The stability of the narrow-gap Taylor-Couette system with an axial flow

Summary The stability of viscous flow between concentric rotating cylinders with an axial flow due to an axial pressure gradient is considered. The governing equations with respect to three-dimensional disturbances are derived and solved by a direct numerical procedure. Results are given for the case of small-gap approximation. Three typical cases =–1,0 and 0.5 are studied, where represents the ratio of angular velocity of the outer cylinder to that of the inner cylinder. The value of the axial Reynolds numberR is up to 100. It is found that the critical disturbance is a non-axisymmetric mode when the value ofR is sufficiently large, and the transition of the onset mode withR is demonstrated in detail. Results for the critical Taylor number, wave number, vortex incline angle, and relative wave velocity are also determined. The present stability analysis is found to be in agreement with previous experimental studies and particularly reveals the stability characteristics with the variation of .     

Existence of extra vortex and twin vortex of anomalous mode in Taylor vortex flow with a small aspect ratio

Summary This experimental work on Taylor vortex flow in a gap with a small aspect ratio is concerned with two extra vortices and a twin vortex system, each of which depends on an anomalous cell of the anomalous mode. Extra vortices are smaller than other vortices such as defined cells. At any Reynolds number and aspect ratio extra vortices can be found at the corner of the end plate and inner rotating cylinder and at the corner of the end plate and outer stationary cylinder. For a one-cell flow (anomalous one-cell mode) in a symmetric system, an outer extra vortex develops and grows to the same size as the main cell, only in an aspect ratio of less than one. A twin vortex is observed to form when two vortices are aligned in the direction of the radius. There are three flow fields on the end plate; two are extra vortex flows and the other is the main cell flow. The flow direction of the anomalous cell is from the inner cylinder to the outer one, at the end plate opposite of the flow direction of the normal cell.Nomenclature R ₁ Radius of inner cylinder (2R ₁=40.19±0.006 mm) - R ₂ Radius of outer cylinder (2R ₂=60.11±0.024 mm) - R _r Radius ratio (R ₁/R ₂=0.669) - d Clearance between cylinders (R ₂–R ₁=9.96±0.025 mm) - L Height of working fluid - Aspect ratio=L/d - Rotational angular speed - Kinematic viscosity - Re Reynolds number=R ₁ d/ Other nomenclature is defined as it appears     

Symmetric and asymmetric Taylor vortex flow in spherical gaps

Summary The rotationally symmetric flow between two concentric rotating spheres is investigated both theoretically and experimentally. The non-uniqueness of the supercritical flow exhibits three different modes with zero, one and two Taylor vortices in each hemisphere. These modes are realized by different accelerations of the inner sphere from the state at rest. A initial value code, based on a finite difference method, is used for the numerical simulation. The existence regions of the different supercritical flows are connected with symmetric and asymmetric transitions. It is found, that a steady state can exist asymmetric with respect to the equator. The flow is analyzed by plotting the size of the Taylor vortices, the depending variables , ,V, the velocity distributions and the torque. A comparison between theory and experiments for the observed modes of flow is given.     

Order parameters and energies of analytic and singular vortex lines in rotating3He-A

We present the expressions of the generalized Ginzburg-Landau (GL) theory for the free energy and the supercurrent in terms of thed vector, the magnetic fieldH, and operators containing the spatial gradient and the rotation. These expressions are then specialized to the Anderson-Brinkman-Morel (ABM) state. We consider eight single-vortex lines of cylindrical symmetry and radiusR=[2m/]^–1/2: the Mermin-Ho vortex, a second analytic vortex, and six singular vortices, i.e., the orbital and radial disgyrations, the orbital and radial phase vortices, and two axial phase vortices. These eight vortex states are determined by solving the Euler-Lagrange equations whose solutions minimize the GL free energy functional. For increasing field, the core radius of the texture of the Mermin-Ho vortex tends to a limiting value, while the core radius of the texture goes to zero. The gap of the singular vortices behaves liker forr 0, where ranges between and . The energy of the radial disgyration becomes lower than that of the Mermin-Ho vortex for fieldsH6.5H*=6.5×25 G (atT=0.99T _c and forR=10L*=60 µm, or=2.9 rad/sec). ForR 2 _T ( _T is the GL coherence length) or _c2 (upper critical rotation speed), the energies of the singular vortices become lower than the energies of the analytic vortices. This is in agreement with the exact result of Schopohl for a vortex lattice at _c 2. Finally, we calculate the correction of order (1 -T/T _c ) to the GL gap for the axial phase vortex.     

Magnetic textural deformation in rotating3He-A

An axial magnetic field is shown to deform the type I lattice of nonsingular vortices in rotating³He-A. At a critical field of orderH _D (30 G), this texture becomes unstable with respect to an array of doubly quantized vortices, seen in experiments at 284 G.     

The exact form of caustics in mixed-mode fracture: A comparison with approximate solutions

Summary The form of caustics created by stress singularities in elastic problems was up-to-now derived from the Sneddon expressions for the components of stresses at the point of singularity, which are based on the first and singular term of the series expansion of the Muskhelishvili complex stress function. In this paper the closed from expression for the Muskhelishvili complex stress function (z) was used to define the exact form of the caustic. Moreover, the forms of the caustics were constructed for several terms, besides the first one, in the Taylor expansion of (z).The approximate forms with the singular term of (z) and several terms of the Taylor expansion of (z) were compared with the form derived from the exact solution. The discrepancies between exact and approximate solutions were evaluated for the case of a slant crack in an infinite plate under in-plane biaxial loading where theK _I andK _II -mode stress intensity factors were compared as derived from the various solutions. It was concluded that, although the method of caustics yields superior results than any other experimental method, it is possible to improve these results by using either the exact solution for the particular problem, or higher order approximations.With 11 Figures     

Exact 3-D piezoelasticity solution for buckling of hybrid cross-ply plates using transfer matrices

Summary. A three dimensional exact piezoelasticity solution is presented for buckling of simply-supported symmetrically laminated hybrid plates with elastic substrate and piezoelectric layers. Buckling is considered under inplane normal strains in open as well as closed circuit conditions of the piezolayers and actuation potentials for movable and immovable inplane end conditions. A mixed formulation is used to form the governing equations for the buckling mode in terms of eight primary variables: displacements u,v,w, potential , stresses _z,_yz,_zx and electric displacement D_z. These entities are expanded in double Fourier series that satisfy the end conditions. The governing equations reduce to eight first-order homogeneous ordinary differential equations in z with constant coefficients dependent on the inplane strains and actuation potentials. The solution has eight constants for each layer. The transfer matrix is derived relating the eight primary variables at the top and bottom of a layer. The eight conditions _z= _yz=_zx=0,/D_z=0 at the top and bottom of the plate are used to set up four homogeneous equations for u,v,w,D_z/ at the bottom. The determinant of their coefficient matrix is set to zero to obtain the buckling strains/potential. Benchmark results are presented for hybrid highly inhomogeneous test plate, cross-ply composite plate and sandwich plates. The effects of thickness parameter, aspect ratio and the electric boundary conditions on the buckling loads are illustrated.     

On the elastic deformation of non-saturated swelling soils

Summary This paper aims at completing and extending the theories as they have been applied to swelling media for the last 50 years, to swelling non-saturated soils. However, having regard to the complicated behaviour of swelling soils, it was thought necessary to keep the state of stress as simple as possible when discussing swelling in cylindrical specimens in which drainage is completely prevented. Definitions of the parameters are attemptedbased on equilibrium thermodynamics. Contributions to swelling stress calculation, when expansion is considered in relation to vapor pressure and moisture content, are also given.Notation u _w hydrostatic pressure of soil water - total stress - effective stress - m _c soil mass - m _w water mass - H=(m _w/m _c) moisture content (gravimetric) - A=(m _w/m _c)_Xi (i=x,y,z orr, ,z) moisture content at constant swelling pressure (gravimetric) - n=(m _w/m _c)_h moisture content at constant vapor pressure (gravimetric) - n _a amount of a constituant phase - h _v vapor pressure of soil water - h ₀ vapor pressure of pure water - h=(h _v/h ₀) relative vapor pressures - M molecular weight of water - specific volume of the soil water vapor - R gas constant (8.3144 Joules/mole·^oK) - T absolute temperature - P swelling pressure of an isotropic soil swelling without constrain - S entropy of a sample of volumeV - O oncotic energy - O _r residual oncotic energy - U internal energy - F Helmholtz free energy - G Gibb's free energy - partial molar Gibb's free energy of constituenta - chemical potential which equals the partial molar Gibb's free energy of constituenta - _v chemical potential of water vapor - _v chemical potential of soil water - V volume - V _H volume at constant moisture content - W _a specific energy for the adsorbed water - W _a0 specific osmotic energy of adsorbed solutes - W ₀ specific osmotic energy of free solutes - x, y, z Cartesian coordinate system - r, ,z cylindrical coordinate system - u, v, z _w displacement components along the strain variablesx, y, andz - z _w height of the water column - P _i (i=x, y, z) loads along the strain variablesx, y, z - P _i (i=x, y, z) orr, ,z mechanical pressure along the strain variables - X _i (i=x, y, z orr, ,z) swelling pressures along the strain variablesx, y, z - P _k hydrostatic swelling pressure defined byp _k=(X _x+X _y+X _z)/3 - P _{k, H} hydrostatic swelling pressure under constant moisture content - l _i (i=x, y, z) extential displacements along the strain variablesx, y, z - s _i ^* (i=x, y, z orr, ,z) differential swelling - s _i (i=x, y, z orx, ,z) differential swelling per unit volume - s _k hydrostatic differential swelling defined bys _k=(s _x+s _y+s _z)/3 - s _h differential swelling under constant vapor pressure - Q effective swelling pressure defined as the energy encompassing all the unknown effects contributing to swelling against the mechanical pressure and the vapor pressure of soil water per unit volume - Q coefficient of swelling - E _i (i=r, ,z orx, y, z) Young's moduli in the directioni - E _{i, h} (i=x, y, z) Young's moduli at constant vapor pressure - B _h Bulk modulus under constant vapor pressure - G _{ij. h} (i, j=x, y, z) Shear modulus under constant vapor pressure - v _ij (i, j=r, ,z orx, y, z) Poisson's coefficient which characterizes the compression in the directioni for tension in the directionj, etc. - e _i (i=r, ,z) final radial, tangential and axial strains - e _{i, s} (i=r, ,z) radial, tangential and axial strains due to swelling - loads along the strain variablesr, ,z - G _ij (i, j=x, y, z) rigidity modulus - g acceleration due to gravity - a radius of the cylinder - r ₁,r ₂ roots of the characteristic equation - W work done by the surroundings on the system     

Phase slip solutions in magnetically modulated Taylor–Couette flow

We numerically investigate Taylor–Couette flow in a wide-gap configuration, with \({r_i/r_o=1/2}\), the inner cylinder rotating, and the outer cylinder stationary. The fluid is taken to be electrically conducting, and a magnetic field of the form \({B_z\approx(1 + \cos(2\pi z/z_0))/2}\) is externally imposed, where the wavelength \({z_0=50(r_o-r_i)}\). Taylor vortices form where the field is weak, but not where it is strong. As the Reynolds number measuring the rotation rate is increased, the initial onset of vortices involves phase slip events, whereby pairs of Taylor vortices are periodically formed and then drift outward, away from the midplane where \({B_z=0}\). Subsequent bifurcations lead to a variety of other solutions, including ones both symmetric and asymmetric about the midplane. For even larger Reynolds numbers, a different type of phase slip arises, in which vortices form at the outer edges of the pattern and drift inward, disappearing abruptly at a certain point. These solutions can also be symmetric or asymmetric about the midplane and co-exist at the same Reynolds number. Many of the dynamics of these phase slip solutions are qualitatively similar to previous results in geometrically ramped Taylor–Couette flows.     

The existence of Taylor vortices and wide-gap instabilities in spherical Couette flow

Summary The flow of a viscous incompressible fluid in the gap between two concentric spheres was investigated for the case where only the inner sphere rotates and the outer one is stationary. Flow visualization studies were carried out for a wide range of Reynolds numbers (Re2·10⁵) and aspect ratios (0.080.5) to determine the instabilities during the laminar-turbulent transition and the corresponding critical Reynolds numbers as a function of the aspect ratio. It was found that the laminar basic flow loses its stability at the stability threshold in different ways. The instabilities occurring depend strongly on the aspect ratio and the initial conditions. For small and medium aspect ratios (0.080.25), experiments were carried out as a function of the Reynolds number to determine the regions of existence for basic flow, Taylor vortex flow, supercritical basic flow and furthermore, to give the best fit for the maximum number of pairs of Taylor vortices as a function of aspect ratio. For wide gaps (0.330.5), however, Taylor vortices could not be detected. The first instability manifests itself as a break of the spatial symmetry and non-axisymmetric secondary waves with spiral arms appear depending on the Reynolds number. For =0.33, secondary waves with an azimuthal wave numbern=six, five and four were found, while in the gap with an aspect ratio of =0.5 secondary waves withn=five, four and three spiral arms exist. Frequencies of these secondary waves were measured, the corresponding critical Reynolds numbers and the transition Reynolds numbers during the transition to turbulence were found. The flow modes occurring at the poles look very similar to those found in the flow between two rotating disks. Effects of non-uniqueness and hysteresis were determined as a function of the acceleration rate.     

Component-transfer equation in column with longitudinal separation

method of integration over the transverse coordinate, a transfer equation in a separating column in obtained, taking the longitudinal enrichment mechanism into account.Notation j_r, j_z density of radial and axial diffusional flows of the heavy component - c heavy-component concentration - ¯c mean (over the cross section) concentration - c₀ concentration is feed cross section, z=L/2 - z longitudinal coordinate - v_z axial component of hydrodynamic mixture velocity - v _z ^* velocity corresponding to condition of maximum enrichment factor - current function - mixture density - p pressure, _p, _t, barodiffusional and thermodiffusional constants - T₂, T₁ temperature of hot axial and cold near-wall chamber regions - W electric power absorbed by gas-discharge plasma - V_ph phase velocity of traveling magnetic wave - dynamic viscosity - =/(K₃+ _zK₂) relative drawoff - L, R column length and radius - M ionic mass of readily ionized component - Q_ni, q_nN effective cross section for collisions of atoms of separation mixture with ions and added neutrals, respectively - D mutual diffusion factor of mixture isotopes Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 81–85, July, 1980.     

Magnetic field dependence of zero-sound attenuation close to the pair-breaking edge in3He-B due toJ=1−,J z =±1 collective modes

Our previous theory yielded for the Zeeman splitting of the imaginaryJ=1 collective mode in³He-B the result =2+0.25J _z ( is the effective Larmor frequency). In this paper we take into account the downward shift of the pair-breaking edge from 2 to 2₂– (₂ and ₁ are the longitudinal and transverse gap parameters). This leads to a complex Landé factor: the frequencies of theJ _z =±1 components become =2+0.39J _z , and the linewidths of these resonances become finite: =0.18. The coupling amplitudes of theJ _z =±1 components to density are found to be proportional to gap distortion, (₁–₂/(/)². Our results for the ultrasonic attenuation due to theJ _z =±1,J=1 modes are capable of explaining the field dependence of the attenuation close to the pair-breaking edge as observed by Dobbs, Saunders, et al. The observed peak is caused by theJ _z =–1 component: its height increases due to gap distortion as the field is increased, and the peak shifts downward in temperature and its width increases with the field due to the complex Landé factor. TheJ _z =+1 component gives rise to a corresponding dip relative to the continuum attenuation.     

The Superfluid Transition and Vortex Dynamics in Submonolayer Superfluid 4He Films in Porous Glass Under Rotation

A combination of a rotating dilution refrigerator and high-Q torsional oscillator technique has been used to study dynamics of vortices in thin ⁴ He films adsorbed on the porous glass (d=1m pore size). Under rotation an additional dissipation peak with the amplitude proportional to the angular velocity is seen at the middle of the superfluid transition, on the low temperature side of the stationary peak which is present even at =0. We attribute this peak to the 3D Type vortices created in multiply connected ⁴ He film by the rotation. Peak shape of the rotation-induced dissipation could be interpreted as a freezing of the 3D vortices well below T _c     

A boundary layer theory for the end problem of a circular cylinder

Summary This paper discusses the nature of an approximate solution for the hollow circular cylinder whose fixed ends are given a uniform relative axial displacement and whose cylindrical surfaces are free from traction. We shall take the solution of this problem to be given by a super-position of the following two problems: problem I considers a finite length cylinder whose ends are given a relative axial displacement, but are no longer fixed; problem II removes the radial displacement at the end of the cylinder obtained in problem I.Nomenclature a mid-surface radius of cylinder - c half-height of cylinder - E, in-plane elastic moduli - E_t, _t, G_t transverse elastic moduli - _z, , _r axial, circumferential, and normal strain - _rz transverse shear strain - h cylinder thickness - _z, , _r axial, circumferential, and normal stress - _rz transverse shear stress - z, r axial and radial coordinates - u_z, u_r axial and normal displacements     

Scaling Equation of State for a Nonequilibrium Solution Under Gravity Close to Phase Interface

Experimental investigations of equilibration kinetics for a methanol-hexane binary solution under gravity have been carried out at temperatures T below the critical consolute temperature, T<T _c , by using a refractometry technique. As a result of the experiment, both equilibrium n _z (z,t _e ) and nonequilibrium n _z (z,t) height dependences of the refractive index gradient n _z at different times t after the beginning of thermal equilibration have been obtained. Analysis of the data shows that the relaxation properties of the system at different fixed heights are determined not by a single relaxation time (z), but by a spectrum of relaxation times _i (z _j ,t). On the basis of the experimental data, the height dependence of the relaxation times has been analyzed for the studied solution in the course of its transition to equilibrium. The average relaxation time has been shown to decrease when nearing the phase interface (z=0). The relaxation time (z,t) at a certain height z has been shown to also decrease when the system approaches an equilibrium state. A dynamic nonequilibrium equation of state has been proposed on the basis of the fluctuation theory of phase transitions for a substance under gravity close to the phase interface of a binary solution. It is based on the assumption that for small solution concentrations, (c–c _c )/c _c <<1, every nonequilibrium height distribution n _z (z,t) corresponds to an equilibrium distribution n _z (z,T) at a certain temperature T=T–T _c . Here, c _c is the critical concentration of the solution.     

On the axisymmetric Mindlin's problem for a semi-space of granular material

Summary The methods of images and Hankel transforms are used to construct solution to an axisymmetric boundary value problem of a semi-space of transversely isotropic (granular) material due to a point force applied at a distanceh beneath its stress free plane boundaryz=0. Exact closed form expressions are determined for the components of displacements and stresses throughout the interior of the granular semi-space. The solution is then used to derive the surface displacements due to a uniformly distributed force over a circle of radius a with centre at (0, 0, –h) in the planez=–h of the semi-space. By a suitable choice of material constants and through a limit process as ₁, ₂ approach 1, the granular semi-space becomes isotropic and the corresponding results derived in this particular case agree with those presented in [14].With 2 Figures     

Theoretical investigation of Josephson tunnel junctions with spatially inhomogeneous superconducting electrodes

The microscopic theory of the Josephson effect in tunnel structures with electrodes having spatially inhomogeneous superconducting properties is formulated. Two mechanisms of inhomogeneity are considered. The first is associated with the presence of a thin transition normal layer located near the tunnel barrier, which is relevant for junctions based on refractory superconductors. The second case is the trapping of Abrikosov vortices by junction electrodes. The tunnel current components are calculated numerically in the whole temperature range 0<T<T _c and magnetic field range 0<H<H _c2. It is shown that the tunnel current is extremely sensitive to the type of smearing of the singularities of the classical tunnel theory ateV=2. The results allow experimental determination of the characteristics of real tunnel junctions.     

Three-dimensional elastic stress fields ahead of blunt V-notches in finite thickness plates

Based on detailed three-dimensional (3D) finite element analyses, elastic fields in front of blunt V-notches in finite thickness plates subjected to uniaxial far-end tensile stress have been investigated. By comparison with the corresponding planar V-notch fields and 3D through-thickness sharp crack fields, various aspects of the 3D fields of the blunt V-notches in finite thickness plates are revealed: (1) The plate thickness and notch angle have obvious effects on the stress concentration factor (SCF) K _t, which is higher in finite thickness plates than in the plane stress and plane strain cases. When the notch angle is smaller than 90°, the SCF is insensitive to the notch angle, but has close relation with the dimensionless plate thickness. With the notch angle increasing further, the SCF decreases and the effect of dimensionless plate thickness on it becomes weaker. (2) For any notch angle considered, the variation of the opening stress _yy normalized by its value _yy0 at the notch-root with the distance x from the root normalized by the root-radius , is insensitive to the plate thickness and coincides well with the two-dimensional (2D) planar solution. (3) The 3D distribution of the out-of-plane constraint factor T _z=_zz/(_yy+_xx) is controlled by the plate thickness (B), the notch-radius () as well as the notch angle (), but for deeper V-notches with 90°, the distribution of T _z coincides well with that of a U-notch as well as a sharp 3D through-thickness crack and an explicit empirical expression of T _z is presented. (4) The distribution of the in-plane stress ratio T _x=_yy/_xx in front of the 3D V-notch is nearly independent of the plate thickness and coincides well with the corresponding 2D solutions when the opening angle is smaller than 120°. (5) The gradient of the out-of-plane strain _zz is significant near the free surface in finite thickness plates. On the free surface, the _zz can be 3.5 times the value on the mid-plane, and the through-thickness gradient of the _zz increases with decreasing notch angle. It is of interest to note that most of the field quantities ahead of V-notches are insensitive to the notch angles when the notch angle is smaller than 90°.     

Fast matrix exponent for deterministic or random excitations

The solution of ż=Az is z(t)=exp(At)z₀=E_tz₀, z₀=z(0). Since z(2t)=E_2tz₀=Ez₀, z(4t)=E_4tz₀=Ez₀, etc., one function evaluation can double the time step. For an n‐degree‐of‐freedoms system, A is a 2n matrix of the nth‐order mass, damping and stiffness matrices M, C and K. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean‐square response P to a white noise random force with intensity W(t) is governed by the Lyapunov differential equation: Ṗ=AP+PA^T+W. The solution of the homogeneous Lyapunov equation is P(t)=exp(At) P₀ exp(A^Tt), P₀=P(0). One function evaluation can also double the time step. If W(t) is given as piecewise polynomials, the mean‐square response can also be obtained analytically. In fact, exp(At) consists of the impulsive‐ and step‐response functions and requires no special treatment. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non‐stationary response is resulted immediately without integration. The method is further extended to modulated noise excitations. The method gives analytical mean‐square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean‐square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non‐white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means‐square response matrix of a simply supported beam by finite element method. Copyright © 2001 John Wiley & Sons, Ltd.     

<Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">z</Emphasis></Subscript> constraints of semi-elliptical surface cracks in elastic plates subjected to uniform tension loading

The out-of-plane constraints T_z around the semi-elliptical surface cracks in an elastic plate subjected to uniform tension loading have been investigated through detailed three-dimensional (3D) finite element (FE) analyses. The distributions of T_z are obtained in the vicinity of the crack border with aspect ratios of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z drops from Poissons ratio at the crack tip to approximate zero beyond certain radial distance in the normal plane of the crack front line, and increases gradually from the free surface to the mid-plane at the same radial distance. By fitting the numerical results, empirical formulae are obtained to describe the 3D distributions of T_z for semi-elliptical surface cracks with a sufficient accuracy in the wide aspect ratio range of 0.2a/c 1.0 except very near the free surface, where T_z is extremely low. T_z, combining with the corresponding K and T or J and Q, can be applied to establish the three-parameter dominated stress field, which can characterize the 3D crack front field completely as an attempt.