Rheology time-independent

Viscous Hquids are classified based on their rheological behavior characterized by the relationship of shear stress with shear rate. Eor Newtonian Hquids, the viscosity represented by the ratio of shear stress to shear rate is independent of shear rate, whereas non-Newtonian Hquid viscosity changes with shear rate. Non-Newtonian Hquids are further divided into three categories time-independent, time-dependent, and viscoelastic. A detailed discussion of these rheologically complex Hquids is given elsewhere (see Rheological measurements). 

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. 

Newtonian flow, and their viscosity is not constant but changes as a function of shear rate and/or time. The rheological properties of such systems cannot be defined simply in terms of one value. These non-Newtonian phenomena are either time-independent or time-dependent. In the first case, the systems can be classified as pseudoplastic, plastic, or dilatant, in the second case as thixotropic or rheopective. 

Rheologically, the flow of many non-Newtonian materials can be characterized by a time-independent power law function (sometimes referred to as the Ostwald-deWaele equation)... 

The dimensionless flow exponent m and the rheological time constant are additional influencing variables that turn the one-dimensional problem for Newtonian fluids into a three-dimensional problem. The rheological time constant , when multiplied by the revolution speed n, forms an independent dimensionless group (Deborah number). 

Flow models have been used also to derive expressions for velocity profiles and volumetric flow rates in tube and channel flows, and in the analysis of heat transfer phenomenon. Numerous flow models can be encountered in the rheology literature and some from the food rheology literature are listed in Table 2-1. Also, here those models that have found extensive use in the analysis of the flow behavior of fluid foods are discussed. Models that account for yield stress are known as viscoplastic models (Bird et al., 1982). For convenience, the flow models can be divided in to those for time-independent and for time-dependent flow behavior. 

In time-independent liquid food products, the flow curve is linear but intersects the shear stress axis at a positive value of shear stress. This value is known as a yield stress. The significance of the yield stress is that it is the stress that must be exceeded before the material will flow. This type of flow can be characterized by the following rheological equation (for the Bingham-Schwedoff model) ... 

Under one-dimensional shear, many Theologically stable fluids of complex structure (whose rheological characteristics are time-independent) have a flow curve other than Newtonian. If the flow curve is curvilinear but still passes through the origin in the plane 7, r, then the corresponding fluids are said to be nonlinearly viscous (often they are said to be purely viscous, anomalously viscous, or sometimes non-Newtonian). 

For the most part, PFDs exhibit shear-thinning (pseudoplastic) rheological behavior that is either time-independent or time-dependent (thixotropic). In addition, many PFDs also exhibit yield stresses. The time-independent flow curves are illustrated in Figure 1. The shear-thinning behavior appears to be the result of breakdown of relatively weak structures and it may have important relationship to mouthfeel of the dispersions. Because the viscosity of non-Newtonian foods is not constant but depends on the shear rate, one must deal with apparent viscosity defined as ... 

Time-Independent Rheology. In the simplest case, the shear stress a is independent of time t and is proportional to the shear rate 7 (equation 1). For this case, the fluid is called Newtonian (line 1 of... 

All the above categories of non-Newtonian behaviour are classified as time-independent, i.e. the shear stress is a unique function of the shear rate and does not depend on the tiriK of shearing. Thus the material responds instantaneously to changes in shear rate. However, some pastes, foods, paints, etc., exhibit marked changes in rheology as the time shearing increases. Such materials are time-dependent and two types of behaviour are possible ... 

Unlike the flows considered in Chapter 3 which were essentially imidirectional, the fluid flows in particulate systems are either two- or three-dimensional and hence are inherently more difficult to analyse theoretically, even in the creeping (small Reynolds number) flow regime. Secondly, the results are often dependent on the rheological model appropriate to the fluid and a more generalised treatment is not possible. For instance, there is no standard non-Newtonian drag curve for spheres, and the relevant dimensionless groups depend on the fluid model which is used. Most of the information in this chapter relates to time-independent fluids, with occasional reference to visco-elastic fluids. 

The rheological properties of concentrated suspensions are often time-dependent. If the apparent viscosity continuously decreases with time under shear, with a subsequent recovery of the viscosity when the flow is stopped, the system is said to be thixotropic. The opposite behaviour is called antithixotropy, or sometimes rheopexy. Thixotropy should not be confused with shear-thinning which is a time-independent characteristic of a system. Systems which show an irreversible decrease in viscosity with shear should be termed shear-destructive and not thixotropic. 

In the power-law models of blood rheology, the indices k and n are primarily dependent on the hematocrit fraction, which is a measure of the number and size distribution of red blood cells in the sample. The above functional dependence may take the following form under time-independent circumstances [7] ... 

The classification of materials given here is based on the rheological properties discussed in the foregoing section. Elastic behavior is independent of the duration of the deformation but with viscous systems time-independent and time-dependent behavior may be distinguished. 

Lescarboura, J. A., Eichstadt, F. J., and Swift, G. W., A general differentiation method for interpreting rheological data of time-independent fluids, AIChE J. 

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