Rheology application shear stress

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. 

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. 

Shear Rate, denoted by the symbol, j>, is the velocity gradient established in a fluid as a result of an applied shear stress. It is expressed in units of reciprocal seconds, s". Shear Stress is the stress component applied tangentially. It is equal to the force vector (a vector has both magnitude and direction) divided by the area of application and is expressed in units of force per unit area (Pa). The nomenclature committee of the Society of Rheology recommends that the symbol a be used to denote shear stress. However, the symbol r that was used to denote shear stress for a long time can be still encountered in rheology literature. 

A flow model may be considered to be a mathematical equation that can describe rheological data, such as shear rate versus shear stress, in a basic shear diagram, and that provides a convenient and concise manner of describing the data. Occasionally, such as for the viscosity versus temperature data during starch gelatinization, more than one equation may be necessary to describe the rheological data. In addition to mathematical convenience, it is important to quantify how magnitudes of model parameters are affected by state variables, such as temperature, and the effect of structure/composition (e.g., concentration of solids) of foods and establish widely applicable relationships that may be called functional models. 

A master curve of the rheological conditions applicable during spreading of lipophilic preparations with force on the skin using a method similar to that of Wood (1968) showed that the range of acceptable apparent viscosity was about 3.9 poise to 11.8 poise, with an optimum value of approximately 7.8 poise (Barry and Grace, 1972). The preferred region was approximately bounded by shear rates 400-700 s and shear stress 2,000-7,000 dyne cm (200-700 Pa, respectively). 

Determination of shear rate vs. shear stress curves by application of the ram extruder allow characterization of the rheological properties of the extruded material according to the basic type of curve, as expressed by Eqs. (8)-(ll). 

Fig. 1 Classes of rheological behavior that can be shown by coal slurries, as they appear when plotted on a shear rate/ shear stress graph. It is desirable for coal slurries to be Bingham plastic or pseudoplastic with yield, as such slurries flow readily at high shear rates (such as during pumping or atomization), while remaining stable against settling at low shear rates because of their yield stress. Dilatant slurries are completely unsuitable for coal slurry applications because they are extremely difficult to pump.

The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, care should be exercised in its application for reliable results, the range of shear stress (or shear rate) expected in the application should not extend beyond the range of the rheological data used to evaluate the model parameters. Both laminar and turbulent pipe flow of highly loaded slurries of line particles, for example, can often be adequately represented by either of these two models, as shown by Darby et al. (1992). 

A convenient way to summarize the flow properties of fluids is by plotting flow curves of shear stress versus shear rate (r versus 7). These curves can be categorized into several rheological classifications. Foams are frequently pseudoplastic that is, as shear rate increases, viscosity decreases. This is also termed shear-thinning. Persistent foams (polyeder-schaum) usually exhibit a yield stress (rY), that is, the shear rate (flow) remains zero until a threshold shear stress is reached, then pseudoplastic or Newtonian flow begins. An example would be a foam for which the stress due to gravity is insufficient to cause the foam to flow, but the application of additional mechanical shear does cause flow (Figure 17). 

Equation (3.41) can now be used directly to give the pressine drop for any pipe diameter if the flow is turbulent. Alternatively, this approach can be used to construct a wall shear stress (T )-apparent wall shear rate (SV/D) turbulent flow line for any pipe diameter. This method is particularly suitable when either the basic rheological measurements for laminar flow are not available or it is not possible to obtain a satisfactory lit of such data. The application of this method is illustrated in example 3.8. 

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