Couette flow apparent viscosity

In order to model the flow behavior of molten silicate suspensions such as magmas and slags, the rheological behavior must be known as a function of the concentration of suspended crystals, melt composition, and external conditions. We have determined the viscosity and crystallization sequence for a Kilauea Iki basalt between 1250°C and 1149°C at 100 kPa total pressure and f02 corresponding to the quartz-fayalite-magnetite buffer in an iron-saturated Pt30Rh rotating cup. viscometer of the Couette type. The apparent viscosity varies from 9 to 879 Pa.s. The concentration of suspended crystals varies from 18 volume percent at 1250°C to 59 volume percent at 1149 C. The molten silicate suspension shows power-law behavior ... 

We are most concerned in this text with the apparent viscosity, which for the steady Couette flow examined is simply the ratio of the shear stress to shear rate... 

Illustrated in Fig. 9.1.1, relative to a Newtonian fluid, are the behaviors of the shear stress versus shear rate in a Couette flow for three principal types of non-Newtonian fluids that can be characterized by the form of the apparent viscosity function in Eq. (9.1.3). A number of empirical functions have been widely employed to characterize the apparent viscosities for these classes of fluids. One termed a Bingham plastic behaves like a solid until a yield stress Tq is exceeded subsequent to which it behaves like a Newtonian fluid with a plastic viscosity lip. The apparent viscosity for this fluid may be written... 

The concept of an apparent viscosity introduced for a Couette flow has been applied empirically to a variety of incompressible inertia free steady shear flows through the generalized relation... 

The apparent viscosity of dilute polymer solutions can be represented by the power-law model over a wide range of shear rates [3,4]. For such fluids, the shear rate depends on, among other factors, the power-law index. The shear rate for a power-law fluid in a co-axial rotational viscometer (Couette flow) is ... 

In the foregoing analysis we assumed ideal Couette flow V = v(0, rO, 0). For concentrated suspensions, some gels, and polymer solutions, a low viscosity layer can develop near Ae cylinder surfaces (note Figure 10.2.1a). This leads to an apparent wall slip. This slip velocity can be determined by making measurements with two different radii bobs, / and Rz, with cups sized to give the same k (Yoshimura and Prud homme, 1988)... 

As pointed out by Currie [59], the symmetric solution to the above problem need not be the only solution, as he discussed via a phase-plane analysis of the general equations (5.121) and (5.122), motivated by his earlier observations for Couette flow [57]. Further studies of the apparent viscosity were also carried out by Currie [60] for various viscometric flows. Under some simplifying assumptions, for example, setting Ki = K2 = Kz and ai = 0, Currie and MacSithigh [61] managed... 

A detailed and intricate scaling analysis for the Couette flow of nematics, similar in style to that contained in Section 5.5.5, has been carried out by Atkin and Leslie [6]. We do not pursue this aspect of the analysis in this text and refer the reader to Reference [6] for comments on possible experimentally determined quantities such as an apparent viscosity. As already mentioned, solutions for unequal elastic constants can also be found in [6]. A more extensive analysis of Couette flow incorporating the effects of an applied magnetic field has been provided by Currie [57], who also comments on other types of solutions which may be possible. 

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