Determination of Flow Curves

The formulas just developed allow the relation of pressure gradient, velocity profile, and volumetric flow rate os long as the shear stress-shear rate relation (flow curve) for the fluid is known. The problem in viscometry is just the reverse How is the flow curve obtained from pressure drop-volumetric flow rate measurements in a cylindrical tube If the mathematical form of the flow curve, that is, a particular constitutive equation, is assumed a priori, the integrated equations as developed above may be used to establish the parameters in the constitutive relation. For example, if it is assumed that the power law represents the flow curve of a fluid under investigation, two readings of dP/dx versus Q in a tube of known R will allow calculation of K and n from (16.13). However, in the general case, the form of the constitutive equation is not known a priori, and must be established by viscometry. This may be done, first by integrating (16.12) by parts 

The first term in the brackets is zero, because at the upper limit u R) 0, and at the lower limit r = 0. Therefore, 

Applying Leibnitz rule to differentiate both sides of (16.17) with respect to Tw gives 

Neglect of the Rabinowitsch correction (the term in brackets in (16.21) and (16.22)) can lead to serious error. To illustrate, differentiation of (16.13) and substitution of (16.5) and (16.23) reveal that for a power-law flui4 (din F)/(d In t ) = 1/n, a constant The Rabinowitsch equation then becomes 

Only for a Newtonian fluid, n = 1, is F equal to the true shear rate at the tube wall 7, For an n of a reasonable value for a polymer melt or solution, the correction term is 1.5, so the apparent shear rate F is 50% lower than the true shear rate at the tube wall Rabinowitsch correction accounts for the fact 

---