Nonstationary process of heat transfer in a tube with longitudinal fins

An analytical solution to the problem of nonstationary thermal interaction of a flow of a heat-transfer agent and a thin-walled tube with longitudinal fins is constructed for variable parameters of heat transfer.Notation u, temperatures of the fins - ,w temperatures of the tube walls - temperature of the flow of the heat-transfer agent - _i ,i= coefficients of heat transfer from the ambient medium to the fins and the tube walls, respectively - _i ,i= temperature distributions for the ambient medium - coefficients of heat transfer from the flow of the heat-transfer agent to the tube walls - q _i density of the heat flux to the corresponding portions of the tube - heat capacity, thermal conductivity, density, and thickness of the fin and tube material - c _p , ,G, F heat capacity, density, and flow rate of the heat-transfer agent, cross-sectional area of the tube - dimensions of the tube Bauman Moscow State Technical University. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 6, pp. 673–680, June, 1994.     

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     

Nonlinear asymptotic theory of hypersonic flow past a circular cone

Summary The hypersonic small-disturbance theory is reexamined in this study. A systematic and rigorous approach is proposed to obtain the nonlinear asymptotic equation from the Taylor-Maccoll equation for hypersonic flow past a circular cone. Using this approach, consideration is made of a general asymptotic expansion of the unified supersonic-hypersonic similarity parameter together with the stretched coordinate. Moreover, the successive approximate solutions of the nonlinear hypersonic smalldisturbance equation are solved by iteration. Both of these approximations provide a closed-form solution, which is suitable for the analysis of various related flow problems. Besides the velocity components, the shock location and other thermodynamic properties are presented. Comparisons are also made of the zeroth-order with first-order approximations for shock location and pressure coefficient on the cone surface, respectively. The latter (including the nonlinear effects) demonstrates better correlation with exact solution than the zeroth-order approximation. This approach offers further insight into the fundamental features of hypersonic small-disturbance theory.Notation a speed of sound - H unified supersonic-hypersonic similarity parameter, - K hypersonic similarity parameter, M - M freestream Mach number - P pressure - T temperature - S entropy - u, v radial, polar velocities - V freestream velocity - shock angle - cone angle - density - density ratio, /() - ratio of specific heats - polar angle - stretched polar angle, / - (), (), () gage functions     

Bounding-surface plasticity: Stress space versus strain space

Summary A bounding-surface plasticity model is formulated in stress space in a general enough manner to accommodate a considerable range of hardening mechanisms. Conditions are then established under which this formulation can be made equivalent to its strain-space analogue. Special cases of the hardening law are discussed next, followed by a new criterion to ensure nesting. Finally, correlations with experimental data are investigated.Notation (a) centre of the stress-space (strain-space) loading surface; i.e., backstress (backstrain) - ^* (a ^*) centre of the stress-space (strain-space) bounding surface - (a ) target toward which the centre of the stress-space (strain-space) loading surface moves under purely image-point hardening - (b) parameter to describe how close the loading surface is to nesting with the bounding surface in stress (strain) space; see (H10) - (c) elastic compliance (stiffness) tensor - (d) parameter to describe how close the stress (strain) lies to its image point on the bounding surface; see (H10) - (D) generalised plastic modulus (plastic compliance); see (1) - function expressing the dependence of the generalised plastic modulus on (plastic complianceD ond) - ^* (D ^*) analogue to (D) for the bounding surface - function expressing the dependence of ^* on (D ^* ond) - () strain (stress) - ' (') deviatoric strain (stress) - ^P ( ^R ) plastic strain (stress relaxation); see Fig. 1 - () image point on the bounding surface corresponding to the current strain (stress) - _iso (f _iso) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change of radius; i.e., fraction of isotropic hardening in the stress-space theory - _kin (f _kin) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change in the backstress (backstrain); i.e., fraction of kinematic hardening in the stress-space theory - _nor (f _nor) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - _ima (f _ima) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - function relating _iso to , , and (f _iso tob,d, andl) - function relating _kin to , , and (f _kin onb,d, andl) - function relating _nor to , , and (f _nor onb,d, andl) - function relating _ima to , , and (f _ima onb,d, andl) - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change of radius - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change in the centre - function relating _iso ^* to (f _iso ^* tod) - function relating _kin ^* to (f _kin ^* tod) - (l) parameter to describe the full extent of plastic loading up to the present, giving the arc length of plastic strain (stress relaxation) trajectory; see (H10) - function relating the direction for image-point translation of the loading surface to various other tensorial directions associated with the current state; see (H5). With 6 Figures     

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.     

The influence of antimony-additions on the creep behaviour of a 25wt% Cr-20wt% Ni austenitic stainless steel

The effect of antimony on the creep behaviour (dislocation creep) of a 25 wt% Cr-20 wt% Ni stainless steel with ~ 0.005 wt% C was studied with a view to assessing the segregation effect. The antimony content of the steel was varied up to 4000 ppm. The test temperature range was 1153 to 1193 K, the stress range, 9.8 to 49.0 MPa, and the grain-size range, 40 to 600m. The steady state creep rate, , decreases with increasing antimony content, especially in the range of intermediate grain sizes (100 to 300m). Stress drop tests were performed in the secondary creep stages and the results indicate that antimony causes dislocations in the substructure to be immobile, probably by segregating to them, reducing the driving stress for creep.Nomenclature _a Creep stress in a constant load creep test without stress-drop - _A Initial applied stress in stress-drop tests - Stress decrement - ( _A-) Applied stress after a stress decrement, - t _i Incubation time after stress drop (by the positive creep) - _C Strain-arrest stress - _i Internal stress - _{s s}-component (= _i- _c) - Steady state creep rate (average value) in a constant load creep test - Strain rate at time,t, in a constant load creep test - New steady state creep rate (average value) after stress drop from _A to ( _A-) - Strain rate at time,t, after stress drop.     

Heat flow model of rapid solidification with nonhomogeneous thermal contact

A heat flow model is presented of the solidification process of a thin melt layer on a heat conducting substrate. The model is based on the two-dimensional heat conduction equation, which was solved numerically. The effect of coexisting regions of good and bad thermal contact between foil and substrate is considered. The numerical results for thermal parameters of the Al-Cu eutectic alloy show considerable deviations from one-dimensional solidification models. Except for drastic differences in the magnitude of the solidification rate near the foil-substrate interface, the solidification direction deviates from being perpendicular to the substrate and large lateral temperature gradients occur. Interruption of the thermal contact may lead to back-melting effects. A new quantity, the effective diffusion length, is introduced which allows some conclusions to be drawn concerning the behaviour of the frozen microstructure during subsequent cooling.Nomenclature _i ,a _i Thermal diffusivity _i = _i /c _{i i} ,a _i = _i / ₁ - c _i Specific heat capacity - d Foil thickness - D Solid state diffusion coefficient - e_x, e_z Unit vectors - H Latent heat of fusion - h ,h Foil-substrate heat transfer coefficients - i Index: 1, melt; 2, solidified foil; 3, substrate - _i ,k _i Thermal conductivityk _i = _i / ₁ - n Normal unit vector - Nu ,Nu Nusselt numbers for regions of badNu(x,) and good thermal contact, respectivelyNu =h Nu _d / ₁,,Nu(x, )=h(x,)d/ ₁ - R Universal gas constant - , s Position of the liquid-solid interface ¯s/d=s=s _xe_x+s _ze_z - Local solidification rate /d = s =s _xe_x +s _ze_z - t Real time - T _i Temperature field - T ₀ Ambient temperature - T _f Melting temperature - u _i Dimensionless temperature fieldu _i (x, z,)=T _i (x,z,)/T _f - u ₀ Dimensionless ambient temperatureu ₀=T ₀/T _f - _i Local cooling rate within the foil _i = du _i /d - W Stefan numberW=H/c ₁ T _f - ,x Cartesian coordinate parallel to the foil-substrate interfacex= /d - ₀,x ₀ Lateral extension of foil sectionx ₀= ₀/d - ₁,x ₁ Lateral contact lengthx ₁= ₁/d - ,z Cartesian coordinate perpendicular to the foil-substrate interfacez= /d - ₀,z ₀ Substrate thicknessz ₀= ₀/d - E Activation energy of diffusion - T Initial superheat of the melt - u Dimensionless initial superheat u=T/T _f - (x) Step function - _eff Dimensionless effective diffusion length - _i Mass density - Dimensionless time=t ₁/d ² - _f, _f(x, z) Total and local dimensionless freezing time, respectively     

Free vibration of rectangular beams of arbitrary depth

Summary The state space approach is extended to the two dimensional elastodynamic problems. The formulation is in a form particularly amenable to consistent reduction to obtain approximate theories of any desired order. Free vibration of rectangular beams of arbitrary depth is investigated using this approach. The method does not involve the concept of the shear coefficientk. It takes into account the vertical normal stress and the transverse shear stress. The frequency values are calculated using the Timoshenko beam theory and the present analysis for different values of Poisson's ratio and they are in good agreement. Four cases of beams with different end conditions are considered.

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Freie Schwingungen rechteckiger Balken beliebiger Höhe

Zusammenfassung Die Zustandsraum-Technik wird auf zweidimensionale elastodynamische Probleme ausgedehnt. Die Formulierung ist besonders geeignet für die Aufstellung von Näherungstheorien beliebigen Grades. Freie Schwingungen von Rechteckbalken beliebiger Höhe wurden mit Hilfe dieser Technik untersucht. Das Verfahren umgeht den Begriff des Schubbeiwertsk. Es berücksichtigt die senkrechte Normalbeanspruchung und die Querkraft. Die Frequenzwerte werden mit Hilfe der Balkentheorie von Timoshenko und der vorliegenden Analyse berechnet, und zwar für verschiedene Werte der Querdehnzahl. Die berechneten Werte befinden sich in guter Übereinstimmung. Vier Fälle von Balken mit verschiedenen Endbedingungen werden untersucht.

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Notation 2h depth of beam - k Timoshenko shear constant - L length of the beam - n mode number - u, v displacement inx, y directions - A area of cross section - A _n coefficient in series representation - E modulus of elasticity - G modulus of rigidity - I moment of inertia aboutz-axis - mass density - Poisson's ratio - r - r×n - _{x y} direct stresses - _xy shear stress - eigenvalue of square matrix - frequency of harmonic vibration - eigenvalue= - frequency parameter= - ^* frequency parameter=× With 1 FigureOn leave from M. A. College of Technology, Bhopal, 462007, India     

Parafluidity in helium near the λ-transition and the generalized time-dependent Landau theory

A phenomenological theory of parafluidity, i.e., an enhancement of fluidity due to order-parameter fluctuations, is presented for helium near the transition. The generalized time-dependent Landau theory of second-order phase transitions is reviewed in general and is applied to the superfluid transition in helium as a particular example. In helium, it is found that parafluidity is manifested in the divergences of the mass diffusivity , the thermal conductivity , the first-sound amplitude attenuation ||^–1, and the second-sound dampling , which are all consistent with the dynamic scaling hypothesis. Here a characteristic relaxation time ₀ ||^–1 is used, where =(T–T _c )/T _c andT _c is the transition temperature. Although there are not enough experimental data to confirm our formulas, the present approach is seen to agree in order of magnitude with available experiments. Finally, the sound absorption above a ferromagnetic transition is calculated by adding a diffusion term to the generalized time-dependent Landau equation. The result thus obtained agrees in order of magnitude with experiments in nickel.Supported in part by the National Science Foundation and the Horace H. Rackham School of Graduate Studies.     

Heat transfer in plane film formation

A mathematical model is obtained for the process of cooling with formation of a planar film. The solution obtained is verified experimentally.Notation mean axial velocity gradient - v_x current axial velocity - v_o initial polymer velocity - v₁ sampling velocity - K draw ratio - deformation rate tensor - x, y, z spatial coordinates - X, Y dimensionless coordinates - L() differential operator - T temperature - T_o initial temperature - T_c temperature of surrounding medium - dimensionless temperature - dimensionless temperature averaged over film thickness - thermal-diffusivity coefficient - 2_o initial film thickness - thermal conductivity - heat-transfer coefficient - f(X) distance function - Bi Biot criterion, Bi_o, Biot criterion calculated for initial film thickness - Gz* modified Graetz criterion - V dimensionless velocity - ₁, ₂, ₃ heat-transfer coefficients produced by radiation, free convection, and forced convection - v_c, _c mean velocity and film half-thickness in formation zone - T₁ calculated temperature value - T₂ experimental temperature value - l formation zone length Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 854–858, November, 1979.     

On the thermodynamics of nonlinear single integral representations for thermoviscoelastic materials with applications to one-dimensional wave propagation

Summary Thermodynamic theory is used to develop single integral constitutive relations for the nonlinear thermoviscoelastic response to arbitrary stress and temperature histories; the thermomechanically coupled energy equation is also obtained. The thermorheologically simple material, modified superposition and the isotropic stress power law are discussed in detail. A modified Fourier heat conduction law is employed to ensure that the propagation of thermal disturbances takes place at a finite velocity. Using the nonlinear thermoviscoelastic stress power law along with the linearized energy equation and modified Fourier law, one-dimensional wave front solutions are obtained.

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Zur Beschreibung des nichtlinearen Verhaltens thermoviskoelastischer Stoffe durch einfache Integrale mit Anwendung auf eindimensionale Wellenausbreitung

Zusammenfassung Mit Hilfe der Thermodynamik werden einfache Integrale enthaltende Werkstoffbeziehungen für das nichtlineare thermoviskoelastische Verhalten unter beliebigen Spannungs- und Temperaturverläufen entwickelt und die thermomechanisch gekoppelte Energiegleichung wird angegeben. Im Detail werden der thermodynamisch-einfache Werkstoff, die modifizierte Überlagerung und das isotrope Spannungs-Potenzgesetz diskutiert. Damit thermische Störungen sich mit endlicher Geschwindigkeit ausbreiten, wird ein modifiziertes Fouriersches Wärmeleitgesetz verwendet. Unter Verwendung des nichtlinearen thermoviskoelastischen Spannungs-Potenzgesetzes, der linearisierten Energiegleichung und des modifizierten Wärmeleitgesetzes werden Lösungen der eindimensionalen Wellenfrontausbreitung erhalten.

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Notation C _ijkl linear elastic compliance, Equation (29) - C _s ,C _ts ,m _s ,m _tk ,v _s ,v _tk inelastic material constant, Equation (38) - C specific heat at constant strain, Equation (57) - C specific heat at constant stress, Equation (29) - E Young's modulus, Equation (34) - F _kl tensor functions of stress, Equation (28) - f _mn ⁽¹⁾ ,f _kl ⁽²⁾ tensor functions of the stress and temperature, Equation (8) - f[(t)] monotonically increasing function of stress, Equation (1) - G Gibbs free energy, Equation (2) - initial Gibbs free energy, Equation (29) - Gibbs free energy due to the instantaneous elastic response of the material, Equation (7) - Gibbs free energy due to memory, Equation (7) - _i temperature gradient , Equation (20) - H Helmholtz free energy, Equation (2) - J ₁ first invariant of the stress tensor, Equation (38) - I ₁ second invariant of the stress deviator tensor, Equation (38) - J(t) creep compliance function, Equation (1) - J _ijkl (t) temperature independent material property, Equation (8) - J _s steady creep compliance function, Equation (38) - J _tk transient creep compliance function, Equation (38) - J _I shear creep compliance function, Equation (34) - J _II bulk creep compliance function, Equation (34) - K isotropic thermal conductivity, Equation (42a) - K _ij thermal conductivity tensor, Equation (3) - M number of nonlinear memory integrals, Equation (36) - N=M+2 number of components of strain, Equation (49) - n steady creep power, Equation (45) - Q one-dimensional heat flux vector, Equation (48 b) - Q _i heat flux vector, Equation (3) - q _i transient creep powers, Equation (46) - S entropy per unit mass, Equation (4) - initial entropy density, Equation (29) - s _ij stress deviator tensor - T temperature - T ₀ constant reference temperature, Equation (4) - t time - V ₁,V ₂ wave speeds - uncoupled elastic mechanical wave speed - uncoupled thermal speed - x _i space coordinate - coefficient of thermal expansion, Equation (34) - _ij thermal strain coefficient, Equation (24) - , positive quantities in base characteristics equation, Equation (55) - one-dimensional strain - _ij strain tensor - ₁ linear elastic strain, Equation (49) - ₂ steady creep strain, Equation (49) - ₁,i=3,...,N transient creep strains, Equation (49) - _T thermal strain, Equation (49) - initial strain, Equation (29) - reciprocal of the isotropic conductivity, Equation (41) - reciprocal of the conductivity - temperature difference betweenT and a constant reference temperature, Equation (3) - ₀ initial temperature discontinuity, Equation (73) - ,µ _i material constants, Equation (45), (46) - ₁, ₂, ₃ functions of the three invariants of the stress tensor, Equation (35) - Lamé constants, Equation (57) - A rate of energy dissipation, Equation (13) - elastic Poisson's ratio, Equation (34) - reduced time, Equation (17) - mass density, Equation (2) - one-dimensional stress - _ij stress tensor - relaxation time of heat conduction, Equation (3) - _i retardation time in transient creep, Equation (39) - shift factor, Equation (37) - [ ] _j ,j=1, 2 indicates a discontinuity across the leading and lagging wave fronts respectively - designates dependent variables in (x _i , ) space This research was supported in part by the Office of Naval Research under Contract No. N00014-75-C-0302.     

Nachweis der Eigenwerte 0 und −1 in den Spektren von Integraloperatoren der ebenen Elastizitätstheorie

Zusammenfassung In den Spektren von Integraloperatoren der ebenen Elastizitätstheorie lassen sich für spezielle Scheibenformen mit Hilfe gezielter Ansätze isolierte Eigenwerte 0 und –1 nachweisen. Die physikalische Bedeutung dieser Eigenwerte und der dazugehörigen Eigenfunktionen läßt vermuten, daß die Eigenwerte auch in den Spektren der Operatoren für beliebig geformte Scheiben auftreten. In dieser Arbeit wird bewiesen, daß die Vermutung richtig ist.

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Proof of eigenvalues 0 and –1 of the spectra of integral operators of two-dimensional theory of elasticity

Summary By means of particular trial functions we can show that the spectra of some integral operators of the two dimensional theory of elasticity contain isolated eigenvalues 0 and –1 for specially shaped slices. From the physical meaning of these eigenvalues and the corresponding eigenfunctions it could be expected that the eigenvalues also occur in the spectra of operators of arbitrarily shaped slices. In this paper the conjecture is proved.

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Formelzeichen +K zu berechnende Scheibe - +S Rand von +K - ⁰ K ins Unendliche ausgedehnte Scheibe - S zu +S kongruente Kurve auf⁰ K - zuS äquidistante Kurve - d Abstand zwischenS und - F; vonS bzw. eingeschlossene Fläche - s, Bogenlängen von Punkten aufS bzw. - x _i , Ortsvektoren von Punkten aufS bzw. oder inF bzw. - a Krümmungsradius vonS - Abstand zwischen Quell- und Aufpunkt - n _i ;t _i Normalen- bzw. Tangenteneinheitsvektor - _i Spannungsvektor - _ij Spannungstensor - U _i Distorsionsvektor - U _ij Distorsionstensor - u _i Verschiebungsvektor - R _i Kräftebelegung - C _i Stufenversetzungsbelegung - M _i Singularitätenbelegung - (R) _ij , (C) _ij , (M) _ij , (UR) _ij , (UC) _ij , (UM) _ij Kerne der Integralgleichungen - (R) _ijk , (C) _ijk , (M) _ijk , (UR) _ijk , (UC) _ijk , (UM) _ijk , (uR) _ij , (uC) _ij , (uM) _ij Einflußfunktionen - _ij Einheitstensor - e _ij schiefsymmetrischer Tensor - e _ij Nablavektor - Eigenwert eines Operators Mit 4 Abbildungen     

Creep behaviour of polyethylene and polypropylene

Tensile creep tests and stress reduction studies during creep have been carried out for polyethylene and polypropylene. The results obtained suggest that a consistent approach for the presentation of creep data for these polymeric materials can be obtained since the creep curves at 293K for polyethylene and polypropylene over a wide stress range can be superimposed by describing the variation of creep strain,, with time,t, as= ₀ + _p [1 – exp (–K t)] + t, where ₀ is the initial strain on loading, _p is the primary creep strain, is the secondary creep rate, andK is a constant.     

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.     

An asymptotic solution for large Prandtl number free convection

Summary The set of ordinary differential equations governing free convection boundary layer flow past an isothermal semi-infinite vertical flat plate is solved for large Prandtl numbers by means of the method of matched asymptotic expansions. The analysis leads to an expression for heat transfer which contains the Prandtl number explicitly and which is very accurate for sufficiently large values of the Prandtl number. On the other hand the analysis also has qualitative assets. Before choosing the mathematical method of solution, the physical aspects of the large Prandtl number free convection boundary layer are investigated. The mathematical solution serves to enlarge our understanding of the physical implications of a free convection boundary layer in a large Prandtl number fluid.Nomenclature a_ij coefficient defined by - b_ij coefficient defined by F_j()=b_0j+b_1j +b_2j ²+.... - c coefficient defined by equation (3) - c_p specific heat - f non-dimensional stream function of inner expansion (7) - f_n n-th perturbation of f - F non-dimensional stream function of outer expansion (15) - g non-dimensional stream function (1) - ¯g acceleration due to gravity - Gr_x local Grashof number:g(T_w–T)x³/v² - h non-dimensional temperature (2) - k coefficient of heat conduction - Nu_x local Nusselt number: - T temperature - T_w wall-temperature - T ambient temperature - u longitudinal velocity - x co-ordinate measuring distance from the leading edge - y co-ordinate measuring distance normal to the plate Greek symbols coefficient of thermal expansion - _i expansion parameter (21) - expansion parameter (22) - _i expansion parameter (33) - expansion parameter (34) - expansion parameter: ^–1/2 - inner similarity co-ordinate (9) - non-dimensional temperature of inner expansion (8) - _n n-th perturbation of - non-dimensional temperature of outer expansion (16) - _n n-th perturbation of - similarity co-ordinate (3) - kinematic viscosity - outer similarity co-ordinate (17) - density - Prandtl number:c_p/k - stream function     

Density Response and Equilibration in a Pure Fluid Near the Liquid-Vapor Critical Point: 3He

We report measurements of the local density response inside a quasi 1-D horizontal ³ He fluid layer to a step-like change T of the boundary temperature, where |T| 80 K and much smaller than |T – T_c| where T_c is the critical temperature. These experiments used a new cell design, described in the text, and were carried out along the critical isochore both above and below T_c. The observed temporal and spatial density response (t, z) and its equilibration time are described adequately by the relations developed from the thermodynamic theory of Onuki and Ferrell. We verify that over the temperature range of low stratification, where computer simulations and closed-form calculations can be compared, they are in exact agreement. The systematic differences of experimental results from predictions can be accounted for by the departure of the cell from the ideal 1-D geometry. The much larger disagreement between the experimental and predicted equilibration time scale in earlier experiments is also explained. Finally, deviations from linearity observed in the density response for steps |T| larger than 90 K are reported and the implications of such nonlinearity for the (t, z) profile and especially the effective relaxation time _eff are analyzed. We also discuss the predicted onset of convection near T_c for the conditions in our experiment. In the Appendix, the likely sources for systematic deviations in the density response function for the experimental cell from calculations in the ideal 1-D geometry are presented and their effects calculated. The so-obtained response function Z_F(, z) is compared with previously published data.     

Zero-point motion and superfluid helium

We propose that He II exhibits macroscopic [ _P /N O(1)] quantum zero-point motion in momentum space, i.e., that a nonzero root-mean-square superfluid velocity exists even in an equilibrium superfluid system at rest. At absolute zero, using coherent states, we relate the uncertainty _P /N in the total momentumP (per particle) to the long-range-order (LRO) part of the phase gradient correlation function, which is proposed as an order parameter. The local equilibrium equation for the superfluid velocity potential derived by Biswas and Rama Rao yields, in the strict equilibrium limit, the equation determining this order parameter in terms of fluctuation correlations that remain to be determined. The order parameter is interaction dependent, nonzero atT=0 if (0)–₀V₀>0, and can vanish at some transition temperatureT when fluctuation terms become comparable to theT=0 value. (HereV _{0 0}, and (0) are the uniform parts of the potential, density, and chemical potential with shifted zero of energy, respectively.) A characteristic length (T), diverging atT=T , appears naturally, with its defining relation reducing to a macroscopic uncertainty relation ( _P /N)(0)=/2 atT=0. With certain assumptions it is shown that atT=0, LRO in the phase gradient correlation function is incompatible with off-diagonal long-range order (ODLRO) in the (r)(r) correlation function, and with nonzero condensate function.     

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.     

The dispersion of3He quasiparticles in He II from neutron scattering

In an inelastic neutron scattering (INS) experiment on³He-⁴He mixtures one observes, besides the photon-roton mode which is barely modified by the admixture of³He, an additional excitation at lower energies which is interpreted as quasi-particle-hole excitations of a nearly free Fermi gas. We reanalyse INS data ofx ₃=1% and 4.5% mixtures at various pressures to extract the mean energy of the fermions. In the momentum range 9<q<17 nm^–1 (above 2k _F ) follows very closely the relation =A ₂ q ²+A ₄ q ⁴ at all concentrations, pressures and temperatures observed. In a 4.5% mixture (T _F 0.3 K), measurements were performed for temperatures in the range 0.07<T<0.9 K. We find bothA ₂ andA ₄ to be strongly temperature dependent. For the interpretation of thermodynamical properties, the single particle energy _k is parametrized as _k =_o+1/(2ms*) ·k ₂ · (1+k ²). Neglecting interactions between fermions, we calculate from the free-particle _k the scattering functionS(q, ) and the mean value of the fermion peak energy _q = S ₃(q, )d/S ₃(q, )d. We find that follows closely _q , deviating at most by 10%. A comparison to the measuredA ₂ andA ₄ directly yieldsms* (x ₃,p, T) and (x ₃,p, T). In the limitx ₃=0,p=0 andT=0, the density and concentration dependence of the inertial mass is in excellent agreement with values found by Sherlock and Edwards. The temperature dependence of the specific heat data from Greywall and Owers-Bradleyet al. are well represented by our model atT<0,5 K.