The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112070000745

The stress system in a suspension of force-free particles

The entanglement concept in polymer rheology

Unperturbed dimensions of stereoregular polymers.- Reinforcement of rubber by carbon black.- Transformations of phenolic antioxidants and the role of their products in the long-term properties of polyolefins.

Properties of polymers

Publisher Summary It is noted that the gel is the most typical state for polysaccharides, both in biological and artificial systems. The polymer chains usually form an interconnected network that gives rise to characteristic texture and properties. Polysaccharide gels have various biological functions essentially in the wall of the young plant-cell, in animal fluids and connective tissues, and in the bacterial capsule. They also have widespread commercial uses, particularly in foodstuffs, cosmetics, paper, and textiles. The gel state is considered characteristic of polysaccharides in the way that polypeptides and globular proteins can characteristically occur as compact particles having a high degree of internal order, or nucleic acids can occur as chain pairs in highly ordered and complementary association. Polysaccharides also make a distinctive contribution to natural-polymer chemistry in showing, in terms of molecular structure, interaction of chain molecules in three-dimensions to provide physical and biological properties of gels.

Structure, Conformation, and Mechanism in the Formation of Polysaccharide Gels and Networks

Experimental and theoretical work on the rheological properties of suspensions are reviewed. Attention is focused on systems consisting of rigid, neutrally buoyant particles suspended in Newtonian fluids; no restrictions, however, are placed on the concentration of the particles or on the forces acting in the suspension. The assumption that an effective viscosity depending solely on the volume fraction of the particles suffices to describe the rheology of suspensions is examined and shown to be inadequate. Indeed, the experimental evidence strongly supports the view that suspensions behave macroscopically as non-Newtonian fluids whose rheological properties are influenced by a large number of factors; these factors are listed. The various theories that have been put forward to explain the flow of suspensions are discussed, with particular emphasis placed on the nature of the approximations made, so that purely empirical formulas can be clearly separated from those having a theoretical basis. Suggestions for future work, both theoretical and experimental, are provided.

The Rheological Properties of Suspensions of Rigid Particles

Deformation of undrawn poly(trimethylene terephthalate) (PTT) fibers

The problem of the mechanism of macromolecule orientation occurring in the formation of fibers from polymer melts has been qualitatively analyzed As a result of birefringence, x-ray, and spinning stress investigations carried out on polycapronamide fibers, the main parameters determining orientation have been established They are the parallel velocity gradient along the spinning length, G = dV/dl, and the relaxation factors The deformation ratio S (the ratio of final to original linear velocity), analogous to the draw ratio λ = 1/10 in the cold-drawing process, has no influence on the degree of orientation The orientation by spinning is not accompanied by any such structural transformations as occur through cold-drawing The fibers spun at high velocity gradients consist of well-oriented β-polycapronamide It is thus assumed that the orientation proceeds in the region of the liquid melt-stream as a result of simultaneous action of velocity gradient and thermal relaxation This mechanism, analogous to that causing the familiar streaming orientation of polymer solutions in capillaries or in a Couette apparatus, is different in principle from that of the cold-drawing process

Studies on the orientation phenomena by fiber formation from polymer melts. Part I. Preliminary investigations on polycapronamide

The mechanism for orientation of macromolecules through the melt-spinning of synthetic fibers is analogous to that in polymer solutions flowing in capillaries or in a Couette apparatus. The differences between the two processes, with regard to the velocity field, molecular state, and time dependencies, have been discussed and the differential equations derived to describe the orientation of two extreme molecular models. The coefficients of orientation for fibers consisting of rigid particles (model I) and flexible, coiled chains (model II) have been defined. The specific conditions occuring in fiber-spinning experiments have been shown, as well as the possibility of applying such experiments to the study of the behavior of macromolecules in the molten state.

Studies on orientation phenomena by fiber formation from polymer melts. Part II. Theoretical considerations

The spatial orientation of rigid ellipsoidal particles was analyzed as proceeding in a dilute solution flowing in a velocity field with parallel gradient, i.e., in a field characterized by the deformation rate tensor: 






On the basis of general relations given by Jeffery, the hydrodynamic equations of motion of a single ellipsoid were obtained as Ψ = 0, φ = 0, θ = −¾qR sin 2θ, where q = ∂Vκ/∂κ is the parallel velocity gradient and R = (a2 − b2)/(a2 + b2) is the shape coefficient of ellipsoids. Considering the action of velocity field and that of Brownian motion (rotational diffusion), a distribution density function ρ(t, θ) was derived, which describes the spatial orientation of the axes of symmetry of the ellipsoids: 






where 






is the steady-state distribution. In a similar way, the axial orientation factor f0 = 1 − 3/2 sin2θ was obtained:

Behavior of polymer solutions in a velocity field with parallel gradient. I. Orientation of rigid ellipsoids in a dilute solution

Flow induced crystallization of polymer systems exhibits strong memory effects. Crystalline structures gradually change when the flow is switched off and the polymer is relaxed prior to crystallization. A simple model based on the multidimensional theory of crystal nucleation [1] is proposed. Steady, potential flow applied to a polymer fluid above melting temperature (T p > T m ) results in molecular orientation of crystallizing units. The flow controls formation of molecular clusters which convert into athermal nuclei when the system is cooled down to crystallization temperature, T c < T m . Orientation effects gradually disappear when the melt is relaxed above T m in the absence of flow or stress.

Andrzej Ziabicki

Papers

Deformation of undrawn poly(trimethylene terephthalate) (PTT) fibers

Studies on the orientation phenomena by fiber formation from polymer melts. Part I. Preliminary investigations on polycapronamide

Studies on orientation phenomena by fiber formation from polymer melts. Part II. Theoretical considerations

Behavior of polymer solutions in a velocity field with parallel gradient. I. Orientation of rigid ellipsoids in a dilute solution

A simple model of flow-induced crystallization memory