrations in an electrical line of infinite length. An investigation is made of the limiting transition σ→ 0 (σ is the fraction of mirror-reflected particles) in an exact solution of the Boltzmann problem. It is shown that, with σ ≪ 1, the rate of slip for any given moment of time is determined from the hydrodynamic equations of motion. An investigation is made of the nonsingularity of the limiting transitions (t→∞, σ→ 0; σ→ 0, t→ ∞) with determination of the rate of thermal slip.

Nonstationary Rayleigh problem for magnetized plasma

A solution is obtained for the problem of onset of oscillatory convection in a horizontal layer of fully ionized plasma located in constant vertical gravitational and magnetic fields. In case of significant anisotropy in the thermal conductivity with isotropic viscosity (only the electrons are magnetized) the onset of convection at large Hartmann numbers is accompanied by oscillations if the ''magnetic Prandtl number'' is much larger than unity. The results obtained here are a natural generalization of the well-known Chandrashekar's criteria and can be used in the theory of MHD wave generation in a strongly magnetized plasma.

Power dissipated by a harmonically bound Brownian particle in an ac field: A regeneralization of the Rayleigh problem

The rate of energy dissipated by a harmonically bound Brownian particle also subjected to an ac external field is considered with the use of a non-Markoffian kinetic equation which results from including memory effects in the description of the particle motion. We find a general expression, Eq. (8), for the power dissipation and also give some numerical results for comparison with earlier results obtained using a Markoffian kinetic equation. The Brownian-particle velocity autocorrelation function is also briefly discussed.

The stability of planetary waves on an infinite beta‐plane

The stability of a planetary wave on an infinite ß‐plane is examined. There are two parameters in the problem, (a) ? = Uκ2/β where U is the velocity amplitude of the planetary wave and κ its wavenumber, and (b) the direction of the wavenumber of the planetary wave. For large M, the problem reduces to the Rayleigh problem for a sinusoidal velocity distribution. The growth rate of the most unstable wave is 0.27 Uκ. 38 % of the energy lost by this wave is transferred to wavenumber 0.59 κ and 61 % is transferred to wavenumber 1.16 κ. As ? decreases and β effects become more important, there are fewer unstable waves, but unstable waves can always be found. For small M, the disturbance comprises two waves which form a resonantly interacting triad with the primary wave. Some geophysical applications are discussed. It is shown, for instance, that for strong currents like the Gulf Stream, this type of instability is much more important than inertial instability. Also, observations indicate that the dominant eddies in the energy spectra of both ocean and atmosphere have ? of order unity. But they also have length scales comparable with the radius of deformation, indicating the importance of more physical processes than those considered here.

Rayleigh waves in a nonhomogeneous layer resting on a half-space: PMM vol. 37, n≗5, 1973, pp. 895–899

We study oscillations of Rayleigh wave type, spreading along the surface of a nonhomogeneous (along the depth) elastic layer resting on an elastic half-space. We assume that the Rayleigh velocity on the diurnal surface is less than the transversal velocity in the layer, but larger than the longitudinal velocity in the half-space. In the neighborhood of high frequencies we compute the phase velocity of the wave, similar to the usual Rayleigh wave in the Rayleigh problem for the homogeneous half-space. It is shown that this phase velocity is not real, as a consequence of which the wave diesout as it propagates along the diurnal surface.

Corrigendum and addendum to 'Rayleigh's problem in collisionless plasmas'

Corrigendum and addendum to 'Rayleigh's problem in collisionless plasmas'

Onset of convection regimes with double period in a periodic field of external forces

We investigate the onset of secondary convective flows in the Rayleigh problem (in a horizontal layer of a viscous incompressible liquid with free boundaries) in the presence of a parameter that varies with time and has a period T, namely the temperature gradient or the intensity of the gravitational field.

A variational solution of the Rayleigh problem for a power law non-Newtonian conducting fluid

An investigation is made of the magnetic Rayleigh problem where a semi-infinite plate is given an impulsive motion and thereafter moves with constant velocity in a non-Newtonian power law fluid of infinite extent. The nonstationary flow of this electrically conducting fluid in a transverse magnetic field is then analyzed.

The rayleigh problem for a non-newtonian electrically conducting fluid

The rayleigh problem for a non-newtonian electrically conducting fluid

Rayleigh's problem in collisionless plasmas

The initial motion of the unsteady plasma boundary layer flow over an infinite flat plate with heat transfer effects under the influence of an applied magnetic field is studied. Based on a plasma kinetic formulation, small-time exact solutions are derived for the linearized Rayleigh's problem in plasmas consisting of collisionless electrons embedded in immobile ions. Detailed discussions are presented concerning the time-dependent distributions of a number of interesting plasma flow properties, such as number density, velocity component in the direction of plate motion, velocity slip at the plate and skin friction coefficient. The present method of solution seems to be applicable to a large class of plasma boundary layer flow problems in which the interactions between moving plasmas and solid surfaces are important.

The Rayleigh problem for a slightly diffusive density-stratified fluid

A doubly-infinite sloping flat plate, initially at rest in a slightly diffusive viscous density-stratified fluid, starts to move impulsively with a constant velocity along the line of greatest slope. The resulting flow is found to be an unsteady motion superimposed on a steady diffusion-induced flow, which is present throughout. Laplace transform methods give solutions which are valid either in an essentially non-diffusive outer layer or in a diffusive inner layer. The impulsive start sets up oscillations in the outer layer. These gradually die out, and a steady diffusive flow develops.A glass plate was towed vertically through stratified brine, into which aluminium particles were introduced. The flow velocities deduced from the particle motions confirmed the theoretical predictions.

Rayleigh's Problem of Non-Newtonian Fluids

Rayleigh's Problem of Non-Newtonian Fluids

Rayleigh problem in slip flow with transverse magnetic field

An analytical continuum solution of the Rayleigh problem in slip flow with applied magnetic field is obtained using a modified initial condition and slip boundary conditions. The results are uniformly valid for all times and show that the velocity slip and the local skin friction coefficient remain almost unaffected by the imposition of the magnetic field for small times. They increase however with the magnetic field for large times. The present results reduce to the corresponding results of the hydrodynamic case when there is no magnetic field.

Energy Transients in Harmonic Oscillator Systems

The approach to energy equilibration is discussed for a system of N isolated classical oscillators, coupled or uncoupled. Exact solutions can be given for the kinetic and potential energies vs time, for a simple boundary condition which distributes the potential energy equally between all the normal modes at zero time and which gives zero kinetic energy to all the normal-mode oscillators at zero time. Most solutions are ergodic but a special class is nonergodic, i.e., the potential energy vs time shows exact recursions. The half-life for approaching the equilibrium value of potential or kinetic energy is of the order of the reciprocal maximum frequency of the oscillators. The fluctuations from equilibrium are treated by an extension of the method of Rayleigh's problem of random walks. The results are of interest in connection with the ergodic theorem of statistical mechanics.

On the application of Luke's weight coefficients for evaluating the linearized rayleigh problem in a rarefied gas flow

On the application of Luke's weight coefficients for evaluating the linearized rayleigh problem in a rarefied gas flow

Nonlinear, rarefied Rayleigh's problem.

Discrete ordinate method applied to nonlinear unsteady rarefied Rayleigh flow problem with heat transfer

Turbulent Rayleigh shear flow

Townsend has derived a relation between mean vorticity and Reynolds stress valid in the wall layer of a turbulent flow. The vorticity Ω appears as a function of the local stress τ and its gradient. Such a relation is better suited for use in the vorticity equation than in the momentum equation. The Rayleigh problem, whose vorticity equation is simply ∂Ω/∂t = ∂2τ/∂y2, is introduced as a setting for Townsend's theory. Certain wall speed programmes are shown to generate Rayleigh layers that are exactly self-similar in the fully turbulent part of the flow. Those layers correspond to Clauser's equilibrium boundary layers. A formal analogy between the two families is found; the analogy becomes quantitatively exact in the limit of infinite Reynolds number. The Rayleigh problem is posed in similarity form. A composite non-linear, ordinary differential equation for the stress profile is deduced from a two-layer model incorporating Townsend's relation for the wall layer and Clauser's constant eddy-viscosity assumption for the outer layer. The profile depends on the wall-speed programme selected and on two empirical constants: the combination λ = √(k)/k of Clauser's k and Kármán's k, and Townsend's constant B. Closed-form solutions for arbitrary λ and B are obtained in two important cases: constant wall stress, analogous to constant pressure above a boundary layer, and zero wall stress, corresponding to continuous separation. The velocity profile in the wall region of a continuously separating Rayleigh layer is found to depend sensitively on B.

Reduction of the Number of Variables in Systems of Partial Differential Equations, with Auxiliary Conditions

Reduction of the Number of Variables in Systems of Partial Differential Equations, with Auxiliary Conditions