Rheological equation

Laun, H.M. (1980) Stresses and recoverable strains of stretched polymer melts and their prediction by means of a single integral constitutive equation. Rheology, Vol.2, (ed. G. Astarita et ai). Plenum, New York, pp. 419-425. 

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Step 4 - update the value of viscosity (r/) using an appropriate rheological equation (e.g. temperature-dependent form of the Carreau model given as Equation (5.4)). 

Rheology of LLDPE. AH LLDPE processiag technologies iavolve resia melting viscosities of typical LLDPE melts are between 5000 and 70, 000 Pa-s (50,000—700,000 P). The main factor that affects melt viscosity is the resia molecular weight the other factor is temperature. Its effect is described by the Arrhenius equation with an activation energy of 29—32 kj/mol (7—7.5 kcal/mol) (58). 

This equation is based on the approximation that the penetration is 800 at the softening point, but the approximation fails appreciably when a complex flow is present (80,81). However, the penetration index has been, and continues to be, used for the general characteristics of asphalt for example asphalts with a P/less than —2 are considered to be the pitch type, from —2 to +2, the sol type, and above +2, the gel or blown type (2). Other empirical relations that have been used to express the rheological-temperature relation are fluidity factor a Furol viscosity P, at 135°C and penetration P, at 25°C, relation of (H—P)P/100 and penetration viscosity number PVN again relating the penetration at 25°C and kinematic viscosity at 135 °C (82,83). 

Equation (7) depicts the viscosity decrease independent of the chemical features of materials. Also for fixed T, Figs. 7 and 8 demonstrate a further example of a poly-amide-TLCP blend with different weight ratios. The rheological data in Fig. 7 were taken from Siegmann et al. [1]. It is obvious that the lowest blend viscosity is obtained at a TLCP loading of only 5%. This result is... 

The mud rheological properties t, n and K are typically calculated based upon tbe data from two (or more)-spee(l rotational viscometer experiments. For these experiments, the following equations are applicable ... 

Indeed, one often observes a more or less direct relationship between the rheological properties of melts and the mechanical strength of the condensed material. This is a commonplace statement in regard of, say, stiffness, since the equations relating the viscosity of heterogeneous materials with their composition... 

Rheological methods of measuring the interphase thickness have become very popular in science [50, 62-71]. Usually they use the viscosity versus concentration relationships in the form proposed by Einstein for the purpose [62-66], The factor K0 in Einstein s equation typical of particles of a given shape is evaluated from measurements of dispersion of the filler in question in a low-molecular liquid [61, 62], e.g., in transformer oil [61], Then the viscosity of a suspension of the same filler in a polymer melt or solution is determined, the value of Keff is obtained, and the adsorbed layer thickness is calculated by this formula [61,63,64] ... 

The above considerations illustrate the difficulties of trying to formulate equations descriptive of rheological behavior of polymer melts with gas bubbles. An optimistic approach to the solution of this task is contained in [60, 61]. The content of these works is revealed by their titles On the Use of the Theory of Viscoelasticity for Describing of the Behaviour of Porous Material and for the Calculation of construction... 

It is often sufficient for the technologist to know the difference between the pressures, which is required to extrude the gas-liquid medium through the channel. This problem can be solved, at least for flows in straight pipes with an unchanging cross section, without resorting to rheological equations of such two-phase media. This idea is based on two concessions ... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The branch of science which is concerned with the flow of both simple (Newtonian) and complex (non-Newtonian) fluids is known as rheology. The flow characteristics are represented by a rheogram, which is a plot of shear stress against rate of shear, and normally consists of a collection of experimentally determined points through which a curve may be drawn. If an equation can be fitted to the curve, it facilitates calculation of the behaviour of the fluid. It must be borne in mind, however, that such equations are approximations to the actual behaviour of the fluid and should not be used outside the range of conditions (particularly shear rates) for which they were determined. 

In a given pipe, R is determined solely by the pressure drop — AP and is completely independent of the rheology of the fluid and, from equation 3.6 ... 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

Equation 3.152 provides a method of determining the relationship between pressure gradient and mean velocity of flow in a pipe for fluids whose rheological properties may be expressed in the form of an explicit relation for shear rate as a function of shear stress. 

Cross, M. M. J. Colloid Sci. 20 (1965) 417. Rheology of non-Newtonian fluids a new flow equation for pseudoplastic systems. 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

Reynolds apparatus for tracing flow patterns 59 Rheogram 105, 197 Rheological equation (Cross) 110 Rheology 105, 195... 

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