The interaction of convection modes in a box of a saturated porous medium

Convection in an infinite layer of a porous medium occurs if the dimensionless Rayleigh number exceeds a critical value. This is also true for a box of a porous medium; however, each discrete modal solution has its own associated critical Rayleigh number. Usually just one mode will be generated at the onset of convection, but there are many critical box dimensions for which up to four modes share the same critical Rayleigh number, and all may be generated at the onset of convection. In such circumstances there will be a slow interchange of energy between the preferred modes. A perturbation method is applied to a system where three modes are generated at onset to yield a system of ordinary differential equations which govern the evolution of the amplitudes of the viable modes. Three unique cases arise, each with a different phase space structure. Critical boxes with ‘moderate’ aspect ratios are systematically categorised into these cases. While two of the examples represent the usual case where just one mode survives in the final state, the third example is a special case where it is possible for the three modes to coexist. The initial conditions determine which mode(s) will survive.     

Effects of wall conduction on natural convection in a porous triangular enclosure

Conjugate natural convection in a two-dimensional triangular enclosure filled with a porous medium is examined in this article. It is assumed that the solid vertical wall is of finite conductivity and that the temperature of the inclined wall is lower than that of the vertical wall, while the horizontal wall is adiabatic. A finite difference method is used to solve the governing equations of convection and conduction for different parameters as Rayleigh number, width of the vertical solid wall, aspect ratio of the enclosure and thermal conductivity ratio between solid and porous media. It is found that heat transfer increases with increasing Rayleigh number and aspect ratio of the triangle, decreasing wall thickness and with the increase of the wall conductivity.     

Effect of a non-constant magnetic field on natural convection in a horizontal porous layer heated from the bottom

An analytical and numerical investigation is conducted to study the effect of an electromagnetic field on natural convection in a horizontal shallow porous cavity filled with an electrically conducting fluid. The magnetic field is assumed to be induced by two long wires, carrying current, parallel to the horizontal boundaries of the system. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. The governing parameters of the problem under study are the thermal Rayleigh number, Ra, Hartmann number, Ha, position of the electrical wires, d, current intensity ratio, r, and aspect ratio of cavity, A. An analytical solution, valid for a shallow layer (A ? 1), is derived on the basis of the parallel flow approximation. The critical Rayleigh number, Ra _c , for the onset of motion is derived in closed form in terms of the parameters of the problem. For finite-amplitude convection the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.     

Freckle formation in a solidifying binary alloy

The effect of solutal solidification on the unidirectional solidification of a binary alloy cooled from below is considered. Soon after the onset of convection a mushy region often forms, and is accompanied by vigorous convection in the molten alloy above. In contrast, the fluid in the mush appears unaffected by the neighbouring flow and remains essentially quiescent. This work considers the nonlinear convective stability of the mush and determines a criterion for channelling in the mush to occur. The basic state is a similarity solution, so that a quasi-static approximation must be applied in order to apply conventional stability theory. Moreover, although the model for solidification is relatively simple, an analytical expression for linear stability is not available. Thus the series of equations arising from the nonlinear stability analysis lead to a complicated set of symbolic and numerical calculations. Stable finite-amplitude solutions are found for Rayleigh numbers larger than critical for all values of the chosen superheat. The nonlinear solutions demonstrate the possibility of channelling within the mush dependent upon the strength of convection. These finite-amplitude solutions are extended further by the calculation of a numerical solution of the model equations. The evolution of the stream function and the mass fraction is followed, the onset of convection and freckling can then be deduced from the numerical simulation. The onset of convection in the mush is found in terms of the mush Rayleigh number, and compares favourably with linear stability theory and experimental data. The onset of freckling is also given in terms of a Rayleigh number, but is sensitive to the initial conditions. This appears to explain the large disagreement found in experiments aimed at finding a criterion for freckling.     

Maximum density effects on natural convection from a discrete heater in a cavity filled with a porous medium

Summary. A numerical study of natural convection in a two-dimensional porous cavity saturated with water which possesses a density maximum in the vicinity of 3.98°C is carried out in the present paper. It is assumed that one of the cavity vertical walls is heated differentially by an isothermal discrete heater. The other vertical wall is cooled to a constant temperature, while the horizontal walls are adiabatic. Non-Boussinesq and Darcy models are used in the mathematical formulations. The effects of the location of the center of the discrete isothermal heater, the length of the heater and the aspect ratio of the porous cavity are studied for a wide range of modified Rayleigh numbers (50 Ra1000). For long heater and low Ra, it is required to place the heater in the middle of the vertical wall in order to get maximum heat transfer. For short heater and high Ra, fixing the heater in the upper half of the vertical wall leads to an enhancement of the heat transfer. When the aspect ratio A=0.5, the heater is far from the cold wall and hence more thermal resistance in this case reduces the average Nusselt number. On the other hand, higher values of average Nusselt numbers are found for specified Ra and L when the aspect ratio A is increased.     

Double-diffusive convection in a circular cylinder

Summary A theoretical study is made of the onset of steady double-diffusive convection in a circular cylinder of small to moderate aspect ratio. An eigenfunction expansion method is used to derive systems of amplitude equations for weakly nonlinear evolution of critical disturbances. It is shown that the nature of the convective solution near criticality depends strongly on the cylinder's aspect ratio; this is particularly the case at or near aspect ratios where two modes become unstable at the same Rayleigh number. There is also a strong dependence on Prandtl number, which is discussed.With 7 Figure     

Magnetomechanical effects in the Rayleigh region

A brief résumé of basic magnetism is followed by a consideration of the magnetization process in the absence of applied mechanical stress, in which the importance of internal stresses and inclusions is emphasized and which leads to the introduction of Néel's disperse field theory. The Rayleigh laws are interpreted in terms of "Preisach loops." Early work on the influence of stress on the magnetization process is discussed, leading on to a detailed consideration of this phenomenon in terms of three mechanisms which, for simplicity, are taken to be essentially distinct. These are, first, the alteration of the pressure on 90° domain walls by the application of stress; second, the alteration in the opposition term, which must clearly be in balance with this pressure for domain-wall equilibrium, and, finally, large-scale alterations in the overall domain structure. Existing experimental observations, on both magnetically soft and hard materials, are interpreted in terms of these three mechanisms.     

Acoustic microscopy and dispersion of leaky Rayleigh waves onrandomly rough surfaces: a theoretical study

A theoretical investigation of the dispersion of leaky Rayleigh waves propagating along one-dimensional (1-D) rough fluid-solid interfaces was carried out by simulating the measurement process of a line-focus beam acoustic microscope. The interface profiles were described in terms of their rms, also known as the roughness of the profile, autocorrelation length, and autocorrelation function. The reflectivity of the interfaces was calculated by using a second-order perturbation approach in the profile roughness. Theoretical V(z) curves were generated and analyzed to yield values of the phase velocity of the Rayleigh waves. The dependence of the Rayleigh wave velocity on the profile and material parameters was examined. Significant variations of the phase velocity were found for values of the roughness which are small compared to the shortest of the wavelengths involved in the scattering. The dispersion relations also showed considerable sensitivity to changes in mechanical properties typical of materials of engineering interest. In the low-frequency range, simulations indicated the dispersion of Rayleigh waves to be rather insensitive to the spectral content of the profile     

The similarity regime for natural convection in a vertical cylindrical well filled with an anisotropic porous medium

An integral method based on Lighthill’s analysis (Q J Mech Appl Math 6 (1953) 398–439) is carried out to study the similarity regime for penetration of convective heat transfer in a vertical cylindrical well filled with an anisotropic porous medium. The porous medium is anisotropic in permeability with its principal axes oriented in a direction that is oblique to the gravity vector. In the limit of the slenderness of the porous matrix, the penetration length of the convective flow and the heat-transfer rate are expressed in terms of the anisotropic properties of the porous medium, the modified Darcy–Rayleigh number and the aspect ratio of the geometrical configuration. A scale analysis is applied to predict the order of magnitudes involved in the similarity regime of the phenomenon. The conditions of existence of the similarity pattern is found to be dependent on the anisotropic parameters. It is demonstrated that both the anisotropic permeability ratio and the orientation angle of the principal axes have a strong influence on the heat-transfer rate and on the vertical penetration length into the well.     

DRBEM solution of natural convection flow of nanofluids with a heat source

This paper presents the dual reciprocity boundary element method (DRBEM) solution of the unsteady natural convective flow of nanofluids in enclosures with a heat source. The implicit Euler scheme is used for time integration. All the convective terms are evaluated in terms of DRBEM coordinate matrix. The vorticity boundary conditions are obtained from the Taylor series expansion of stream function equation. The results report that the average Nusselt number increases with the increase in both volume fraction and Rayleigh number. It is also observed that an increase in heater length reduces the heat transfer. The average Nusselt number of aluminum oxide-water based nanofluid is found to be smaller than that of copper-water based nanofluid. Results are given in terms of streamlines, isotherms, vorticity contours, velocity profiles and tables containing average Nusselt number for several values of Rayleigh number, heater length, volume fraction, and number of iterations together with CPU times.     

Convection in a shallow laterally heated cavity with conducting boundaries

Shallow cavity flows driven by horizontal temperature gradients are analysed over a range of Rayleigh numbersR and Prandtl numbers , whereR is comparable in size to the aspect ratioL(1). Eigenvalue calculations show the existence of a critical Prandtl number R > R _c (), below which the parallel core-flow structure is destroyed for Rayleigh numbersR>R _c(). For other Rayleigh numbers and Prandtl numbers the horizontal scale of influence of the end walls of the cavity is determined.     

Dispersionless single-mode fibers with trapezoidal-index profiles in the wavelength region near 1.5 microm

The transmission performance of trapezoidal-index profile single-mode fibers that can operate with zero dispersion in the 1.5-microm region has been investigated. The exact numerical calculations of propagation characteristics for these lightguides established a variety of relationships in terms of the parameters of a trapezoidal-index profile. One of the profile parameters that defines the shape of a trapezoid is the aspect ratio S, which ranges between 0 and 1. It was found that as the aspect ratio decreases from 1.0 to 0.3 the optimum core radius of the fiber gradually increases. But when S is <0.3, it approaches a constant equal to the value of the triangular-index profile (S = 0).     

Transfert de chaleur par convection naturelle dans une couche poreuse limitée par deux cylindres coaxiaux horizontauxHeat transfer by natural convection in a porous layer bounded by two coaxial horizontal cylinders

Natural convection in a porous medium is studied for the geometrical configuration defined by two concentric horizontal cylinders with isothermal and impermeable surfaces, representing a cylinder or a pipe insulated by a layer of porous material. Two methods of investigation are used to approach the phenomenon: experiments on a cell of large aspect ratio (cylinder length to outside diameter), and a numerical model on a two-dimensional plane area. The experimental results reveal three different regimes according to the value of the Rayleigh number. For a finite radial gradient, thermoconvective phenomena develop in the annular layer, but as long as these are not amplified too much, the isotherms remain practically concentric with the cylinders, and heat transfer occurs essentially through conduction. At higher Rayleigh numbers natural convection develops in the form of two-dimensional counter-rotational vortices which are symmetrical with respect to a vertical plane containing the axis common to both cylinders. The third regime, observed at higher Rayleigh numbers, corresponds to a fluctuating evolution of the temperature field which becomes three-dimensional. As the inside cylinder is warmer, instabilities occur in the upper part of the model causing some periodicity of the temperature distribution along a generatrix, with a period of about the radius difference. A relative increase in the heat transfer between the cylinders due to the instabilities is observed. A two-dimensional numerical model, using the method of finite differences, enables us to describe the first two regimes observed experimentally, and to define the temperature field and the streamlines for a given Rayleigh number. The total heat transfer calculated is represented by a correlation between the Nusselt and the rayleigh numbers for many values of the ratio of the radii.The different results are then compared and analysed from a physical point of view.     

Review Nonlinearity in piezoelectric ceramics

The paper presents an overview of experimental evidence and present understanding of nonlinear dielectric, elastic and piezoelectric relationships in piezoelectric ceramics. This topic has gained an increasing recognition in recent years due to the use of such materials under extreme operating conditions, for example in electromechanical actuators and high power acoustic transducers. Linear behaviour is generally confined to relatively low levels of applied electric field and stress, under which the dielectric, elastic and piezoelectric relationships are described well by the standard piezoelectric constitutive equations. Nonlinear relationships are observed above certain threshold values of electric field strength and mechanical stress, giving rise to field and stress-dependent dielectric (), elastic (s) and piezoelectric (d) coefficients. Eventually, strong hysteresis and saturation become evident above the coercive field/stress due to ferroelectric/ferroelastic domain switching. The thermodynamic method provides one approach to describing nonlinear behaviour in the intermediate field region, prior to large scale domain switching, by extending the piezoelectric constitutive equations to include nonlinear terms. However, this method seems to fail in its prediction of the amplitude and phase of high frequency harmonic components in the field-induced polarisation and strain waveforms, which arise directly from the nonlinear dielectric and piezoelectric relationships. A better fit to experimental data is given by the empirical Rayleigh relations, which were first developed to describe nonlinear behaviour in soft magnetic materials. This approach also provides an indication of the origins of nonlinearity in piezoelectric ceramics, in terms of ferroelectric domain wall translation (at intermediate field/stress levels) and domain switching (at high field/stress levels). The analogy with magnetic behaviour is also reflected in the use of Preisach-type models, which have been successfully employed to describe the hysteretic path-dependent strain-field relationships in piezoelectric actuators. The relative merits and limitations of the different modelling methods are compared and possible areas of application are identified.An erratum to this article can be found at     

Numerical solutions for the flow near the end of a shallow laterally heated cavity

The flow near the end of a shallow laterally heated cavity enters a nonlinear convective regime when the Rayleigh number R, based on cavity height, is of the same order of magnitude as the aspect ratio L (length/height). In the case of thermally insulated horizontal boundaries the end-region solution determines a correction to the flow and temperature fields throughout the cavity. Numerical solution are obtained for the end-region flow for several different Prandtl numbers and for a range of values of the scaled Rayleigh number R/L using a Dufort-Frankel multigrid method. The results are compared with asymptotic predictions of the motion in the conductive limit R/L 0 and the boundary-layer limit R/L .     

Simple approach to determining the minimum measurable stress length and stress measurement accuracy in distributed Brillouin sensing

A simple approach is proposed for quantifying the errors in measuring the Brillouin frequency shifts associated with stresses whose lengths are shorter than the pulse length. The smallest detectable Brillouin frequency shift is thus determined with respect to the size of the stressed sections and the frequency resolution. The lowest detectable frequency shift is found to be approximately 42% of the Brillouin gain natural linewidth. A worst-case iso-error curve that associates the minimum frequency shift to the length of the stressed region is derived. A minimum resolvable frequency shift and minimum detectable stress length are defined with an approach based on a Rayleigh equivalent criterion.     

Steady convection in porous media—II. The case of low rayleigh numbers and asymptotic expansions

In [6] we have proved that for sufficiently low Rayleigh numbers uniqueness holds; here we show that this only solution is analytic and regular with respect to the Rayleigh number. In connection with these general results, we study by the perturbation method the case of the domain confined between two concentric spheres which are maintained at different but uniform temperatures. Using the first three terms of the asymptotic expansions, two special values of the Rayleigh number are pointed out; they separate different types of flows—with one, two or three cells.     

Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh–Ritz technique

Axial buckling characteristics of single-walled carbon nanotubes (SWCNTs) including thermal environment effect are studied in this paper. Eringen’s nonlocal elasticity equations are incorporated into the classical Donnell shell theory to establish a nonlocal elastic shell model which takes small-scale effects into account. The Rayleigh–Ritz technique is implemented in conjunction with the set of beam functions as modal displacement functions to consider the four commonly used boundary conditions namely as simply supported–simply supported, clamped–clamped, clamped–simply supported, and clamped-free in the buckling analysis. Selected numerical results are presented to demonstrate the influences of small scale effect, aspect ratio, thermal environment effects and boundary conditions in detail. It is found that the value of aspect ratio has different effects on the critical axial buckling loads of SWCNTs in low and high temperature environments. Also, it is observed that the difference between the thermal axial buckling responses of SWCNTs relevant to various boundary conditions is more prominent for higher values of nonlocal elasticity constant.     

An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching

This is the third in a series of four papers in which problems of dynamic crack propagation are examined experimentally in large, thin sheets of Homalite-100 such that crack growth in an unbounded plate is simulated. In the first paper crack initiation resulting from stress wave loading to the crack tip as well as crack arrest were reported. It was found that for increasing rates of loading in the microsecond range the stress intensity required for initiation rises markedly. Crack arrest occurs abruptly without any deceleration phase at a stress intensity lower than that which causes initiation under quasi-static loading.In the second paper we analyze the occurrence of micro cracks at the front of the running main crack which control the rate of crack growth. The micro cracks are recorded by real time photography. By the same means it is shown that these micro cracks grow and turn away smoothly from the direction of the main crack in the process of branching.In the present paper we report results on crack propagation and branching. It is found that crack propagation occurs at a constant velocity although the stress intensity factor changes markedly. Furthermore, the velocity is determined by the stress wave induced intensity factor at initiation. The terminal velocity in Homalite-100 was found to be about half the Rayleigh wave speed (0.45 C _r ). These observations are analyzed in terms of a microcrack model alluded to in the second paper of this series. A mechanism for crack branching is proposed which considers branching to be a natural evolution from a cloud of microcracks that accompany and lead the main crack. These results are believed to apply to quasi-brittle materials other than Homalite-100 and the reasons for this belief are discussed briefly in the first paper of this series.In the final paper of the series the effect of stress waves impinging on the tip of a rapidly moving crack is examined. Waves affect the velocity and the direction of propagation as well as the process of crack branching.     

High-order perturbation theory for light scattering from a rough metal surface.

The angular distribution of the mean diffuse intensity scattered from a metal surface with one-dimensional roughness is studied with perturbation theory. From an approach based on the reduced Rayleigh equations in p polarization, exact perturbation terms up to eighth order in the height parameter are developed for surface roughness consistent with a stationary Gaussian process. The theory is evaluated for a number of cases in which surface plasmon polariton excitation is significant and produces effects such as backscattering enhancement. For surface roughness having a wide Gaussian power spectrum, it is found that the high-order terms lead to roughness-induced broadening of the backscattering peak. For rectangular spectra, two cases are studied in which backscattering effects are due to sixth- and eighth-order terms; both cases provide good comparisons with previously unexplained experimental results. Further, because of an eighth-order term, the diffuse intensity is shown to contain a specular peak that also relies on polariton excitation. This new effect is studied in detail and is found to arise from the constructive interference of contributions produced by multiple-scattering processes, although the time-reversed paths that produce backscattering enhancement are not essential to the specular effect.