A modified tube dilatation theory for the relaxation of entangled linear polymer melts

In this paper, the “tube dilatation” or “tube Enlargement” concept introduced by Marrucci, is revisited in the case of broad entangled linear polymer melts. Using the tube fluctuation relaxation function of Doi and a linear mixing rule, the model implicitly contains both the features of double reptation and time modification by tube renewal. It has been shown that theoretical arguments of both double reptation with tube renewal and tube dilatation can be used to take into account the effect of polydispersity on the distribution of relaxation times. The model has been tested for some polymers with various N/Ne values. However, experiments indicate that the loss of only one entanglement does not systematically induce a change of the relaxation times through a constraint release and tube renewal process. The freeing of a critical volume, larger than the volume of a tube segment, is required to induce an efficient dilution effect on relaxation times.     

Development of the “binary interaction” theory for entangled polydisperse linear polymers

Based on the theoretical principles previously described in the literature, the development of the “naïve” binary interaction model is detailed in this paper. The new theory is effectively a sweeping generalization of the “Double Reptation” model. The “switch function” has been shown to be an essential feature of any constraint release model for Doi–Edwards type molecular models that invoke the concept of a discrete slip-link tube and is used in our formulation. Using the assumption of a constant entanglement density, a slip link linear density evolution equation is derived to rigorously count matrix entanglements. This function has no counterpart in the conventional Doi–Edwards theory, or its derivatives, and is absolutely required to properly generalize the “Double Reptation” model so that nonlinear flows can be modeled. The binary interaction polydispersity model is complex mathematically but can be rigorously and justifiably simplified by suppressing the tube coordinate dependence using a boundary layer analysis. The simplification process can be continued to the continuum level to create a hierarchy of approximate binary interaction models, thereby making large-scale numerical simulations of complex flows viable, indeed straightforward.     

On Infinitesimal Shear

The setting for this note is the theory of infinitesimal strain in the context of classical linearized elasticity. As a body is subjected to a deformation the angle between a pair of material line elements through a typical point P is changed. The decrease in angle is called the shear of this pair of elements. Here, we determine all pairs of material line elements at P which are unsheared in a deformation. It is seen, in general, that corresponding to any given material line element in a given plane through P, there is one corresponding “companion” material line element such that the given element and its conjugate are unsheared in the deformation. There are two exceptions. If the plane through P is a plane of central circular section of the strain ellipsoid, then every material line element through P in this plane has an infinity of companion elements in this plane – all pairs of material line elements in the plane(s) of central circular section of the strain ellipsoid are unsheared. If the plane through P is not a plane of central circular section of the strain ellipsoid, then there are two exceptional material line elements through P such that neither of them has a companion material line element forming an unsheared pair with it. The directions of these exceptional elements in the plane are called “limiting directions”. It is seen that it is the pair of elements along the limiting directions in a plane which suffer the maximum shear in that plane. A geometrical construction is presented for the determination of the extensional strains along the pairs of elements which are unsheared. Also, it is shown that knowing one unsheared pair in a plane and their extensions is sufficient to determine the principal extensions and the principal axes in this plane. Expressions for all unsheared pairs in a given plane are given in terms of the normals to the planes of central circular sections of the strain ellipsoid. Finally, for a given material line element, a formula is derived for the determination of all other material line elements which form an unsheared pair with the given element.     

On the evolution of rotation of a solid under inelastic collisions with a plane

The motion of a solid in a homogeneous gravity field under inelastic collisions with an immovable absolutely smooth horizontal plane is considered. The body is a homogeneous ellipsoid of revolution. There exists a motion in which the ellipsoid symmetry axis is directed along a fixed vertical, the ellipsoid itself rotates about this axis at a constant angular velocity, and the ellipsoid bounce height over the plane decreases from impact to impact because of the collisions. We study the motion of the ellipsoid in a small neighborhood of the motion corresponding to this infinite-impact process. The main goal is to compute the angle between the ellipsoid symmetry axis and the vertical at the discrete time instants corresponding to the collisions. The problem is solved in the first (linear) approximation. The analysis is based on the canonical transformation method used earlier in [1] to solve problems with absolutely elastic collisions. There are quite a few studies dealing with infinite-impact processes (e.g., see the monographs [2, 3]). A method for continuous representation of systems with inelastic collisions was proposed in [4] and efficiently used in [3–5] when analyzing specific mechanical systems.     

Confined creeping flow around an axisymmetric body: increase of the shape effect by a tube wall

Using a specially adapted experimental technique, associated with a visualization method based upon solid tracers, we have obtained the flow pattern induced by the very slow uniform translation of an axisymmetric body along the axis of a vertical tube filled with a viscous liquid, both in a fixed frame (the frame is attached to the tube) and in a “relative” frame (the frame accompanies the body in its translation). The body, whose shape evolves from a sphere to a cylinder frustum, is free from any attachment or interaction with any other body; only the tube wall interaction is relevant. In these conditions, the upstream-downstream symmetry, relative to the creeping regime hypotheses, has been very well verified and quantitative information concerning, in particular, the velocity field has been deduced with sufficient precision (better than 2%) to exercise the control of a numerical process capable of giving all the details of the hydrodynamic field including those not directly available from the experiments. By comparison with the unbounded flows around the same bodies, the strong increase of the shape effect by the presence of the confining tube wall has been pointed out and evaluated, on the drag as well as on the surface vorticity and pressure distributions.     

Pressure field around a well in a stratified inhomogeneous bed

The steady axisymmetric flow of an incompressible fluid into a vertical well hydrodynamically perfectly drilled into a stratified inhomogeneous half-space consisting of three layers with different permeabilities is considered. The boundaries of the layers are assumed to be horizontal planes and the roof of the upper layer is assumed to be impermeable. The flow obeys a linear Darcy’s law. The pressure distribution on the well is assumed to be given, which is the main obstacle to finding an exact solution of the problem. Beginning with the classical studies of Muskat and Charnyi [1, 2], approximate solutions of such problems have been constructed as a superposition of flows generated by point sources with given intensities, distributed along the well axis in accordance with a fairly simple law. In the present study, this approach is used to obtain an integral equation for the source density distribution, which is then solved numerically. Comparison with the known exact solution for a thin elongated ellipsoid (“needle”) shows that this approach makes it possible to ensure an accuracy which at any rate is sufficient for applications.     

Elastic structural response of prismatic metal sandwich tubes to internal moving pressure loading

The analytical and numerical modeling of the structural response of a prismatic metal sandwich tube subjected to internal moving pressure loading is investigated in this paper. The prismatic core is equivalent to homogeneous and cylindrical orthotropic solids via homogenization procedure. The sandwich tube with the “effective” homogenized core is modeled using multi-layer sandwich theory considering the effects of transverse shear deformation and compressibility of the core; moreover, the solutions are obtained by using the precise integration method. Several dynamic elastic finite element (FE) simulations are carried out to obtain the structural response of the tube to shock loading moving at different velocities. The comparison between analytic solutions and FE simulations demonstrates that the transient analytical model, based on the proposed sandwich model, is capable of predicting the critical velocity and the dynamic structural response of the sandwich tube with the “effective” homogenized core with a high degree of accuracy. In addition, the critical velocity predicted using FE simulations of the complete model is not in agreement with that of the effective model. However, the structural response and the maximum amplification factors obtained using FE simulations of the complete model are nearly similar to that of the effective model, when the shock loading moves at the critical velocity. The influences of the relative density on the structural response are studied, and the capabilities of load bearing for sandwich tubes with different cores are compared with each other and with the monolithic tube. The results indicate that Kagome and triangle-6 are preferred among five topologies.     

Experimental investigation of the interaction between a pulsating bubble and a rigid cylinder

In this paper, the behavior of a bubble near a rigid cylinder is studied experimentally as the positions of bubble induction change, and several cylinders with different diameters are used in the experiment. The main results are as follows. The behavior of a bubble near a rigid cylinder is distinct from that near a rigid plate. When the cylinders are laid in deep water, there will occur three kinds of typical bubble shapes as the distance between bubble and cylinder increases. And the bubble shapes are different as the diameter of cylinder varies. When the cylinders are laid near a free surface, the behaviors of bubble near cylinders with different diameters are similar. For a certain distance between bubble and free surface, as the distance between bubble and cylinder increases, "double jet", "inclined jet" and "downward jet" will take place successively.     

Wave propagation in physiological collapsible tubes and a proposal for a Shapiro number

In this Brief Communication, a few introductory comments are made first, regarding wave propagation in collapsible tubes and flow limitation, as well as what is commonly referred to as the “speed index”, namely the ratio of the flow velocity in the collapsible tube divided by the wave-propagation velocity. It is then proposed that it is fitting for this speed index to be henceforth named as the Shapiro number.     

Shock Hugoniots based on the self-consistent average atom (SCAA) model. Theory and experiments. (Second revision)

We use a “self-consistent average atom” (SCAA) model to compute shock Hugoniots for aluminum, iron, molybdenum, strontium, barium and thulium. The pressures and energies include relativistic effects. We make comparisons with the Thomas-Fermi-Dirac (TFD) model and with the available experimental data including pressures, shock and particle speeds and energy deposition. The connection between the usage of the “average atom” (AA) model and “detailed configuration accounting” (DCA) is discussed in the Appendix.     

Combined stress waves with phase transition in thin-walled tubes简

The incremental constitutive relation and governing equations with combined stresses for phase transition wave propagation in a thin-walled tube are established based on the phase transition criterion considering both the hydrostatic pressure and the deviatoric stress. It is found that the centers of the initial and subsequent phase transition ellipses are shifted along the σ-axis in the στ-plane due to the tension-compression asymmetry induced by the hydrostatic pressure. The wave solution offers the "fast" and "slow" phase transition waves under combined longitudinal and torsional stresses in the phase transition region. The results show some new stress paths and wave structures in a thin-walled tube with phase transition, differing from those of conventional elastic-plastic materials.     

Capillary flow of linear polyethylene melt: sudden increase of flow rate

During the oscillatory flow of linear polyethylene (HDPE) melt through a capillary, the shape of the extrudate varies periodically with time: sharkskin, plug and rough. This paper deals with the transition between increasing and decreasing pressure. At that transition, the flow rate through the tube is suddenly and intensively increased. We present a theoretical analysis which is in good agreement with experiment. The “slip” at the wall is held to be a consequence of and not the cause of phenomenon. The radial profile of the shear rate is described by means of Dirac's delta function.     

Longitudinal diffusion in a flow through a tube

The results of research concerned with a fluid mixing during the movement in a tube, are given. A method of definining the one-dimension theory of matter transfer, accounting for the difference of mixture component velocities is presented. The longitudinal transfer in a zone of “passive” fluids contact is discussed in detail. It has been possible to formulate the theory, which generalises the well-known Taylor and Aris models. The theory presented is based on the integro-differential equation, accounting for the delay effects. It has been possible to describe the experimental facts, which had no explanation so far, in bounds of the given theory.     

Determination of the size of the representative volume element for random quasi-brittle composites

A representative volume element (RVE) is related to the domain size of a microstructure providing a “good” statistical representation of typical material properties. The size of an RVE for the class of quasi-brittle random heterogeneous materials under dynamic loading is one of the major questions to be answered in this paper. A new statistical strategy is thus proposed for the RVE size determination. The microstructure illustrating the methodology of the RVE size determination is a metal matrix composite with randomly distributed aligned brittle inclusions: the hydrided Zircaloy constituting nuclear claddings. For a given volume fraction of inclusions, the periodic RVE size is found in the case of overall elastic properties and of overall fracture energy. In the latter case, the term “representative” is discussed since the fracture tends to localize. A correlation factor between the “elastic” RVE and the “fracture” RVE is discussed.     

Axisymmetric cavitation flow around a body in a tube

The problem of the axisymmetric flow around a body in a circular tube with arbitrary shape of the meridian section is reduced to the numerical solution of a system of two integral equations to determine the shape of the cavern and the intensity of the vortex rings arranged on the solid boundaries and the cavern boundary. Results of computations of the cavitation flow around a sphere, ellipsoid of revolution, and cone in a cylindrical tube, and also for a cone in converging and expanding tubes and in a hydrodynamic tunnel with the actual shape of the converging and working sections, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 50–55, July–August, 1976.     

Particle sizing by capillary hydrodynamic chromatography

Liquid borne ultrafine particles have elsewhere been shown to pass through a packed column at rates varying with particle size. The process is termed “hydrodynamic chromatography”, since a separation of the particles according to size can thereby be achieved. Similar results are reported herein for larger particles suspended in liquid when the passageway, instead of being a packed column is a long, completely open, capillary tube. Particle transit time is a function of the logarithm of particle diameter. The underlying mechanism for the latter phenomena is believed to be that previously called the “tubular pinch effect” which arises from a balance of lateral forces within the flow field, but experimental results are not adequately described by any theory yet proposed. The open-tube discrimination process is designated “capillary hydrodynamic chromatography”.     

The presence of slug flow in horizontal two-phase flow

One of the flow regimes occurring in horizontal two-phase flows is characterized by periodic large waves “surging” along the tube. This flow, called “slug” flow, has been frequently observed in low and high pressure gas liquid systems, but it has been noticed that slugging is absent in certain liquid-liquid two-phase systems. A method is developed giving the necessary conditions for the presence of slug flow. This method quantitatively explains the observed absence of slugging in certain liquid-liquid flows.     

Mixing Measurements in a Supersonic Expansion-Ramp Combustor

This paper reports results on molecular mixing for injection via an expansion-ramp into a supersonic freestream with M ₁ = 1.5. This geometry produces a compressible turbulent shear layer between an upper, high-speed “air” stream and a lower, low-speed “fuel” stream, injected through an expansion-ramp at α = 30° to the high-speed freestream. Mass injection is chosen to force the shear layer to attach to the lower guide wall. This results in part of the flow being directed upstream, forming a recirculation zone. Employing the hypergolic hydrogen-fluorine chemical reaction and pairs of “flip” experiments, molecular mixing is quantified by measuring the resulting temperature rise. Initial experiments established the fast-chemistry limit for this flow in terms of a Damköhler number (Da). For Da ≥ 1.4, molecularly mixed fluid effectively reacts to completion. Parameters varied in these experiments were the measurement station location, the injection velocity of the (lower) “fuel” stream, the stoichiometry for the flip experiments, and the density ratio of the fuel and air streams. As expected, mixing increases with increasing distance from the injection surface. The mixed fluid fraction increases by 12% when changing the fuel-to-air stream density ratio from 1 to 0.2. Comparisons with measurements at subsonic (high-speed) “air” stream velocities show that the trend of decreasing mixing with increasing speed documented in free-shear layer flows is also encountered in these flows. The current geometry produces higher mixing levels than do free shear layers.     

Interaction of a heavy fluid flow in a channel with a transverse jet barrier

An experimental and numerical analysis of the interaction between a plane horizontal water flow in a rectangular channel (free water current) and a plane thin water jet (water jet curtain) is presented; the jet flows out vertically from either a slot nozzle in the bottom of the channel or the crest of a rigid spillway at a velocity appreciably (several times) greater than the water velocity in the channel. Numerical calculations were carried out using the STAR-CD software package preliminarily tested against the experimental data obtained. The dependence of the water level in the channel at a certain distance ahead of the jet barrier on the main jet parameters and the water flow rate in the horizontal channel is studied. It is found that in the region of the interface between the flows both steady and unsteady (self-oscillatory) flow patterns can be realized. Steady stream/jet interaction patterns of the “ejection” and “ejection-spillway” types are distinguished and a criterion separating these regimes is obtained. The notion of a rigid spillway equivalent to a jet curtain is introduced and an approximate dependence of its height on the relevant parameters of the problem is derived. The possibility of effectively controlling the water level ahead of a rigid spillway with a sharp edge by means of a plane water jet flowing from its crest is investigated. The boundary of transition to self-oscillation interaction patterns in the region of the flow interface is determined. The structure of these flows and a possible mechanism of their generation are described. Within the framework of the inviscid incompressible fluid model in the approximate formulation for a “thin” jet, an analytical dependence of the greatest possible depth of a reservoir filled with a heavy fluid at rest and screened by a vertical jet barrier on the jet parameters is obtained.     

Ellipsoidal model of the rise of a Taylor bubble in a round tube

The rise velocity of long gas bubbles (Taylor bubbles) in round tubes is modeled by an ovary ellipsoidal cap bubble rising in an irrotational flow of a viscous liquid. The analysis leads to an expression for the rise velocity which depends on the aspect ratio of the model ellipsoid and the Reynolds and Eötvös numbers. The aspect ratio of the best ellipsoid is selected to give the same rise velocity as the Taylor bubble at given values of the Eötvös and Reynolds numbers. The analysis leads to a prediction of the shape of the ovary ellipsoid which rises with same velocity as the Taylor bubble.