On the effect of a high-frequency magnetic field on the instability of a plasma

The action of a high-frequency magnetic field on low-frequency instabilities of a plasma is considered. The harmonics of the high-frequency field that appear under these conditions are taken into account. It is shown that their effect on the reduction of the growth rate of the instability is weak. However, in analyzing the oscillation spectrum consideration of these harmonics is necessary, since they have the same growth rate as does the suppressed instability. It is well known that in a plasma situated in a strong magnetic field H_0Z unstable oscillations develop, the most dangerous being electrostatic oscillations that propagate almost perpendicular to the magnetic field (k k_¦). These oscillations have the form of troughs extended almost along H_0Z. If it were possible by some method to create conditions under which particles of one species (either electrons or ions) would have time to traverse the distance between the humps of the troughs in a time considerably shorter than the period 2/ of the unstable oscillations, then the potential of the instability would be smoothed out (i.e., the instability would be suppressed). This may be achieved by exciting a high-frequency magnetic field H₁(t) = H₁ cos t( ) in the plasma, which is oriented perpendicular to the constant field H_0z. Then the crossing of the humps by the particles is achieved as a result of motion with thermal velocities along the resultant curved magnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 10–14, March–April, 1971.The author thank A. V. Gordeev for fruitful discussions.     

Study of convection in a liquid layer with vibrating forces

We consider convection in a gravity force field in a liquid enclosed in a vessel which is vibrating along the vertical axis according to the lawa/ sin t ( ).Time-averaged convection equations describing the basic motion in first approximation are derived in [1]. In addition, criteria are introduced in [1] which determine the initiation of convection in this case, and a model problem is considered: the case of spatially periodic disturbances. It is found that in this case high frequency vibrations stabilize the state of relative rest.In the present paper we consider convection in a liquid layer between two horizontal planes whose equations are z=±l/2.The temperatures T₁ and T₂ are specified on the upper and lower surfaces, respectively. It is shown (§1) that for small values of the vibrational criterion the principle of stability variation is satisfied. In this case there is a variational principle (§2) from which it follows that high frequency vibrations prevent the occurrence of convection in the horizontal liquid layer. With the aid of the variational principle a calculation is made of the dependence of the values of Rayleigh criterion on the vibrational criterion.The author wishes to thank I. B. Sinomenko and V. I. Yudovich for their continued interest in this study.     

On mass transfer in an acoustic field

Chemical processes governed by the laws of diffusion kinetics can be intensified by elastic oscillations. It is also known that the rate of combustion of liquid and solid fuels changes markedly with the onset of acoustic vibrations in the combustion chamber. Despite the extensive application of vibrational processes in technology, the mechanisms of heat and mass transfer in the presence of vibrations are not well known. The aim of this research was to analyze the mass transfer from a sphere in an acoustic field.Notation angular frequency of oscillations - wavelength - R characteristic dimension of axisymmetric body - s amplitude of displacement of fluid particles in a plane acoustic wave - B amplitude of oscillation velocity - x, y longitudinal and transverse coordinates - u, v longitudinal and transverse velocity components - v kinematic viscosity - U — A(x) cos t velocity of potential flow - ⁺ thickness of momentum boundary layer - ^– thickness of diffusion boundary layer - m dimensionless concentration - m_* concentration of diffusing component at surface of vaporization - t time - D diffusion coefficient - average density of mixture - erf error function - r radius of axisymmetric body - R Reynolds number - P diffusion Prandtl number - time average - N, N_d Nusselt numbers based on radius and diameter respectively - pulsating component of velocity or concentration - o stationary component of velocity or concentration In conclusion, the authors wish to thank S. S. Kutateladze and I. A. Yavorskii, who supervised the present work.     

The effects of temperature-dependent fluid properties on free convective flow along a semi-infinite vertical plate by the presence of radiation

The effects of temperature-dependent density, viscosity and thermal conductivity on the free convective steady laminar boundary layer flow by the presence of radiation for large temperature differences, are studied. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected due to the large temperature difference between the plate and the fluid. The nonlinear boundary layer equations, governing the problem under consideration, are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the density/temperature parameter n, the thermal conductivity parameter , the viscosity/temperature parameter _r and the radiation parameter F are examined on the velocity and temperature fields as well as the coefficient of heat flux and the shearing stress at the plate.     

Convection in an Obliquely Heated Thin Porous Elliptic Ring

The problem of convection in a thin porous elliptic ring located in an impermeable rock mass is considered. The geothermal gradient is assumed to deviate by a certain angle from the vertical axis of the ellipse. An analytic solution of the problem is obtained in the hydraulic approximation. The stability of the solutions obtained is investigated for a circular ring. The profile of the thermal anomaly in the outer mass due to convection in the contour is given.     

Stabilization of the interface instability of moving magnetizable fluids

A study is made of the stabilization of the interface between two moving magnetizable fluids by means of an external magnetic field H={H₀ cos t, H₀ sin t, 0} of circular polarization. The fluids are assumed to be ideal, incompressible, nonconducting, and electrically neutral. An equation of motion is derived for the perturbed interface. The Borg criterion is used to obtain sufficient conditions of stability of small perturbations of the interface; these conditions relate the amplitude H₀ and the frequency of the external magnetic field to the characteristic parameters of the problem and the wave vector k. The dependence of H₀ and on the modulus of the wave vector is investigated. The obtained results are compared with the results of [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 118–122, July–August, 1980.     

Heat transfer in a thin layer of a viscous heavy noncompressible liquid under wavy flow conditions

The article describes a method for calculating the flow of heat through a wavy boundary separating a layer of liquid from a layer of gas, under the assumption that the viscosity and heat-transfer coefficients are constant, and that a constant temperature of the fixed wall and a constant temperature of the gas flow are given. A study is made of the equations of motion and thermal conductivity (without taking the dissipation energy into account) in the approximations of the theory of the boundary layer; the left-hand sides of these equations are replaced by their averaged values over the layer. These equations, after linearization, are used to determine the velocity and temperature distributions. The qualitative aspect of heat transfer in a thin layer of viscous liquid, under regular-wavy flow conditions, is examined. Particular attention is paid to the effect of the surface tension coefficient on the flow of heat through the interface.Notation x, y coordinates of a liquid particle - t time - v and u coordinates of the velocity vector of the liquid - p pressure in the liquid - c_v, , T,, andv heat capacity, thermal conductivity coefficient, temperature, density, and viscosity of the liquid, respectively - g acceleration due to gravity - surface-tension coefficient - c phase velocity of the waves at the interface - T_w wall temperature - h₀ thickness of the liquid layer - u₀ velocity of the liquid over the layer Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 147–151, July–August, 1970.     

Nonlinear theory of the propagation of pulses of electromagnetic waves through a plasma boundary

A solution is obtained for the problem of the propagation of electromagnetic waves of arbitrary form through a plasma boundary on condition that the length of the wave train is much greater than the wave length. A solution is found both for the case of a wide spectrum of width much greater than the plasma frequency ₀, as well as for a narrow spectrum. The results obtained enable us to draw conclusions about the time and space variation of the shape of electromagnetic pulses in a plasma.The passage of high frequency electromagnetic waves through a plasma is similar to that of a beam of charged particles [1, 2]. This is associated with the fact that decay processes are similar to Cerenkov radiation effects. The dynamics of the development of transverse wave instabilities in a uniform Isotropic plasma were studied in [2] assuming that the wave phase behaves stochastically. It was calculated here that instabilities develop quite differently in the case of a wide frequency spectrum than in the case of a narrow monochromatic spectrum. If we can speak of transverse quanta diffusion effects in the field of the generated longitudinal quanta in the first case, and if the resulting effects are closely similar to the nonlinear effects arising when beam instability develops [3, 4], then the development of instabilities in the case of a narrow spectrum leads to the appearance of red satellites in the transverse wave spectrum differing from the basic frequency by a quantity ₀ (=1, 2, 3,...). In this case the development of the instability corresponds to a tendency for a plateau over the satellites to appear.Attention should however be drawn to the fact that the dynamics of instability development in a semibounded plasma may be quite different. This is associated first with the different values of group velocities of transverse and longitudinal waves, and what is also important, with the effect of longitudinal wave accumulation in the boundary region if the length of the wave train is sufficiently large. The treatment of a similar problem for beam instabilities in paper [5] showed that a narrow transition layer may arise with a transverse wave energy density greatly in excess of the energy density of the injected beam. In what follows we examine the part played by boundary effects in the passage of pulses of electromagnetic waves through the boundary of the plasma. The cases of both narrow and wide spectra are considered. We note that in the case of narrow spectra the wave train must necessarily be greatly in excess of ^–1, and the effects of the accumulation of oscillations will be appreciable.The phases of both transverse waves, and also generated longitudinal waves are assumed to be stochastic quantities. The boundary effects which have been treated may be applied both in the generation of longitudinal waves necessary for the effective acceleration of particles in a plasma as well as in the modulation and alteration of the initial transverse wave spectrum. It should also be stressed that these effects which have been considered could be applied for turbulent plasma diagnostics, as has already been pointed out in [2].The authors are grateful to Ya. B. Fainberg, M. S. Rabinovich, I. S. Danilkin, and M. D. Raizer for their interest in the paper and for valuable criticisms.     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

The problem of the electric field of an electrode with a pre-electrode potential drop in a medium with tensor conductivity

We investigate the nonlinear current distribution in an electrode of finite dimensions with a pre-electrode layer in which the potential locally depends on the current density. The electrode is in contact with a medium of anisotropic conductivity caused by the Hall effect. The problem is reduced to the solution of a nonlinear integrodifferential equation. It is shown that the structure of the field is determined by the Hall parameter and the form of the volt-ampere characteristic in the pre-electrode layer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 19–23, January–February, 1971.     

Experimental snap-through boundaries for acoustically excited,thermally buckled plates

This paper presents some recent experimental results on the dynamic snap-through behavior of a clamped, rectangular plate subject to thermal loading and intense acoustic excitation. The likelihood of snap-through oscillations is characterized in terms of boundaries separating regions of snap-through and no snap-through in the parameter space. Two scenarios are considered. First, using tonal inputs, the regions of snap-through are mapped in the sound pressure level—input frequency domain ((SPL, ) plane). Second, random acoustic inputs are used, and the effect of varying the overall sound pressure level and frequency bandwidth are investigated ((SPL, ) plane). Several nonlinear characteristics are evident and discussed.     

A new technique for ultrasonic nondestructive evaluation of adhesive joints: Part I. Theory

In this paper we proposed a new technique for ultrasonic nondestructive evaluation (UNDE) of adhesively bonded joints. We report an exact solution to the problem of reflection and tarnsmission of a plane, time-harmonic, longitudinal wave through anN-layered medium. The solution is valid for perfectly elastic as well as linear-viscoelastic materials, and for isotropic as well as anisotropic materials (for example, fiber-reinforced composite) so long as the wavepropagation vector coincids with one of the material coordinates. The transfer function,H ^*() is defined as the trans-mitted (or reflected) field normalized with respect to the incident field. A closed-form solution forH ^*() for the case of an adhesive joint (consisting of two adherends joined by an adhesive layer) immersed in an elastic fluid is derived. A detailed analysis of the sensitivity ofH ^*() to the wave speed and thickness of the adherends and the adhesive is carried out. An experimental verification of the analysis is the subject of Part II of this paper.Paper was presented at the 1991 SEM Spring Confernce on Experimental Mechanics held in Milwaukee, WI on June 9–13.     

Boundary layer on a vaporizing surface

The article discusses the laminar boundary layer on the vaporizing surface of liquid hydrogen over which there is a flow of molecular hydrogen. The pressure p in the boundary layer corresponds to the saturation temperature of the oxygen T₀, which is lower than the temperature of the oncoming flow T, but higher than the temperature of the surface of the liquid hydrogen. Under such conditions, within the boundary layer there is condensation of oxygen in the volume, which leads to the formation of drops of liquid oxygen of different sizes. It is proposed in the article that, with the condensation of gas in the volume, drops of one size are formed. Drops of a chosen mean size are regarded as molecules of a heavy gas. The gas drop is a third component present in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 48–53, September–October, 1973.The author thanks G. I. Petrov for his interest in and direction of the work.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (\(Ra_\mathrm{H}\)), solutal Rayleigh number (\(Ra_\mathrm{S}\)), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (\(Ra_\mathrm{cr}\)) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.     

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, = _R + _D . The deformational vorticity, _D , arises from shear deformation and is frame invariant while the rigid vorticity, _R , is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers

The spatial development of disturbances with small and moderate amplitudes in a two-dimensional (2-D) supersonic flat-plate boundary layer at Mach 4.8 is investigated using direct numerical simulations based on the compressible 3-D Navier-Stokes equations. Disturbances are introduced into the boundary layer by blowing and suction within a narrow disturbance strip at the wall. In response to the timewise periodic forcing, two types of disturbance waves are generated, a first-mode wave and a multiple-viscous-solution. The multiple-viscous-solution was described by Mack (1969, 1984) but was not seen before in a direct numerical simulation. The results of the simulations are compared with results of linear stability theory, and the agreement is very good. In simulations for larger amplitudes, fundamental resonance is observed, where both types of 3-D waves are nonlinearly amplified and synchronize their phase velocities with the 2-D disturbance waves. Subharmonic resonance is found for 3-D waves with large wave numbers, where the phase velocities of the linear 2-D and 3-D waves are nearly the same.This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, as part of SFB 259.     

Instability of Sedimenting Bidisperse Particle Gas Suspensions

A linear stability analysis for a sedimenting bidisperse gas-solid suspension (or gas fluidized bed) is performed. Mass, momentum and energy conservation equations for each of the two species are derived using constitutive equations that are valid at high Stokes numbers, (St₁ 1). The homogeneous suspension becomes unstable at sufficiently large St₁ to waves of particle volume fraction with the wave number in the vertical direction. Numerical calculations of the growth rate in an unbounded suspension indicate that the marginal stability limits are controlled by the small wave number (k 1) behavior. Depending on the Stokes number and the volume fractions ₁ and ₂ of the two species, the suspension becomes unstable due to O(k) or O(k²) contributions to the growth rate. The O(k) term corresponds to an instability due to kinematic waves similar to that predicted for bidisperse suspensions of particles in viscous liquids [22]. The O(k²) contribution represents an instability to dynamic waves similar to that obtained from an analysis of averaged equations for monodisperse fluidized beds [4].     

Unsteady one-dimensional flow in a saturated porous medium with a volume heat source

The one-dimensional process of the heating of a saturated porous medium by a volume heat source as a result of the absorption of the energy of a high-frequency (frequency R~ 10¹–10³ MHz) electromagnetic wave is investigated. It is assumed that in the initial state the saturating (second) component is in the high-viscosity liquid or solid state. Under the action of the heat it is heated, melts, expands, becomes less viscous and under the pressure head created may flow relative to the stationary rock skeleton (first component). On the basis of the mathematical model proposed the basic laws of the process are analyzed and numerically investigated in the case of one-dimensional axisymmetric motion. It is shown that under actual conditions the dimensions of the thermal influence zone may be very considerable. Thus, by varying certain external factors it is possible to modify the dynamics of the process and the distributions of the temperature, pressure and phase velocity fields.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–124, July–August, 1991.