Computation of particle settling speed and orientation distribution in suspensions of prolate spheroids

A numerical technique for the dynamical simulation of three-dimensional rigid particles in a Newtonian fluid is presented. The key idea is to satisfy the no-slip boundary condition on the particle surface by a localized force-density distribution in an otherwise force-free suspending fluid. The technique is used to model the sedimentation of prolate spheroids of aspect ratio b/a=5 at Reynolds number 03. For a periodic lattice of single spheroids, the ideas of Hasimoto are extended to obtain an estimate for the finite-size correction to the sedimentation velocity. For a system of several spheroids in periodic arrangement, a maximum of the settling speed is found at the effective volume fraction (b/a)²04, where is the solid-volume fraction. The occurence of a maximum of the settling speed is partially explained by the competition of two effects: (i) a change in the orientation distribution of the prolate spheroids whose major axes shift from a mostly horizontal orientation (corresponding to small sedimentation speeds) at small to a more uniform orientation at larger , and (ii) a monotonic decrease of the the settling speed with increasing solid-volume fraction similar to that predicted by the Richardson–Zaki law (1–)⁵⁵ for suspensions of spheres.     

A combinatorial result on Gröbner Fans with an application to universal Gröbner bases

We prove a regularity theorem on the Gröbner Fan of a polynomial ideal, and apply the regularity to obtain a filling algorithm for the direct enumeration of all the reduced Gröbner bases of the ideal.These results were announced at a presentation given at the MAGMA conference on computational algebra, Queen Mary and Westfield College, London, August 23^rd to 27^th, 1993     

On internal stress and activation volume in polymers

The two-site model is developed for the analysis of stress relaxation data. It is shown that the product of d In (– )/d and (- _i) is constant where is the applied stress, _i is the (deformation-induced) internal stress and = d/dt. The quantity d In ( )/d is often presented in the literature as the (experimental) activation volume, and there are many examples in which the above relationship with (- _i) holds true. This is in apparent contradiction to the arguments that lead to the association of the quantity d In (– )/d with the activation volume, since these normally start with the premise that the activation volume is independent of stress. In the modified theory presented here the source of this anomaly is apparent. Similar anomalies arise in the estimation of activation volume from creep or constant strain rate tests and these are also examined from the standpoint of the site model theory. In the derivation presented here full account is taken of the site population distribution and this is the major difference compared to most other analyses. The predicted behaviour is identical to that obtained with the standard linear solid. Consideration is also given to the orientation-dependence of stress-aided activation.     

Flow visualization glass-ceramic: Preliminary experimental and modelling results

A glass-ceramic material was developed to act as a flow visualization material. Preliminary experiments indicate that aperiodic, thermally induced, convective flows can be sustained at normal processing conditions. These flows and the stress and temperature gradients induced are most likely responsible for the anomalous behaviour seen in these materials and the difficulties encountered in their development and in their production on industrial and experimental scales. A simple model describing the dynamics of variable-viscosity fluids was developed and was shown to be in qualitative agreement with more sophisticated models as well as with experimental results. The model was shown to simulate the dependence of the critical Rayleigh number for the onset of convection on the viscous properties of the fluid at low T, and also to simulate quenching behaviour when the temperature differences were high.Nomenclature C _p Heat capacity - D, E, F Expansion coefficients - H Height of the roll cell - Pr Prandtl number - R _a Rayleigh number - R _c Critical Rayleigh number for the onset of convection in a constant-viscosity fluid - S Dimensionless stream function - T Temperature - T _m Mean temperature - T ₀ Bottom surface temperature - T _r Reference temperature - a Aspect ratio of cell - g Acceleration due to gravity - k Thermal conductivity - k ₁ Function related to ²v/T ² - k ₂ Function related to ⁴v/T ⁴ - r Rayleigh number ratioR _a/R _c - t Time - w Dimensionless vertical coordinate - w _m Mean cell height - x Horizontal coordinate - y Dimensionless horizontal coordinate - z Vertical coordinate - , Constants - _t Thermal expansion coefficient - Constant in viscosity function - T Temperature difference between top and bottom surfaces - _i Viscosity coefficients - Kinematic viscosity - _m Mean kinematic viscosity - Dimensionless kinematic viscosity - Thermal diffusivity - Non-linear temperature function - Dimensionless non-linear temperature function - _o - Stream function - Dimensionless time - Eigenvalues     

Distribution characteristics of mechanical properties and correlation between the respective properties on S35C carbon steel

Mechanical properties of tensile strength, , upper yield stress, _SU, lower yield stress, _SL elongation, , area reduction, , Vickers hardness, H _v, and impact absorbed energy, E, were examined using 50 specimens of S35C carbon steel, which were machined from two bars supplied from the same charged and heat-treated material. Distribution characteristics of these properties are discussed, and the correlation between each pair of them is investigated from a statistical viewpoint. The main conclusions obtained are summarized as follows; distribution characteristics of _B, _SL, , , H _v and E are well approximated by a normal distribution, but those of asu are not approximated as well by this type of distribution. In the latter case, a Weibull distribution is preferable to represent the distribution pattern. No significant correlation was observed between each pair of the above mechanical properties. Consequently, individual properties have the inherent distribution characteristics independent of the other properties.     

The contributions of microrotation of lubricant molecules in a thin film journal bearing

Summary Characteristics of a journal bearing were computed for thin film lubrication accounting for microrotation of the lubricant molecules using both the half-Sommerfeld and Reynolds boundary conditions. Although the Reynolds boundary conditions produced higher pressure and loads, the effects of microrotation studied by both schemes showed similar trends. Primary characteristics that effect the contributions of microrotation to the load carrying capacity of the journal bearing were identified. These characteristics were varied and their effects on the load capacity of the journal bearing are shown.Nomenclature a circumference of the shell, 2R - a ₁,a ₂,a ₃ constants - c radial difference between the shaft and shell, [R-r] - c ₁,c ₂,c ₃,c ₄ solving partial differential equations constants - e eccentricity - F ^* body force per unit mass - G substitute function for integration - h film thickness - j microinertia constant of the fluid - K ₁,K ₂ defined after Eq. (2.27) - l material length, - L ^* body couple - m - N coupling number, - p pressure - p ₀ ambient pressure - Q fluid flux flow through the cavity cross section - r shaft radius - R shell radius - R _h modified Reynold number, - R _l Reynolds number, - t time - u, v velocity components inx-andy-direction, respectively - u ₀ velocity of the shaft surface - V velocity vector - W load carrying capacity - W load component resulting from pressure parallel to the line of centers - W load component resulting from pressure perpendicular to the line of centers - x, y, z Cartesian coordinates - , , , micropolar viscosity coefficients - ₁, ₂ parameters of boundary conditions for the microrotation vector at the shell and shaft, respectively - the deviate angle of the load direction from the line of centers - e/c - , Newtonian viscosity coefficients - microrotation velocity vector - microrotation velocity component in thez-direction - angular velocity of the shaft - ^* thermodynamic pressure - mass density of the lubricant fluid - polar angle around the journal bearing - ^* angle which satisfies Reynold's B.C.     

Nonlinear axisymmetric response of orthotropic thin truncated conical and spherical caps

Summary An approximate analytical solution of the large deflection axisymmetric response of polar orthotropic thin truncated conical and spherical shallow caps is presented. Donnell type equations are employed. The deflection is approximated by a one term mode shape satisfying the boundary conditions. The Galerkin's method is used to get the governing equation for the deflection at the hole. Nonlinear free vibration response and the response under uniformly distributed static and step function loads are obtained. The effect of various parameters is investigated.Notations A, A ^* Inward and outward amplitudes - a, b, h Base radius, inner radius and thickness of the cap - D M h ³/[12(–v ² )] - E ,E Young's moduli - H ^*,H Apex height, dimensionless apex heght:H ^*/h - N _, Stress resultants - p ^1/2 - q Uniformly distributed load - Q,Q₀ Dimensionless load: , dimensionless step load - Q, Q ₀ Dimensionless load: , step load - t, Time, dimensionless time: t - T _A Ratio of nonlinear periodT for inward amplitudeA and the linear periodT _L - w ^* Normal displacement at middle surface - w Dimensionless displacement:w ^*/h - ₁ Linear parameter of static response - Orthotropic Parameter:E /E - Mass density - ₂,₃ Quadratic and cubic nonlinearity parameters - b/a - v ,v Poisson's ratios - Dimensionless radius:r/a - ^*, Stress function, dimensionless stress function: - ₀ ^* ,₀ Linnear frequency, dimensionless frequency: With 7 Figures     

Thermomechanic behavior of a polymer upon its formation in a crystallization process: Theoretical principles,hypotheses, and mathematical model

A model of thermomechanic behavior of a polymer upon its formation in a crystallization process is proposed. Based on methods of nonequilibrium thermodynamics governing relationships are obtained which make it possible to establish the dependence of the final degree of crystallicity of the material on the history of the crystallization process and to explain the mechanism of formation of the remanent stresses in a polymer article.Notation u translation vector - v velocity vector - acceleration vector - absolute temperature - density - c specific heat capacity - deformation tensor - strain tensor - specific enthropy - U ^* internal energy - z specific free enthalpy - _i internal parameters of state - t time - q heat flux vector - matrix of heat conduction coefficients - W ^* energy dissipation - F vector of mass forces - the 4th rank tensor of elastic pliabilities - matrix of heat expansion coefficients - tensor of contribution of structural variations to deformation - function of equilibrium value ^* - p mean pressure - deviator of the tensor of deformations - spherical part of the deformation tensor - deviator of the tensor of stresses - K volume modulus - unity tensor - Q enthalpy of the crystallization process - Q _eq enthalpy of the equilibrium crystallization process - _g glass transition temperature - ^*() the curve obtained in the equilibrium crystallization process - f final degree of crystallicity Institute of Mechanics of Continuous Media of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Institute of Technical Chemistry of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 479–485, May–June, 1995.     

Uniform suction/blowing effect on forced convection about a wedge: Uniform heat flux

Summary The effect of constant suction/blowing on steady two-dimensional laminar forced flow about a uniform heat flux wedge is numerically analyzed. The nonlinear boundary-layer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution, the temperature distribution, the local skin friction coefficient and the local Nusselt number are presented for various values of Prandtl number Pr, pressure gradient parameterm and suction/blowing parameter . In general, it has been found that the local skin friction coeffcient and the local Nusselt number increase owing to suction of fluid. This trend reversed for blowing of fluid. In addition to, as the blowing effect is strong enough, i.e. –0.65, the flow separation only occurred in the case ofm=0.0.Notation C Constant defined in Eq. (4.2) - C _f Local skin friction coefficient, 2v(u/y) _y=0/U ² - f Dimensionless stream function defined in Eq. (5.3) - g Gravitational acceleration - h Local heat transfer coefficient - k Thermal conductivity - m Pressure gradient parameter, /(2–) - Nu _x Local Nusselt number,hx/k - Pr Prandtl number,v/ - q _w Wall heat flux - Re _x Local Reynolds number,U x/v - T Temperature - T _w Wall temperature - T Temperature of ambient fluid - u Velocity component in thex-direction - U Potential flow velocity,Cx ^m - v Velocity component in they-direction - V _w Surface mass transfer - x Coordinate along the wedge surface - y Coordinate normal to the wedge surface - Thermal diffusivity - Angle factor of the wedge - Pseudosimilarity variable defined in Eq. (5.2) - Suction/blowing parameter defined in Eq. (5.1) - Total angle of the wedge - Dimensionless temperature defined in Eq. (5.4) - Kinematic viscosity - Stream function     

Thermal transport properties of helium near the superfluid transition. II. Dilute3He-4He mixtures in the superfluid phase

Measurements of the average thermal conductivity _exp hQ/T and of the thermal relaxation time to reach steady-state equilibrium conditions are reported in the superfluid phase for dilute mixtures of³He in⁴He. Hereh is the cell height,Q is the heat flux, andT is the temperature difference across the fluid layer. The measurements were made over the impurity range 2×10^–9<X(³He)<3×10^–2 and with heat fluxes 0.3<Q<160 µW/cm². Assuming the boundary resistanceR _b , measured forX<10^–5, to be independent ofX over the whole range ofX, a calculation is given for _exp. ForQ smaller than a well-defined critical heat fluxQ _c (X) X ^0.9, _exp is independent of Q and can be identified with the local conductivity _eff, which is found to be independent of the reduced temperature = (T–T)/T for –10^–2. Its extrapolated value at T is found to depart forX10^–3 from the prediction X ^–1 , tending instead to a weaker divergence X ^–a witha0.08. A finite conductivity asX tends to zero is not excluded by the data, however. ForQ >Q _c (X), a nonlinear regime is entered. ForX10^–6, the measurements with the available temperature resolution are limited to the nonlinear conditions, but can be extrapolated into the linear regime forX2×10^–7. The results for _exp(Q),Q _c (X), and _eff(XX) are found to be internally consistent, as shown by comparison with a theory by Behringer based on Khalatnikov's transport equations. Furthermore, the observed relaxation times (X) in the linear regime are found to be consistent forX>10^–5 with the hydrodynamic calculations using the measured _eff(X). ForX<10^–5, a faster relaxation mechanism than predicted seems to dominate. The transport properties in the nonlinear regimes are presented and unexplained observations are discussed.     

Shear viscosity of liquid4He and3He-4He mixtures,especially near the superfluid transition

High-resolution measurements of are reported for liquid⁴He and³He-⁴He mixtures at saturated vapor pressures between 1.2 and 4.2 K with particular emphasis on the superfluid transition. Here is the mass density, the shear viscosity, and in the superfluid phase both and are the contributions from the normal component of the fluid ( _n and _n ). The experiments were performed with a torsional oscillator operating at 151 Hz. The mole fraction X of³He in the mixtures ranged from 0.03 to 0.65. New data for the total density and data for _n by various authors led to the calculation of . For⁴He, the results for are compared with published ones, both in the normal and superfluid phases, and also with predictions in the normal phase both over a broad range and close to T. The behavior of and of in mixtures if presented. The sloped/dT near T and its change at the superfluid transition are found to decrease with increasing³He concentration. Measurements at one temperature of versus pressure indicate a decreasing dependence of on molar volume asX(³He) increases. Comparison of at T, the minimum of _n in the superfluid phase and the temperature of this minimum is made with previous measurements. Thermal conductivity measurements in the mixtures, carried out simultaneously with those of , revealed no difference in the recorded superfluid transition, contrary to earlier work. In the appendices, we present data from new measurements of the total density for the same mixtures used in viscosity experiments. Furthermore, we discuss the data for _n determined for⁴He and for³He-⁴He mixtures, and which are used in the analysis of the data.     

hp-version finite elements for the space-time domain

A bilinear formulation of elasto-dynamics is offered which includes, as a special case, Hamilton's law of varying action. However, the more general bilinear formulation has several advantages over Hamilton's law. First, it admits a larger class of initial-value and boundary-value problems. Second, in its variational form, it offers physical insight into the so-called trailing terms of Hamilton's law. Third, numerical applications (i.e., finite elements in time) can be proven to be convergent under correct application of the bilinear formulation, whereas they can be demonstrated to diverge for specific problems under Hamilton's law. Fourth, the bilinear formulation offers automatic convergence of the natural velocity end conditions; while these must be constrained in present applications of Hamilton's law. Fifth, the bilinear formulation can be implemented in terms of a Larange multiplier that gives an order of magnitude improvement in the convergence of velocity. This implies that, in this form, the method is a hybrid finite-element approach.List of symbols b arbitrary constant - A _i, A _i vector of integrals, i = 0, j - A() linear operator on - () Hamilton's form of A - B (u, ) bilinear operator u, - B (u, ) Hamilton's form of B - B _i,j , B _ij , B _ij matrix of integrals - C constant, N/m - c number of floating-point operations per coef. evaluation - f, f(x) force per unit length, N/m - F, F ₀, F _L forces, N - J number of functions in series for û - k spring rate per unit length, N/m² - K spring rate, N/m - K _max maximum value of K - L _a Lagrangian, non-dimensional - L length of beam, m - m mass per unit length, kg/m - M mass, kg - M _max maximum value of M - n number of functions in series for - N number of elements in domain - p momentum density, kg/sec - P, P ₀, P _T momentum, kg-m/sec - q _i generalized coordinates - r _j coefficients of _j - t time, sec - t ₀, t ₁ limits of action integral, Hamilton's law - T end of time period, sec - u solution for displacement, m - û approximation to u, m - u ₀ initial value for u, m - test function, m - limited class of , m - x spatial coordinate, m - flapping angle, rad - Lock number - time increment, sec - Lagrange multiplier - longitudinal stiffness EA, N (Eqs. 1–18) - advance ratio of rotor (Eqs. 33–34 and figures) - _i , _r polynomial functions - non-dimensional time, azimuth angle - () variation of ( ) - W virtual work - ( ) d ( )/dx - ( .) d ( )/dt - (*) d/d - [ ] matrix - { } column vector - row vector     

Analysis of the tensile properties and fractography of as-cast and heat treated 359-SiC-20p composite

An experimental study of the heat treatment of 359-SiC 20p composite and its base alloy was made to determine the strength-ductility characteristics under varying conditions of heat treatment. Microstructural observations revealed that addition of the SiC_p reinforcement to the base alloy produced a more uniform and refined interdendritic microstructure compared to the latter. The tensile data obtained was analysed in terms of the theoretical models existing in the composite literature. Ultimate tensile strength (UTS)-log elongation relationships were obtained to test the applicability of the quality index parameter,Q, to the present composite. From this analysis, it was found that all data points in the ageing temperature range 140–210 °C could be represented by a single line (cf. two lines in the case of 359 alloy), indicating the important fact that the tensile properties of this composite can be predicted/determined over the entire temperature range. The presence of the SiC particles was seen to accelerate the Mg₂Si precipitation kinetics, but not to alter it. Fracture mechanisms were determined from both the fracture surfaces and their longitudinal sections beneath the fracture surface, employing both optical and scanning electron microscopy.Nomenclature a Particle diameter - b Burger's vector - b _ii Numerical constant relating P _ii ^E _m andP ₃₃ ^A - E _c Young's modulus of the composite - E _m Young's modulus of the matrix - E _p Young's modulus of SiC particles - El Elongation (%) - f _p SiC volume fraction - P ₃₃ ^A Applied stress - P _ii ^E Long range back stress developed by elastic misfit - P _m ^F Change in matrix flow stress - <P _ii ^P >_m Back stresses due to plastic deformation - P _c ^ps Proof strain of a composite - q _ii Plastic misfit - Q Quality index - R Statistical correlation coefficient - RE Rockwell E hardness value - S SiC particle aspect ratio - S _c Critical aspect ratio for the SiC particles - UTS Ultimate tensile strength of the alloy or composite - YS Yield strength of the alloy or composite - Critical misfit strain - Constant, 1.25 for aluminum alloys - Plastic strain - ^ps Plastic strain at whichP _c ^ps is required - Work hardening rate at a given plastic strain - Work hardening rate as a function of total strain - Shear modulus - Dislocation density - _c ^O Yield stress of the composite - _CTE Increase in yield stress due to coefficient of thermal expansion (CTE) - _m ^O Yield stress of the matrix - _p Particle strength - _i Interfacial shear strength     

Microstructure and mechanical properties of rapidly quenched L20 and L20+L12 alloys in Ni-Al-Fe and Ni-Al-Co systems

Ductile L2₀-type wires and+L1₂-type duplex wires with high strengths and large elongation in the Ni-Al-Fe and Ni-Al-Co ternary systems have been manufactured directly from the liquid state by an in-rotating-water spinning method. The wire diameter was in the range 80 to 180m and the average grain size was 2 to 4m for the wires and 0.2 to 1.0m for the+ wires. _y, _f and _p of the wires were found to be about 360 to 760 MPa, 560 to 960 MPa, and 0.2 to 5.5%, respectively, for the Ni-Al-Fe system, those of the+ wires were about 395 to 660 MPa, 670 to 1285 MPa, and 3.5 to 17%, respectively, for the Ni-Al-Fe system, and about 260 to 365 MPa, 600 to 870 MPa, and 4.0 to 7.0%, respectively, for the Ni-Al-Co system. Cold-drawing caused a significant increase in _y and _f and the values attained were about 1850 and 2500 MPa, respectively, for Ni-20Al-30Fe and Ni-25Al-30Co wires drawn to about 90% reduction in area. The high strengths, large elongation and good cold-workability of the melt-quenched and+ compound wires have been inferred to be due to the structural change into a low-degree ordered state containing a high density of phase boundaries, suppression of grain-boundary segregation and refinement of grain size.     

Knudsen molecular flow in a channel with a small temperature difference at its ends

The thermomolecular pressure difference (TPD) of helium, argon, and krypton is measured in a packet of glass capillaries for temperatures 273 and 293 K at their ends in a 10–100 range of Knudsen numbers.Notation exponent of the thermomolecular pressure difference effect - Kn Knudsen number - rarefaction parameter - Q_T reduced thermal creep flux - Q_P reduced Poiseuille flux - C(_t) Cercignani-Lampis scattering kernel - R specular scattering kernel - (1 – ) fraction of specular reflection - _t accommodation coefficient of the tangential momentum - P_c, P_h gas pressure in the cold and hot volumes, respectively - coefficient of dynamic viscosity - m mass of gas molecules - k Boltzmann constant - D, L the capillary diameter and length, respectively Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 5, pp. 719–724, May, 1988.     

A nonlinear composite shell element with continuous interlaminar shear stresses

A numerical model for layered composite structures based on a geometrical nonlinear shell theory is presented. The kinematic is based on a multi-director theory, thus the in-plane displacements of each layer are described by independent director vectors. Using the isoparametric apporach a finite element formulation for quadrilaterals is developed. Continuity of the interlaminar shear stresses is obtained within the nonlinear solution process. Several examples are presented to illustrate the performance of the developed numerical model.List of symbols reference surface - convected coordinates of the shell middle surface - ⁱ coordinate in thickness direction - ⁱ h thickness of layer i - X_o position vector of the reference surface - ⁱX_o position vector of midsurface of layer i - t _k orthonormal basis system in the reference configuration - ⁱ a _k orthonormal basis system of layer i - ⁱW axial vector - R_o orthonormal tensor in the reference configuration - ⁱ R orthonormal tensor of layer i - ⁱ Cauchy stress tensor - ⁱ P First Piola-Kirchhoff stress tensor - ⁱ q vector of interlaminar stresses - ⁱ n, ⁱ m vector of stress resultants and stress couple resultants - v _x components of the normal vector of boundary - ⁱ N, ⁱ Q, ⁱ M stress resultants and stress couple resultants of First Piola-Kirchhoff tensor - stress resultants and stress couple resultants of Second Piola-Kirchhoff tensor - ⁱ , ⁱ , ⁱ strains of layer i - _K transformation matrix - u_o displacement vector of layer 1 - ⁱ local rotational degrees of freedom of layer i     

Optimization of multilayer thermal insulation

An iteration method is developed for determination of the thicknesses of layers of a multilayer thermal insulation with minimum mass, with consideration of temperature limitations. The penalty function method is employed.Notation M(h) target function - _i thickness of the i-th layer - p_i density of material in i-th layer - n number of layers of thermal insulation - y spatial coordinate - t time - Y_i, i = 0, 1, 2, ..., n coordinates of layer boundaries - Cⁱ(T) volume heat capacity of material in i-th layer - ⁱ(T) thermal conductivity coefficient of material in i-th layer - (y) initial temperature distribution - q thermal flux - t_c right-hand value of time interval - T _max ⁱ , i = 1, 2, ..., n maximum admissible temperatures on i-th boundary - penalty function - penalty parameters - g_i function considering temperature limitations - transformed function - k number of successive unconditional minimization problem - l number of iteration in search for local minimum - ,, s parameters of conjugate gradient method Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 286–291, August, 1980.     

Bounds for traces in complete intersections and degrees in the Nullstellensatz

In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f₁,...,f n–rk[X₁,...,X_n] be a regular sequence with d:=max_j deg f_j. Denote byA the polynomial ringk [X₁,..., X_r] and byB the factor ring k[X₁,...,X_n]/(f₁,...,f_n r); assume that the canonical morphism AB is injective and integral and that the Jacobian determinant with respect to the variables X_r+1,...,X_n is not a zero divisor inB. Let finally B^*:=Hom_A(B, A) be the generator of B^* associated to the regular sequence.We show that for each polynomialf the inequality deg (¯f) dⁿ r(+1) holds (¯fdenotes the class off inB and is an upper bound for (n–r)d and degf). For the usual trace associated to the (free) extensionA B we obtain a somewhat more precise bound: deg Tr(¯f) dⁿ r degf. From these bounds and Bertini's theorem we deduce an elementary proof of the following effective Nullstellensatz: let f₁,..., f_s be polynomials in k[X₁,...,X_n] with degrees bounded by a constant d2; then 1 (f₁,..., f_s) if and only if there exist polynomials p₁,..., p_sk[X₁,..., X_n] with degrees bounded by 4n(d+ 1)ⁿ such that 1=_ip_if_i. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndⁿ is obtained.Partially supported by UBACYT and CONICET (Argentina)     

The effect of a magnetic field on fourth sound in 3He

The influence of a magnetic field on the propagation of fourth sound in superfluid ³He is studied. The field and temperature dependences of the average superfluid density \- _s / and fourth sound Q are measured. The field dependence of \- _s / is very different in a porous medium than predicted by Ginzburg-Landau theory applied to bulk liquid. In particular, a magnetic suppression of \- _s / is observed in the temperature and pressure ranges corresponding to the A phase in bulk liquid. There is strong evidence of a magnetic suppression of T _c itself. The measured \- _s / has a slight history dependence in a magnetic field, but none in zero field. The fourth-sound Q values are compared to the theoretical work of Smith, Jensen, and Wolfle. Quantitative confirmation of their work is problematic.