Effect of alkali metal ions on the viscoelasticity of concentrated kappa-carrageenan and agarose gels

The effect of the addition of the monovalent cations Li⁺, Na⁺, K⁺, and Cs⁺ on the gelation of agarose and kappa-carrageenan aqueous gels has been studied by the measurement of longitudinal vibration. The dynamic Youngs's modulusE of 2% w/w agarose and 0.4–6% w/w kappa-carrageenan gels containing the alkali metal salt LiCl, NaCl, KCl or CsCl of various concentrations from 0 to 4.5 mol/l has been measured at various temperatures. By the addition of the alkali metal salt, the value ofE for agarose gels is influenced only slightly, while for kappa-carrageenanE is increased substantially. Kappa-carrageenan has many sulphate groups. The addition of the alkali metal ions screens the electrostatic repulsion between these groups. As a result of this, the helical structure of kappa-carrageenan is stabilised and the helices may form densely packed aggregates, so increasingE. In contrast, agarose has a naturally stable molecular structure and therefore, the structure and henceE is not sensitive to added ions. The K⁺ and Cs⁺ ions increaseE more than Li⁺ and Na⁺ for kappa-carrageenan gels. This is interpreted on the basis that these ions are either structure ordering or structure disordering ions for water.     

Deposition of droplets from turbulent stream

A model of particle deposition from a turbulent stream is presented. It is based on the modified stopping distance concept which allows for the difference between particle and eddy diffusivities, recognizing that the particle has equal probability to move toward the wall or back into the turbulent core. The theory is tested against the data which cover a wide range of droplet size and duct Reynolds number for different surface configuration. A quite satisfactory agreement has been found in all examined cases.     

Determination of the body shape for minimum radiative heating along a flight trajectory

Space vehicles are subject to intense aerodynamic heating in planetary entry. According to estimates in [1], the heat shield mass for entry of a probe into the atmospheres of the outer planets can make up 20–50% of its total mass; here the radiative component predominates in the aerodynamic heating. It is therefore interesting to investigate methods of reducing the heat flux to the nose region of a vehicle. Analysis shows [2–6] that, for a given atmospheric composition, the heat-shield weight is determined by the trajectory, the body shape, the heat-protection method, and the chemical composition and the thermophysical and optical properties of the heat shield material. In such a general statement of the problem, optimization of the heat-shield mass depends on many parameters, and has not been solved hitherto. A number of papers have examined simpler problems, associated with reducing spacevehicle heating: optimization of the trajectory from the condition that the total heat flux to the body stagnation point should be a minimum for given probe parameters [2, 3], optimization of the characteristic probe size for a given trajectory [2–4], and optimization of the probe shape in a class of conical bodies at a given trajectory point [3, 5, 6J. In [7] a variational problem was formulated to determine the shape of an axisymmetric body from the condition that the radiative heat flux to the body at a given trajectory point should be a minimum for the entire surface, and an analytical solution was found for this in limiting cases. The present paper investigates a more general variational problem: determination of the shape of an axisymmetric body from the condition that the total radiative influx of heat to the body along its atmospheric trajectory should be a minimum. A solution has been obtained for a class of slender bodies for different isoperimetric conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–100, March–April, 1978.     

Viscosity of particle filled polymer melts

Summary Viscosities of particle filled polymer melts were measured at fairly low shear rate. Particles were glass beads, glass balloons, and silas balloons. Polymers were polyethylene and polystyrene. Flow curves were superimposed with respect to concentration of filler. The relative viscosity defined as the ratio of viscosity of filled polymer to that of unfilled polymer at the same shear stress is obtained as an asymptotic value even for highly filled material. The relation between relative viscosity and volume fraction of filler was represented by the equations derived byMaron andPierce orMooney. Only the distribution of particle size had influence on relative viscosity at a defined filler concentration. Yield stresses were estimated, and found to increase exponentially in the range of volume fraction from 0.1 to 0.5.

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Zusammenfassung Viskositäten von Polymerschmelzen, die mit Teilchen gefüllt worden waren, wurden bei mittleren Schergeschwindigkeiten bestimmt. Als Teilchen wurden Voll- und Hohlkugeln aus Glas, sowie sog. silas balloons, als Polymere Polyäthylen und Polystyrol verwendet. Die Fließkurven für verschiedene Füllstoffkonzentrationen wurden überlagert. Die relative Viskosität, definiert als das Verhältnis der Viskositäten von gefülltem und ungefülltem Polymer bei gleicher Schubspannung, besitzt einen asymptotischen Wert selbst für hoch gefülltes Material. Die Beziehung zwischen relativer Viskosität und Füllstoff-Volumenkonzentration läßt sich durch eine vonMaron undPierce oder eine vonMooney abgeleitete Gleichung beschreiben. Nur die Teilchengrößenverteilung hat bei einer definierten Füllstoffkonzentration einen Einfluß auf die relative Viskosität. Fließspannungen werden abgeschätzt und dafür in einem Konzentrationsbereich zwischen 0,1 und 0,5 ein exponentieller Anstieg gefunden.

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With 13 figures and 2 tables     

Harmonic non-linear response of beck's column to a lateral excitation

The non-linear response of a column with a follower force (Beck's column) subjected to a distributed periodic lateral excitation, or to a support excitation, is determined. An analytical solution for the response amplitude in terms of the loading and system parameters is obtained by a perturbation analysis of the differential equations of motion. Non-linear inertia and non-linear curvature terms are taken into account in the formulation of the differential equations.     

Closed form solutions for small deformations superimposed upon the symmetrical expansion of a spherical shell

For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are super-imposed upon the symmetrical expansion of a spherical shell, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the n=1 buckling criterion for thick-walled spherical shells which are subjected to uniform external pressure. They are also used to deduce an upper bound to the force deflection relation for small superimposed translational deflections of bonded pre-compressed spherical rubber bush mountings.

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Zusammenfassung Für den Fall kleiner, axialsymmetrischer Deformationen isotropischer inkompressibler hyperelastischer Materialien, die der symmetrischen Dehnung einer sphärischen Schale überlagert sind, werden Lösungen in geschlossener Form abgeleitet, ohne einschränkende Bedingungen für die Deformationsenergiefunktion. Mit Hilfe dieser Lösungen wird das Knick-Kriterium (n=1) für dickwandige sphärische Schalen gewonnen, die gleichförmigem, äusserem Druck, ausgesetzt sind. Weiterhin wird mit Hilfe der Lösungen eine obere Grenze gewonnen für die Kraft-Ablenkungs-Relation im Falle überlagerter kleiner Translationen von gebundenen, vorgespannten sphärischen Gummibuchsen.

     

Postbuckling of angle-ply and anisotropic plates

Summary An analysis based on the von Kármán-type large-deflection equations of plates is presented for the postbuckling behavior of homogeneous and symmetrically laminated anisotropic rectangular plates. Boundary conditions for both simply supported and clamped edges are considered. A solution is obtained by expressing the force function and transverse deflection as double series involving the appropriate beam eigen-functions. Numerical results are presented for graphite-epoxy angle-ply laminates and homogeneous anisotropic plates. In the cases of buckling of anisotropic plates and postbuckling of isotropic and orthotropic plates, the present solution is in good agreement with existing solutions.

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Übersicht Es wird das Nachbeulverhalten von homogenen, symmetrisch geschichteten, anisotropen Rechteckplatten auf der Grundlage der von Kármán angegebenen Platten-Gleichungen für große Auslenkungen untersucht. Dabei werden sowohl einfach aufliegende, wie auch eingespannte Ränder als Randbedingungen berücksichtigt. Lösungen werden dadurch gewonnen, daß die Belastungsfunktion und die Querauslenkung als doppelte Reihen von geeigneten Balken-Eigenfunktionen angesetzt werden. Für Graphit-Epoxid-Schichtplatten und anisotrope homogene Platten werden numerische Ergebnisse mitgeteilt. Die erhaltenen Lösungen stimmen in den Fällen des Beulens von anisotropen Platten sowie des Nachbeulens von isotropen und orthotropen Platten gut mit bekannten Lösungen überein.

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The results presented in this paper were obtained in the course of research sponsored by the National Research Council of Canada.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Numerical investigation of nonequilibrium flows of mixtures of fusible metal particles and gases in a laval nozzle

There are numerous papers [1–11] on the determination of the parameters of condensed oxide particles which are formed during combustion of metallized fuels. The ambiguity, and sometimes the contradictoriness, of the test results obtained [3–5, 9–11] indicate the difficulties in conducting correct experimental investigations. In this connection, numerical studies using mixtures of calibrated liquid-metal particles and different gases are of practical interest. Different probes can be calibrated by using calibrated two-phase flows, the two-phase flow around models and probes can be studied, as can the interaction between liquid-metal particles and the front of an aerodynamic compression shock, their intrusion in different entraining media, the interaction between fine particles (particle-projectiles) and large size particles (particle-targets), etc. In many cases, the prehistory of the flow and the parameters of the gas mixture with the particles in the area of the nozzle exit section must be known to investigate the above-mentioned phenomena. The parameters of different nonequilibrium flows of mixtures of gallium particles and gases in a Laval nozzle are investigated numerically in this paper; the maximum diameter (upper boundary of the spectrum) of the particles (d_s = 30 ) which are not destroyed in the nozzle under the effect of the aerodynamic forces and are suitable for use in a calibrated two-phase stream is determined. The computations were carried out in a one-dimensional approximation according to [12–14].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 86–91, March–April, 1976.The authors are grateful to V. K. Starkov and U. G. Pirumov for discussing the results of the research and to N. M. Alekseev for aid in constructing the graphs.     

Heat exchange between a selectively emitting liquid and a laminar gas flow in the presence of an external source of radiation

An investigation is conducted in the solution of a number of practical problems of the radiative and combined heat exchange in nonuniform systems having widely different physical properties. The processes of thermal interaction between the ocean and the atmosphere have been treated in the paper [1], the effect of thermal radiation on the melting and solidification of semitransparent crystals has been investigated in [2], the flow of a selectively emitting gas around the lateral surface of an object evaporating under the action of radiative heating has been discussed in [3], and heat transfer from a jet to the molten mass of glass in a glassmaking furnace tank has been investigated in [4]. The radiative and combined heat exchange between a selectively emitting liquid and a transparent heat-conducting laminar gas flow in the case of a specified external thermal radiation field is discussed in this paper. The energy conservation equations are set up taking into account the heat transfer by radiation, convection, and molecular thermal conduction. A differential approximation is used in calculating the values of the radiation fluxes. A system of fundamental computational equations is solved by the method of finite differences and iterations and by the Runge-Kutta method. The results of the calculations are presented in the form of graphs.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 116–122, May–June, 1976.