Viscoelastic properties of semidilute poly(methyl methacrylate) solutions

Viscoelastic properties were examined for semidilute solutions of poly(methyl methacrylate) (PMMA) and polystyrene (PS) in chlorinated biphenyl. The number of entanglement per molecule, N, was evaluated from the plateau modulus, G _N . Two time constants, _s and ₁, respectively, characterizing the glass-to-rubber transition and terminal flow regions, were evaluated from the complex modulus and the relaxation modulus. A time constant _k supposedly characterizing the shrink of an extended chain, was evaluated from the relaxation modulus at finite strains. The ratios ₁/ _s and _k / _s were determined solely by N for each polymer species. The ratio ₁/ _s was proportional to N ^4.5 and N ^3.5 for PMMA and PS solutions, respectively. The ratio _k / _s was approximately proportional to N ^2.0 in accord with the prediction of the tube model theory, for either of the polymers. However, the values for PMMA were about four times as large as those for PS. The result is contrary to the expectation from the tube model theory that the viscoelasticity of a polymeric system, with given molecular weight and concentration, is determined if two material constants _s and G _N are known.     

Testing viscometric predictions of the internal viscosity model,using dilute viscous theta solutions

A stress-symmetrized internal viscosity (I.V.) model for flexible polymer chains, proposed by Bazua and Williams, is scrutinized for its theoretical predictions of complex viscosity ^* () = – i and non-Newtonian viscosity (), where is frequency and is shear stress. Parameters varied are the number of submolecules,N (i.e., molecular weightM = NM _s ); the hydrodynamic interaction,h ^*; and/f, where andf are the I.V. and friction coefficients of the submolecule. Detailed examination is made of the eigenvalues _p (N, h ^*) and how they can be estimated by various approximations, and property predictions are made for these approximations.Comparisons are made with data from our preceding companion paper, representing intrinsic properties [], [], [] in very viscous theta solutions, so that theoretical foundations of the model are fulfilled. It is found that [ ()] data can be predicted well, but that [ ()] data cannot be matched at high. The latter deficiency is attributed in part to unrealistic predictions of coil deformation in shear.     

The Preston tube in adiabatic compressible flow

A new procedure for the reduction of Preston tube data is introduced, based on the van Driest transformation. It appears to give results agreeing with the better calibration experiments, although a significant assumption in its derivation is violated.List of Symbols M _s Mach number sensed by Preston tube - M Friction Mach number (=u/wall sound speed) - R Gas constant - T _w Wall temperature - d Diameter of Preston tube - h Height of effective centre of Preston tube - p Preston tube pressure difference reading - p _i Equivalent incompressible Preston tube reading - p _w Wall pressure - r Recovery factor (=0.896) - u Friction velocity (=[_w/wall density]^1/2) - Empirical constant allowing for departure from Crocco temperature-velocity correlation (=0.975) - Specific heat ratio - Fluid kinematic viscosity - _w Wall shear-stress     

On the boundary conditions at the macroscopic level

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium. Roman Letters _sf interfacial area of the s-f interface contained within the macroscopic system m² - A _sf interfacial area of the s-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - d _s , d _f microscopic characteristic length m - g vector that maps to _s, m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l REV characteristic length, m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for the f-phase at the f-s interface - n _e outwardly directed unit normal vector at the dividing surface - T ^* macroscopic temperature field obtained by solving the macroscopic equation (3), K - V averaging volume, m³ - V _s , V _f volume of the considered phase within the averaging volume, m³ - volume of the macroscopic system, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² Greek Letters _s , _f volume fraction - ratio of thermal conductivities - _s , _f thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding to T ^* , K - spatial deviation temperature K - error on the temperature due to the macroscopic boundary conditions, K - spatial average - ^s , ^f intrinsic phase average     

Calculation of the pressure on the side surface of bodies consisting of a spherical segment joined to an inverted cone in a supersonic stream

The results of investigations of inviscid flow over inverted cones with nose consisting of a spherical segment were published for the first time in Soviet literature in [1–4]. In the present paper, a numerical solution to this problem is obtained using the improved algorithms of [5, 6], which have proved themselves well in problems of exterior flow over surfaces with positive angles of inclination to the oncoming flow. It is shown that the Mach number 2 M , equilibrium and nonequilibrium physicochemical transformations in air (H = 60 km, V = 7.4 km/sec, R₀ = 1 m), and the angle of attack 0 40° influence the investigated pressure distributions. A comparison of the results of the calculations with drainage experiments for M = 6, = 0-25° confirms the extended region of applicability of the developed numerical methods. Also proposed is a simple correlation of the dependence on the Mach number in the range 1.5 M of the shape of the shock wave near a sphere in a stream of ideal gas with adiabatic exponent = 1.4.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 178–183, January–February, 1981.     

Fixed point theorem of nonexpansive mappings in convex metric spaces

Let X be a convex metric space with the property that every decreasing sequence of nonenply dosed subsets of X with diameters tending to has menemptyintersection. This paper proved that if T is a mapping of a elosed conver nonempty subset K of X into itself satisfying the inequality:d(T_x,T_y)≤ad(x,t)+b{d(x,T_x)+d(y,T_y)}+c{d(x,T_x)+d(y,T_y)}for all x,y in K,where 0≤a<1,b≥0,c≥0,a+c≠0 and a+2b+3c≤1, then T has a unique fixed point in K.     

Bifurcation Phenomena from Standing Pulse Solutions in Some Reaction-Diffusion Systems

Bifurcation phenomena from standing pulse solutions of the problem is considered. (>0) is a sufficiently small parameter and is a positive one. It is shown that there exist two types of destabilization of standing pulse solutions when decreases. One is the appearance of travelling pulse solutions via the static bifurcation and the other is that of in-phase breathers via the Hopf bifurcation. Furthermore which type of destabilization occurs first with decreasing is discussed for the piecewise linear nonlinearities f and g.     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

Memory effects during the flow of thixotropic fluids in pipes

Summary The paper presents the phenomenon of thioxotropy from the point of view of the theory of fluids with fading memory. In the first part of the paper the mechanism of thixotropy was discussed in order to justify the application of the concept of structural parameter (this parameter occurs in the previously presented rheological model of thixotropic materials). In the second part of the paper an equation was derived, which enables the prediction of the mean value of the friction factor during the flow of a thixotropic fluid in a pipe. According to the obtained equation the friction factor is a function of three dimensionless numbers: the generalized Reynolds number, a modified Deborah number and a new dimensionless number which may be called a structural number. The preliminary experimental results confirmed the applicability of the obtained equation.

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Zusammenfassung Die Veröffentlichung behandelt das Phänomen der Thixotropie vom Standpunkt der Theorie der Flüssigkeiten mit schwindendem Gedächtnis. Im ersten Teil wird der Mechanismus der Thixotropie untersucht und die Einführung eines sog. Strukturparameters begründet (dieser Parameter kommt bereits in dem früher behandelten rheologischen Modell eines thixotropen Körpers vor). Im zweiten Teil wird dann eine Formel abgeleitet, welche die Voraussage des mittleren Wertes des Widerstandskoeffizienten bei der Strömung einer thixotropen Flüssigkeit durch ein Rohr ermöglicht. Dieser Formel gemäß ist der Widerstandskoeffizient eine Funktion von drei dimensionslosen Zahlen: einer verallgemeinerten Reynolds-Zahl, einer modifizierten Deborah-Zahl und einer neuen dimensionslosen Zahl, die als Struktur-Kennzahl bezeichnet werden kann. Die vorläufigen Versuchsergebnisse bestätigen die Brauchbarkeit der abgeleiteten Formel.

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a rheological parameter in eq. [1], s^–1 - A rheological parameter in eq. [1]; function defined in eq. [15] - b rheological parameter in eq. [1] - B constant in eq. [15] - c rheological parameter in eq. [4] - c function defined in eq. [4] - C function defined in eq. [48] (see also eq. [43]) - D pipe diameter,m - K ₁,K₂ coefficients of proportionality in eq. [6] - k rheological parameter in eq. [12], Nsⁿ/m² - k ^* rheological parameter in eq. [1], Ns^m/m² - L pipe length, m - m rheological parameter in eq. [1] - n rheological parameter in eq. [12] - N number of particles in unit volume - p pressure, Pa - p ₀ pressure at the pipe entrance, Pa - r radial coordinate, m - R pipe radius, m - s rheological parameter in eq. [1] - t time, s - u _z axial local velocity in the pipe, m/s - v mean linear velocity in the pipe, m/s - z axial coordinate, m - rheological parameter in eq. [5], = 1 s - shear rate, s^–1 - nominal shear rate defined by eq. [39], s^–1 - structural parameter - substantial derivative of structural parameter, s^–1 - _e equilibrium structural parameter in eqs. [2] and [5] - _en nominal structural parameter - ₀ initial value of structural parameter - function of natural time - mean value of natural time, s - shear stress, Pa - ₀ shear stress field atZ = 0 (at pipe entrance) - _y0 equilibrium yield stress, Pa - shear stress field atz - fluid density, kg/m³ - v number of bonds in an average aggregate - mean value of the friction factor - De modified Deborah number defined by eq. [46] - Re generalized Reynolds number defined by eq. [45] - Se structural number defined by eq. [41a] With 4 figures and 1 table     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Spectral and correlation characteristics of the turbulent boundary layer on an extended flexible cylinder

In marine geophysical seismological prospecting extensive use is made of towed receiving systems consisting of extended flexible cylinders containing acoustic sensors over which the water flows in the longitudinal direction. The boundary layer pressure fluctuations on the cylinder surface are picked up by the sensors as hydrodynamic noise. This paper is concerned with the study of the energy and spacetime characteristics of the pressure fluctuations in the turbulent boundary layer on an extended flexible cylinder in a longitudinal flow. The pressure fluctuations on the cylinder surface have been investigated experimentally for Re_X=(2–4)·10⁷, a dimensionless diameter of the pressure fluctuation sensors d _p ⁺ =d_pu^*/=500, where d_p is the sensor diameter, u^* the dynamic viscosity, and the viscosity coefficient, and frequencies 0.02311.259 (=^*/U). The spectral and correlation characteristics of the pressure fluctuations on the surface of the flexible cylinder are found to differ from the corresponding characteristics for a rigid cylinder [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i aza, No, 5, pp. 49–54, September–October, 1989.     

The eddy viscosity of turbulent equilibrium layers

Summary Similarity laws for the mean flow and scaling laws for the turbulent motion are used in an attempt to obtain a general expression for the eddy viscosity of equilibrium layers. It is found that =0.09 w ² /w*, in which w ² is a Reynolds stress representative for the region of overlap between the law of the wall and the velocity-defect law, while w* is the logarithmic slope of the mean velocity profile in that region. The distinction between w and w* is related to the strong inhomogeneity of the mean rate of strain in the inner layer. The results of the theory agree with experimental evidence obtained from transpired equilibrium layers.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Wärmeübergang bei erzwungener turbulenter Strömung in Ringspalten und örtlich beliebig veränderlichen Randbedingungen zweiter Art

Zusammenfassung Das Problem des Wärmeübergangs bei turbulenter Strömung in konzentrischen Ringspalten wird für den dreidimensionalen Fall theoretisch gelöst, wobei die Wandwärmestromdichte sowohl in azimutaler als auch in axialer Richtung beliebig variiert. Die Lösung der Energiegleichung erfolgt mit der klassischen Methode der Superposition und Trennung der Variablen, wobei das dabei auftretende Sturm-Liouvillesche Eigenwertproblem numerisch gelöst wird. Zur Lösung werden Verteilungen für die Geschwindigkeit und anisotropen turbulenten Austauschgrößen verwendet, die mit dem phänomenlogischen Turbulenzmodell von Ramm berechnet wurden. Ergebnisse werden über einen weiten Bereich der Reynolds-Zahl (10⁴ Re 10⁶), der Prandtl-Zahl (0 Pr 100) und für verschiedene Radienverhältnisse diskutiert.

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Turbulent forced convection heat transfer in annuli with arbitrarily varying boundary conditions of second kind

The problem of turbulent flow heat transfer in concentric annuli is analysed for the general threedimensional case in which the wall heat flux varies arbitrarily in both the circumferential and axial directions. The energy equation is solved using the classical method of superposition and separating variables, where the resulting Sturm-Liouville problem are evaluated numerically. The solution is based on velocity profiles and anisotropic thermal turbulent transport properties evaluated by Ramm's phenomenological turbulence model. Results are discussed over a wide range of Reynolds number (10⁴ Re 10⁶), Prandtl number (0 Pr 100) and radius ratio.

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Bezeichnungen a,b Fourierkoeffizienten - B geometrische Funktion, [s(1-r) + r]/(1–s) - C Koeffizienten - D hydraulischer Durchmesser, 2(r₂ – r₁) - E Energietransportfunktion - f axiale Wärmestromdichteverteilung - F azimutale Wärmestromdichteverteilung - g radiale Temperaturfunktion - l Kanallänge - L dimensionslose Kanallänge, 1/D - M axialer Temperaturgradient im thermisch ausgebildeten Bereich - n harmonischer Parameter - Nu Nusselt-Zahl - Pe Péclet-Zahl - Pr Prandtl-Zahl - q Wärmestromdichte - Q dimensionslose Wärmestromdichte, q/q₀ - r dimensionslose radiale Koordinate, (R-r₁)/(r₂-r₁) - r₁,r₂ innerer und äußerer Ringspaltradius - R radiale Koordinate - Re Reynolds-Zahl - s Ringspaltverhältnis, r₁/r₂ - T dimensionslose Temperatur, 2· · (-_E/(D· q₀ - u dimensionslose Geschwindigkeit, U/U_m - U Geschwindigkeit - x dimensionslose axiale Koordinate, X/D - X axiale Koordinate - Wärmeübergangskoeffizient - _un modifizierter Eigenwert - halber Segmentwinkel - turbulente Austauschgröe - Temperatur - dimensionslose Temperaturdifferenz, T - T_m - Wärmeleitfähigkeit - _un Eigenwerte - kinematische Viskosität - azimutale Koordinate - Eigenfunktionen Indizes e thermischer Einlauf - E Eintritt bei x=0 - H Wärme - i Bedingung an der i-ten benetzten Oberfläche (i=1 – Innenrohr, i=2 - Außenrohr) - j Bedingung, wenn nur an der j-ten Oberfläche des Ringspaltes die Wärme übertragen wird (j=1,2) - ij Bedingung an der i-ten Oberfläche, wenn nur an der j-ten Oberfläche des Ringspaltes die Wärme übertragen wird (ij=11, 12, 22, 21) - m mittel - n Ordnung der Harmonischen - r radiale Richtung - u Ordnung des Eigenwertproblems - azimutale Richtung - 0 umfangskonstant - thermisch ausgebildet     

Analysis of melt spinning transients in Lagrangian coordinates

Transients in melt spinning of isothermal power law and Newtonian fluids were found to be governed by an extremely simple partial differential equation ² ( ^1/n )/() = 0 in Lagrangian coordinates where is the cross-sectional area,n the power law exponent, the time and the the time at which a fluid molecule constituting the spinline left the spinneret. The general integral ^1/n =f() +g () of the above governing equation containing two arbitrary functions represents physically attainable spinline transients. Hitherto unknown analytical transient solutions of the above governing equation were obtained for the response of isothermal constant tension spinlines to a stepwise change in tension, spinneret hole area, extrusion speed or extrusion viscosity and for the starting transient in gravitational spinning. Linearized perturbation solutions and the stability limit of the spinline derived from the above new found nonlinear solutions were in agreement with previous findings and the above nonlinear response of the spinline to a step increase in the spinneret hole area was found to be equivalent to Orowan's tandem cylinder model of dent growth in filament stretching.     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.     

On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method

In this paper, the compactness of quasi-conforming element spaces and the—convergence of quasi-conforming element method are discussed. The well-known Rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties, and the generalized Poincare inequality. The generalized Friedrichs inequality and the generalized inequality of Poincare-Friedrichs are proved true for them. The error estimates are also given. It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity, and satisfy the rank condition of element and pass the test IPT. As practical examples, 6-parameter, 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and 21-paramenter quasi-conforming elements are shown to be convergent, and their L²²()-errors are O(h), O(h), O(h ² ), O(h ² ), O(h ), and O(h ⁴ ) respectively.     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

The onset of Taylor-like vortices in the flow induced by an impulsively started rotating cylinder

The onset of instability in the flow by an impulsively started rotating cylinder is analyzed under linear theory. It is well-known that at the critical Taylor number T_c=1695 the secondary flow in form of Taylor vortices sets in under the narrow-gap approximation. Here the dimensionless critical time _c to mark the onset of instability for TT_c is presented as a function of the Taylor number T. Available experimental data of water indicate that deviation of the velocity profiles from the primary flow occurs starting from a certain time 4_c. It seems evident that during _c4_c the secondary flow is very weak and the primary state of time-dependent annular Couette flow is maintained.     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).