The Power Law or Ostwald de Waele Model

This was conceived as a means of predicting the flow behavior of time-independent fluids, particularly shear-thinning and shearthickening fluids. It is really nothing more than an equation to fit the plot of shear stress versus shear rate for such a fluid when both are plotted on logarithmic scaled paper. 

The smaller the value of n, the more shear thinning the liquid will be. The opposite is true for a shear-thickening liquid. 

Please note that values of k cannot be compared for fluids with d i fferent n values. This is because the value and units of the consistency index k depend upon the value of the power law index n. (In fact k actually has dimensions ML T , whereas n is dimensionless.) 

The areas where this model (equation) will be the most unrehable are at very low and very high rates of shear. It has been suggested that this model is typically at its most reliable over shear-rate ranges between 10 and 10 s T 

In general, even with its limitations, the power law model is usually the first choice for engineering calculations because it is simple, and when used with care, over a narrow or medium range of shear rates, it provides adequate and quite reasonable predictions. 

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In this case, p is an arbitrary constant, chosen as the zero shear rate viscosity. The expression for the non-Newtonian viscosity is a constitutive equation for a generalized Newtonian fluid, like the power law or Ostwald-de-Waele model [6]... 

The power law, or Ostwald-de Waele, model is the simplest and most widely used rheologic empiricism. The power law states... 

Otherwise, bear in mind that very often the range of shear rates to which a process fluid is exposed in pip>e flow is often cjuite limited, and because of this it is often possible to adecjuately represent the flow behavior of a process fluid over a limited range of shear rates by the power law or Ostwald de Waele model, and the Metzner-Reed Reynolds number, which we shall shortly discuss. Meanwhile, in the absence of any lab equipment to provide you with shear-stress versus shear-rate data, we suggest that you vary the flow rate to provide several flow rates, and run pressure drop surveys over the pipeline in question. Tabulate the data and use this to develop a "power law" relationship (see following sections) as a first approximation for your pijjeUne and process fluid. Most likely you will not need to perform any more elaborate study than that... 

In many situations, t]o rioc Ky l,and 77,is small. Then the Cross equation (with a simple change of the variables K and m) reduces to the well-known power-law (or Ostwald-de Waele) model, which is given by... 

A useful two-parameter model is the power-law model, or Ostwald-de Waele law to identify its first proponents. The relation between shear stress and shear rare is given by ... 

This equation predicts a straight line on a log-log plot of viscosity and shear rate with a slope of [n-1]. A horizontal line or n s 1 indicates a Newtonian fluid. The fluids that obey the Ostwald-de Waele model are called power-law fluids and n is referred to as the power-law index. 

On the other hand, measurements of shear stress versus shear rate, when plotted on logarithmic scales, may often be represented by a linear expression over a limited range of shear rate (in this case, from 7 to 158 s ). The equation which represents this behavior is usually called the power-law or the Ostwald-De Waele model ... 

One of the most common empirical models used to describe the behaviour of pseudoplastic fluids is the the Ostwald-de Waele [374,375] model or, more colloquially, the power law model ... 

In this case, the system does not show a yield value rather, it shows a limiting viscosity ri 6) at low shear rates (that is referred to as residual or zero shear viscosity). The flow curve can be fitted to a power law fluid model (Ostwald de Waele)... 

The polymer solutions or base gels and suspensions exhibited pseudoplastic non-Newtonian behavior, and they were characterized by the following Ostwald-de Waele or power law fluid model. 

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