Rheological expressions

Now, in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas ... 

All of these rheological expressions (equations 13.16, 13.17, and 13.18) can be used to analyze the flow under the doctor blade in tape casting. Using the momentum balance equation 13.14 and one of the preceding equations for the shear stress, the differential equation which governs the velocity V,., can be determined. For Newtonian fluids, the solution is given by... 

The pertinent rheological expressions for the cone and plate are given in Appendix 1, equations B-1 to B-5. 

Pseudoplastic, in which the shear stress depends on the shear rate alone. Power law, in which the shear stress is not a linear but an exponential function of shear rate. The rheological expression for a power-law fluid is... 

Fluids in which no deformation occurs until a certain threshold shear stress is applied, in which upon the shear stress x becomes a linear function of shear rate y. The characteristics of the function are the slope (viscosity) and the shear stress intercept (yield value) Xy. The rheological expression for this type of material, known as a Bingham solid, is... 

Casson equation n. Rheology expression used to relate share rate, viscosity, and at infinite shear rate for dispesrsions (pigment coatings, etc.). Patton TC (1979) Paint flow and pigment dispersion a... 

Casson Equation n Rheology expression used to relate share rate, viscosity, and at infinite shear rate for... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

Filler particle si2e distribution (psd) and shape affect rheology and loading limits of filled compositions and generally are the primary selection criteria. On a theoretical level the influence of particle si2e is understood by contribution to the total energy of a system (2) which can be expressed on a unit volume basis as ... 

Viscosity is equal to the slope of the flow curve, Tf = dr/dj. The quantity r/y is the viscosity Tj for a Newtonian Hquid and the apparent viscosity Tj for a non-Newtonian Hquid. The kinematic viscosity is the viscosity coefficient divided by the density, ly = tj/p. The fluidity is the reciprocal of the viscosity, (j) = 1/rj. The common units for viscosity, dyne seconds per square centimeter ((dyn-s)/cm ) or grams per centimeter second ((g/(cm-s)), called poise, which is usually expressed as centipoise (cP), have been replaced by the SI units of pascal seconds, ie, Pa-s and mPa-s, where 1 mPa-s = 1 cP. In the same manner the shear stress units of dynes per square centimeter, dyn/cmhave been replaced by Pascals, where 10 dyn/cm = 1 Pa, and newtons per square meter, where 1 N/m = 1 Pa. Shear rate is AH/AX, or length /time/length, so that values are given as per second (s ) in both systems. The SI units for kinematic viscosity are square centimeters per second, cm /s, ie, Stokes (St), and square millimeters per second, mm /s, ie, centistokes (cSt). Information is available for the official Society of Rheology nomenclature and units for a wide range of rheological parameters (11). 

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). 

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. 

For a Newtonian fluid, the data for pressure drop may be represented on a pipe friction chart as a friction factor = (R/pu2) expressed as a function of Reynolds number Re = (udp/n). The friction factor is independent of the rheological properties of the fluid, but the Reynolds number involves the viscosity which, for a non-Newtonian fluid, is... 

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. 

Actinomycetes Large surface area to volume ratio should favour protein export Widely used in industrial microbiology Good expression systems being developed Promoters/gene regulation still poorly understood Rheology of fermentations important... 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

The gelation transition is observable for Ng > 10. Otherwise, the material behaves as a liquid (Ng < 1). Little is known about materials near Ng = 1. For the following, we consider only materials with iVg 1 and treat them just like chemical gels. The expression T(n + 1, (t — t )/Xpg)/T(n + 1) in Eq. 5-2 approaches a value of one in this case of Ng g> 1, and the critical gel equation, Eq. 4-1, is recovered. However, much work is needed to understand the role of non-permanent physical clusters on network formation and rheological properties. 

For correlation with most physical properties (mechanical strength, optical scattering), mass average molecular mass of a polymer Mw" appears more satisfactory. Higher-power averages like z-average molecular mass "Mz" seem to better correlate with rheological properties. Expressions of M , Mw and Mz are... 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

Inasmuch as the rheological properties are very difficult to measure for very dilute solutions (100 ppm or less), a simplified expression was developed by Darby and Pivsa-Art (1991) in which these rheological parameters are contained within two constants, /q and k2 ... 

In the case of polymer samples, it is expected that, at the temperatures and frequencies at which the rheological measurements were carried out, the polymer chains should be fully relaxed and exhibit characteristic homo-polymer-like terminal flow behavior (i.e., the curves can be expressed by a power-law of G oc co2 and G" oc co). 

Dynamic rheological measurements have recently been used to accurately determine the gel point (79). Winter and Chambon (20) have determined that at the gel point, where a macromolecule spans the entire sample size, the elastic modulus (G ) and the viscous modulus (G") both exhibit the same power law dependence with respect to the frequency of oscillation. These expressions for the dynamic moduli at the gel point are as follows ... 

Activation Energy. The gel times, determined by dynamic rheological tests, can also be utilized to calculate an apparent activation energy. We can obtain the gel times over the temperature range of interest and if the extent of reaction at these temperatures are constant, an apparent activation energy can be determined. First, the polymerization reaction can be represented by a generalized kinetic expression of the type (24)... 

Most characterisation of non-linear responses of materials with De < 1 have concerned the application of a shear rate and the shear stress has been monitored. The ratio at any particular rate has defined the apparent viscosity. When these values are plotted against one another we produce flow curves. The reason for the popularity of this approach is partly historic and is related to the type of characterisation tool that was available when rheology was developing as a subject. As a consequence there are many expressions relating shear stress, viscosity and shear rate. There is also a plethora of interpretations for meaning behind the parameters in the modelling equations. There are a number that are commonly used as phenomenological descriptions of the flow behaviour. 

Taking into account the symmetrical behavior of both the reduced volume fraction axes, a rheological equation expressing the mid-point of the phase inversion region may be written ... 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

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