Rayleigh Problem Research Articles

On the Rayleigh problem for a viscoelastic fluid

On the Rayleigh problem for a viscoelastic fluid

Hydromagnetic Rayleigh problem in a rotating fluid

An analysis of the MHD Rayleigh problem of a rotating fluid above a plate is carried out, when the magnetic Reynolds number of the flow is small and the magnetic field is fixed on the plate. An exact solution is obtained with the help of the Laplace transform technique. It is found that the effect of the magnetic field increases the primary flow and decreases the secondary flow.

Experimental method of measuring energy in the Bénard–Rayleigh problem

The energy content, introduced by the thermodynamics of steady states, is measured in the Bénard–Rayleigh problem using a heat fluxmeter connection. We describe the experimental apparatus, which can turn around a horizontal axis, thus inducing or destroying convective structures. After calibration, we were able to measure the heat flux through the fluid layer as well as the structural energy of the convective cells. These structural energy values confirm that the energy content is a steady-state function.

A Few Applications of the Nee-Kovasznay Model for Shear Flow Turbulence

The Nee-Kovasznay model is applied to the three turbulent shear flows ; the wall region of the turbulent boundary layer, the plane mixing layer and the turbulent Rayleigh problem. The model equation is solved analytically, partly by the use of a technique analogous to the integral method for turbulence prediction. The former two applications are intended to determine the two unknown parameters in the model by comparing the theory with experiment, the results of which are A =0.20 and B =2.2. In the third problem approximate solutions for the mean velocity and the turbulent viscosity are obtained and compared with experiments. Agreements are satisfactory. The theory shows also some disagreements, which may be due to the lack of the concept of `coherent structure' in the model.

Corrections and extensions to “propagation of a vortex sheet in viscoelastic liquids—the rayleigh problem”,J. non-newtonian fluid mech., 8 (1981) 337–347

Corrections and extensions to “propagation of a vortex sheet in viscoelastic liquids—the rayleigh problem”,J. non-newtonian fluid mech., 8 (1981) 337–347

Determination of the density perturbation at the wall for the Rayleigh problem

The Rayleigh problem, a fundamental time-dependent problem of gas kinetics, is studied in the context of the constant collision frequency BGK model. An analytical result, expressed in terms of the stationary solution, is obtained and numerically evaluated. Asymptotic solutions in both the small and large time limit are also presented and compared to results of other theories.

Rayleigh's Problem in Magnetohydrodynamics for a Non‐Conducting Plate

Rayleigh's Problem in Magnetohydrodynamics for a Non‐Conducting Plate

Homogenization in Stochastic Differential Geometry

It is well known that a diffusion process on Euclidean space can be rigorously considered as the limit of a sequence of transport processes. This idea, which dates at least to Rayleigh's problem of random flight [3] has now received a general treatment by modern probabilistic methods [2, 10]. In addition, we have shown that a suitable transport approximation remains valid for the Brownian motion of any complete Riemannian manifold [11]. In another direction several authors [1, 4, 5, 6, 7, 8] have considered parallel displacement, i.e. parallel displacement of vectors along Brownian motion curves in a manifold. The purpose of this paper is to show that the stochastic parallel displacement can be rigorously considered as the limit of parallel displacement along the paths of a transport process. The transport approximation introduces an extra velocity variable which disappears in the limit, hence the term homogenizatio7i. In addition to its elementary geometric appeal, our approach has the advantage of producing the following coordinate-free definition of the infinitesimal operator of the stochastic parallel displacement:

Propagation of a vortex sheet in viscoelastic liquids—the Rayleigh problem

It is shown by an example that unsteady shearing motions can give rise to vortex sheets in a special class of viscoelastic liquids only. Further, the congruence between the quantitative predictions of an initial—boundary value problem, namely the Reyleigh problem, and the theory of propagation of singular surfaces is demonstrated.

New experimental apparatus for the study of the Bénard–Rayleigh problem

A new experimental system to measure the equivalent thermal conductivity of a liquid with regard to the Bénard–Rayleigh problem was constructed. The liquid is enclosed within walls of polymethylmethacrylate between two copper plates in which there are thermocouples to measure the difference in temperature between the lower and upper surfaces of the layer of liquid. Heat flux is measured by means of a linear heat fluxmeter consisting of 204 thermocouples in series. The fluxmeter was calibrated and the linear relationship that exists between the heat flux and the emf generated was verified. The thermal conductivity of the polymethylmethacrylate employed was measured and measurements of the equivalent conductivity in cylindrical boundaries of two silicone oils were made. The critical value of the temperature difference and the contribution of the convective process to the transmission of heat were determined.

On the Rayleigh problem of gas kinetics

On the Rayleigh problem of gas kinetics

Effects of mass transfer on the free convective flow in the Stokes' problem for an infinite vertical limiting surface

An exact analysis of the mass transfer effects on the free convection flow of an incompressible viscous fluid past an impulsively started infinite vertical (wall) limiting surface (Stokes's or Rayleigh's problem) has been carried out. Expressions for the velocity, temperature, species concentration and skin friction are obtained by using the Laplace transform technique. The velocity field and the skin friction are shown graphically for air (P=0.71) and mercury (P=0.025). The effects ofG (Grashof number),Gc (the modified Grashof number) andSc (Schmidt number) are considered qualitatively during the course of discussion.

Hydromagnetic free convection effects on the stokes problem for an infinite vertical plate

The extension of the problem of Stokes (also called Rayleigh's problem) to magnetohydrodynamic for the flow past an infinite, non-conducting and non-magnetic, vertical plate, is studied. The plate is assumed to move after receiving an initial impulse. The magnetic Reynold's number is taken small enough so that the induced magnetic field is negligible. Expressions, in closed form for the velocity and the skin friction are obtained by applying the Laplace transform technique and the results obtained for various values of the parameters G (Grashof number) and M (Magnetic parameter) are given in graphical form. The paper is concluded with a discussion of the results obtained.

Effects of mass transfer and free convection currents on MHD Stokes' problem for a vertical plate

An exact analysis is made of the effects of mass transfer and free convection currents on MHD Stokes' (Rayleigh's) problem for the flow of an electrically conducting, incompressible, viscous fluid past an impulsively started vertical plate, under the action of a transversely applied magnetic field. The heat due to viscous and Joule dissipation and the induced magnetic field is neglected. During the course of the discussion, the effects of heating (Gr < 0, Gr = Grashof number) or cooling (Gr > 0) of the plate by the free convection currents, Gr m (modified Grashof number), Sc (Schmidt number) and M (Hartmann number) and the velocity and skin-friction are studied.

Free convection effects on MHD Stokes problem for a vertical plate

An exact analysis of MHD Stokes problem (also Rayleigh's problem) for the flow of an electrically conducting, incompressible, viscous fluid past an impulsively started vertical plate, under the action of transversely applied magnetic field, is carried out. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The effects of the external heating or cooling of the plate by the free convection currents are studied. It is observed that reverse flow occurs when the plate is being heated by the free convection currents. The skin-friction increases owing to greater heating of the plate and decreases due to greater cooling of the plate.

Magnetohydrodynamic, unsteady, one-dimensional viscous flows—I: A general solution in the presence of non-magnetic body-forces

The problem of the unsteady, laminar, magnetohydrodynamic flow over an infinite flat plate is considered, in the presence of non-magnetic body forces and of a transverse magnetic field. The conditions for the existence of a regular flow are determined and the general solution is given. As an application, a generalized solution to the classical Rayleigh's problem is carried out and a contra-example is reported.

Generalized calculus and its applications

This paper gives a simple development of generalized differentiation and integration with examples and applications. The concept of symmetric derivatives is discussed with examples and applications to complex potential functions. A comparison is made between ordinary and symmetric derivatives. Some important examples of continuous non‐differentiable functions are stated. The calculus of generalized functions and its physical significance are also presented. With a short historical development, the fractional calculus is discussed with examples and simple applications. The different approaches to derivatives and integrals of fractional orders are elaborated with illustrations. The use of the Laplace transforms for determining the fractional derivatives and integrals is discussed with simple examples. Some simple and important applications of fractional calculus to Abel's problem of isochrones, volume flux of water through a river dam, the heat conduction problem and the Rayleigh problem of unsteady boundary flows are investigated in detail.

Optimum load capacity of a slider bearing lubricated by a fluid with couple stress

The slider profile which gives the maximum load capacity of a slider bearing lubricated by an incompressible fluid with couple stress is considered. The optimum profile is a step function with riser location and step height ratio depending on the couple stress parameter. The results reduce to those of the classical Rayleigh problem when the couple stress parameter tends to zero. The optimum load capacity increases with the couple stress parameter.

Hydromagnetic Rayleigh problem with Hall effect

An exact solution of the governing equations of motion for the flow, induced by an impulsive start of a long flat plate, under the action of a strong transverse magnetic field with Hall currents, has been obtained in a closed form. Some interesting results along with the asymptotic cases of very small and very large times are discussed, accounting for the influence of the governing parameters on the flow character. It is seen that the time taken by the flow to attain the steady state increases due to Hall effect. The final steady flow is characterised by a boundary layer, of which the thickness increases due to Hall current.

Exact transform solution of the one-dimensional special Rayleigh problem

We examine the full-range initial-value problem for velocity relaxation in a one-dimensional ensemble of Rayleigh test-particles in an arbitrary heat-bath at unit mass-ratio (the special Rayleigh problem). A Laplace transformation method is used to obtain an explicit solution in two independent parity components. Of these, the even component, giving the speed relaxation is obtained exactly and agrees with our previous result obtained by the singular eigenfunction method; the odd component, explicit to within a Laplace inverse provides the time-dependent flux of particles with given velocity. By numerical inversion of the latter, we obtain both time-dependent velocity distributions and the velocity autocorrelation function for equilibrium fluctuations. In keeping with the singular character of the transport equation, the velocity ACF proves to be appreciably non-exponential in character, behaving asymptotically as t−5/2e−t independently of the heat-bath distribution. The complex admittance of a system of charged Rayleigh test-particles is also derived and shown to lead to expected behaviour for dissipation to the heat-bath and phase-lag of response to the applied field.