Effect of viscosity on the current layers emerging upon propagation of the Alfvén pulse in a hyperbolic magnetic field

The propagation of the Alfvén pulse in the vicinity of the X-point in the presence of viscosity is studied for the first time. It is shown that, in contrast to the case of magnetosonic perturbation, where the dynamic viscosity η (the point is that we are dealing with dimensionless quantities), which is small compared to the magnetic plasma viscosity ν, does not affect the flow, this influence is of primary importance in the Alfvén case. The magnitude of the steady-state current density is proportional to (vνη)^-1/4. It is also shown that at large times the distribution of the z-component of a magnetic field that is close to the distribution obtained in solving a linear problem is established in this significantly nonlinear problem. The effect of the heat conduction on this process is studied. Institute of Computational Technologies, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 11–16, November–December, 1999.     

Influence of conicity on the stress–strain state of a flexible orthotropic conical shell in a nonstationary magnetic field

A problem of magnetoelasticity for a flexible conical shell in a nonstationary magnetic field is solved. The effect of conicity on the stress–strain state of the shell is analyzed     

Crack tip field in a linear elastic–plastic strain-hardening material

The strip necking model for strain-hardening materials is studied in this paper, in which the stress distributed over the strip necking zone is assumed to be ultimate stress. The bi-linear stress–strain relation which can model certain features of plastic flow is adopted in this model. The stress and strain fields are calculated based on this model in this paper. The size of the strip necking region is determined by balancing the stress intensity factor due to remote loading with that due to assumed closing forces equal to the ultimate tensile strength of the material distributed over the strip necking zone. It is interesting that the strip necking region size and the crack tip opening displacement depend not only on the remote load, but also the material hardening parameters, which is different from the results of strip yield model. The results agree with experiments well, and the model has wider application.     

Stress–strain state of flexible ring plates of variable stiffness in a magnetic field

A nonlinear two-dimensional model of a magnetoelastic flexible current-carrying ring plate is developed.Asystem of nonlinear equations describing the stress–strain state of flexible current-carrying plates in nonstationary mechanical and electromagnetic fields is derived. The stress state of a flexible plate of variable stiffness in a magnetic field is determined     

Low-Prandtl-number Bénard–Marangoni convection in a vertical magnetic field

The effect of a homogeneous magnetic field on surface-tension-driven Bénard convection is studied by means of direct numerical simulations. The flow is computed in a rectangular domain with periodic horizontal boundary conditions and the free-slip condition on the bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. Two- and three-dimensional flows are computed for either vanishing or small Prandtl number, which are typical of liquid metals. The main focus of the paper is on a qualitative comparison of the flow states with the non-magnetic case, and on the effects associated with the possible near-cancellation of the nonlinear and pressure terms in the momentum equations for two-dimensional rolls. In the three-dimensional case, the transition from a stationary hexagonal pattern at the onset of convection to three-dimensional time-dependent convection is explored by a series of simulations at zero Prandtl number.     

Approximate analytical solution to oscillations of a conductor in a magnetic field

An approximate analytical solution to a system exhibiting oscillations of a conductor in a magnetic field which is controlled by a discrete waveform is sought by means of multiple scales. The system involves the use of a solenoid driven by a RLC circuit, coupled with a solid state relay (SSRL), to generate large electromagnetic forces acting on a conductor, which oscillates within the solenoid. The steady state response of the metal bar, in terms of oscillations is described. This solution is expressed in terms of system and circuit parameters, valid in the weakly nonlinear region, which is identified to be small oscillatory displacement near the center of the solenoid. By analyzing different cases of resonance, period-1 and period-2 like motions are identified and validated through experimental studies. The solution provides a guideline to design an effective control strategy so as to guide the system to a desirable attractor.     

On the stability of the simple shear flow of a Johnson–Segalman fluid

We solve the time-dependent simple shear flow of a Johnson–Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress/shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.     

Stability of resonant interfacial waves in the presence of uniform magnetic field

The aim of this work is to study the instability of interacting waves between two immiscible magnetic liquids. The effects of gravitation and a uniform normal magnetic field are taken into account. The method of multiple scales is used to determine the stability criteria of the considered problem. The various stability criteria are discussed both analytically and graphically. According to the numerical examples, we have remarked that the increase of the ratio of the permeability of the liquids appears to be the destabilizing effect of the magnetic field. The short waves below the critical wavenumbers are stable whereas a number of long waves are unstable. The viscosity effect on the stability criteria is a dual-role one, depending on the strength of the applied magnetic field.     

Simultaneous velocity and concentration measurements in the near field of a turbulent low-pressure jet by digital particle image velocimetry–planar laser-induced fluorescence

The main purpose of this work is to develop a method for simultaneous measurement of velocity and passive scalar concentration by means of digital particle image velocimetry and planar laser-induced fluorescence. Details of the implementation of the method are given, and the technique is applied to measurements of concentration and velocity in the centre-plane of a liquid jet with a Reynolds number of 6,000. The measurements are compared with large eddy simulations. Mean velocities and concentrations, fluctuating velocities and concentrations, and correlation between fluctuating velocities and concentrations are analysed for the first six diameters downstream of the jet exit. The general agreement between measured and simulated results was found to be good, in particular for mean quantities. Mean profiles are also found to be in good agreement with other experimental work on jets reported in the literature. The “whole-plane” measurement method was found to be very useful for detailed comparisons of turbulent statistics with simulated data. The inadequacy of models for turbulent mass transport based on the standard gradient diffusion concept is demonstrated through the experimental data. Received: 4 October 2000/Accepted: 27 November 2000     

Influence of the spatial variation of Poisson’s ratio upon the elastic field in nonhomogeneous axisymmetric bodies

The effect of a nonconstant Poisson’s ratio upon the elastic field in functionally graded axisymmetric solids is analyzed. Both of the elastic coefficients, i.e. Young’s modulus and Poisson’s ratio, are permitted to vary in the radial direction. These elastic coefficients are considered to be functions of composition and are related on this basis. This allows a closed form solution for the stress function to be obtained. Two cases are discussed in this investigation: first, both Young’s modulus and Poisson’s ratio are allowed to vary across the radius and the effect of spatial variation of Poisson’s ratio upon the maximum radial displacement is investigated; secondly, Young’s modulus is taken as constant and the change in the maximum hoop stress resulting from a variable Poisson’s ratio is calculated.     

Numerical simulation of counter-current flow field in the downcomer of a liquid–solid circulating fluidized bed

A comprehensive study on the hydrodynamics in the downcomer of a liquid–solid circulating fluidized bed (LSCFB) is crucial in the control and optimization of the extraction process using an ion exchange LSCFB. A computational fluid dynamics model is proposed in this study to simulate the counter-current two-phase flow in the downcomer of the LSCFB. The model is based on the Eulerian–Eulerian approach incorporating the kinetic theory of granular flow. The predicted results agree well with our earlier experimental data. Furthermore, it is shown that the bed expansion of the particles in the downcomer is directly affected by the superficial liquid velocity in downcomer and solids circulation rate. The model also predicts the residence time of solid particles in the downcomer using a pulse technique. It is demonstrated that the increase in the superficial liquid velocity decreases the solids dispersion in the downcomer of the LSCFB.     

Superposition of finite-amplitude waves propagating in a principal plane of a deformed Mooney–Rivlin material

Here we consider finite-amplitude wave motions in Mooney–Rivlin elastic materials which are first subjected to a static homogeneous deformation (prestrain). We assume that the time-dependent displacement superimposed on the prestrain is along a principal axis of the prestrain and depends on two spatial variables in the principal plane orthogonal to this axis. Thus all waves considered here are linearly polarized along this axis. After retrieving known results for a single homogeneous plane wave propagating in a principal plane, a superposition of an arbitrary number of sinusoidal homogeneous plane waves is shown to be a solution of the equations of motion. Also, inhomogeneous plane wave solutions with complex wave vector in a principal plane and complex frequency are obtained. Moreover, appropriate superpositions of such inhomogeneous waves are also shown to be solutions. In each case, expressions are obtained for the energy density and energy flux associated with the wave motion.     

Investigating the effects of shear rate on the collapse time in a gas–liquid system by lattice Boltzmann

In order to evaluate the effects of shear rate on the time of collapse process in a two-phase; one component system, a free energy lattice Boltzmann model is developed. Considering a system consists of two droplets in the surrounding vapor it is observed that, for various physical and geometrical conditions, imposing shear flow has similar effects on the collapse time. Actually, for each case there is an optimum shear rate of collapse that is corresponded to highest collapse rate and smallest time. In all test cases, for shear rates smaller than the optimum value, increasing the shear rate raise collapse rate and reduce its time. But for higher shear rates, increasing the shear rate has reverse effect and decreases the collapse rate. The results also show that, higher droplet radius ratio, viscosity ratio, surface tension and interface thickness lead to a higher optimum shear rate of collapse while higher density ratio of liquid and vapor decreases that.     

Influence of a homogeneous axial magnetic field on Taylor–Couette flow of ferrofluids with low particle–particle interaction

Experimental results concerning the stability of Couette flow of ferrofluids under magnetic field influence are presented. The fluid cell of the Taylor–Couette system is subject to a homogeneous axial magnetic field and the axial flow profiles are measured by ultrasound Doppler velocimetry. It has been found that an axial magnetic field stabilizes the Couette flow. This effect decreases with a rotating outer cylinder. Moreover, it could be observed that lower axial wave numbers are more stable at a higher axial magnetic field strength. Since the used ferrofluid shows a negligible particle–particle interaction, the observed effects are considered to be solely based on the hindrance of free particle rotation.     

Wall material and roughness effects on transmittable shear stresses of magnetorheological fluids in plate–plate magnetorheometry

A direct comparison of plate–plate magnetorheometry results for nonmagnetic (titanium/brass) and ferromagnetic plates is presented, using a modified Anton Paar magnetocell MRD180/1T. Necessary corrections to derive the true flux density in the magnetorheological fluid (MRF) from the online Hall probe reading and to account for the gap opening effect caused by normal forces on shear stress and flux density are addressed. The measured shear stress versus magnetic flux density characteristics agree in the low flux density regime <0.1 T but yield distinctly higher transmittable shear stresses for ferromagnetic plates at elevated flux densities (49% increase at 1 T for 90% by weight carbonyl iron powder (CIP) and 84% for 85% by weight CIP). Remarkably, the normal force, if corrected for its magnetostatic part, remains independent of the type of plates up to about 0.6 T. We address the role of normal forces, of magnetic interactions between CIP and wall, as well as the role of wall roughness in a solid body friction model. A systematic variation of wall properties and materials was achieved by introducing both a modular rotor and stator, which ease the variation of the walls in contact to the MRF. The transmittable shear stress of nonmagnetic plates (e.g., brass) may be increased up to the level of ferromagnetic disks by a higher wall roughness or by grooves. No shear stress increase is obtained for grooves in ferromagnetic plates, which is explained by the different local flux density modulation at the grooves for ferromagnetic compared to nonmagnetic plates. Finally, we address the effect of ferromagnetic and nonmagnetic coatings on brass and steel disks, and show that, e.g., a layer of CIP on brass efficiently increases the transmittable shear stress.     

Generation of the tollmien–schlichting wave in a supersonic boundary layer by two sinusoidal acoustic waves

The problem is solved using parabolized equations of stability for threedimensional perturbations of a compressible boundary layer on a flat plate. Nonlinearity is taken into account by quadratic terms that are most significant in estimates of the viscous critical layer of the stability theory. These terms are determined by the total field of two acoustic perturbations, and the equations become linear and inhomogeneous. The calculations are performed for one acoustic wave being stationary in the reference system fitted to the plate for Mach numbers M=2 and 5. Solutions are presented, which are identified very accurately with Tollmien–Schlichting waves at a rather large distance from the plate edge.     

The effect of the irregular interface geometry in deformation and fracture of a steel substrate–boride coating composite

Presented in this paper is a computational analysis of the mechanisms involved in plastic deformation and fracture of a composite with coating under compressive and tensile loading. Using a steel specimen surface-hardened by diffusion borating, a role of the irregular geometry of the interface between the base material and hardened surface layer is investigated. In order to describe the mechanical behavior of the steel substrate and brittle coating, use is made of a plastic flow model including isotropic strain hardening and a fracture model, respectively. Using the Huber fracture criterion, the model takes into account the difference in the critical strength values for different types of local compressive and tensile states. It is shown that the irregular, serrated shape of the substrate–coating interface retards propagation of a longitudinal crack into this coating and prevents it from spalling under external compression of this composite. It is found out that even in the case of a simple uniaxial compression of the mesovolumes of this composite the boride “teeth” are subjected to tensile stresses, whose values are comparable with those of the external compressive load, and the direction of crack propagation and the general fracture behavior largely depend on the external loading conditions.     

Simulations of vertical jet penetration using a filtered two-fluid model in a gas–solid fluidized bed

The influence of a vertical jet located at the distributor in a cylindrical fluidized bed on the flow behavior of gas and particles was predicted using a filtered two-fluid model proposed by Sundaresan and coworkers. The distributions of volume fraction and the velocity of particles along the lateral direction were investigated for different jet velocities by analyzing the simulated results. The vertical jet penetration lengths at the different gas jet velocities have been obtained and compared with predictions derived from empirical correlations; the predicted air jet penetration length is discussed. Agreement between the numerical simulations and experimental results has been achieved.