Weissenberg-Rabinowitsch equation

The calculation of the shear rate at the capillary wall, 7 , is computed from the function slope of Fig 3.18 and the apparent shear rate using Eq. 3.36. The derivative of the function appears relatively constant over the shear stress range for Fig. 3.18. Many resin systems will have derivatives that vary from point to point. The corrected viscosity can then be obtained by dividing the shear stress at the wall by the shear rate i ,. Equation 3.36 is known as the Weissenberg-Rabinowitsch equation [9]. 

EitherEq. E3.1-9or Eq. E3.1-10, known as the Rabinowitsch or Weissenberg-Rabinowitsch equations, can be used to determine the shear rate at the wail yw by measuring Q and AP or r and Tw (21). Thus, in Eq. E3.1-4 both xw and yw can be experimentally measured for any fluid having a shear rate-dependent viscosity as long as it does not slip at the capillary wall. Therefore, the viscosity function can be obtained. 

The combination of Equations 3.86 and 3.88 for and >/w(7w) e known as the Mooney-Weissenberg-Rabinowitsch equations. They show how a rj/y curve may be extracted from capillary flow data (i.e. pressure drop/flow rate measurements) for a non-Newtonian fluid in steady shear flow and these are the equations already presented without proof in Figure 3.14. 

However, there is a condition which must be fulfilled for the above analysis to be valid. It is that the shear rate at a given radius in the tube is a unique function radius. This will normally be so if the tube radius is large compared with the molecular dimension of the polymer. However, for very narrow capillaries, this may not be the case and the solution may become depleted in polymer molecules close to the capillary wall through the depleted layer effect (see Chapters 6 and 7). Thus, the concentration may vary across the capillary, and hence the constitutive model relating rj and y must also depend on local concentration and there is not a unique inversion of the rj/y relationship. This will be discussed in detail in Chapter 6, which will refer back to the development of the Mooney-Weissenberg-Rabinowitsch equations in this context (Sorbie, 1989, 1990). 

Parallel-plate torsional flow is a second choice. Assuming incompressible flow, the viscosity can be calculated from the total torque needed to turn one disk while keeping the other immobile. Following a derivation similar to that used for the Weissenberg-Rabinowitsch equation and using Leibnitz rule, it is straightforward to get the viscosity at the rim of the disk ... 

This equation is the Weissenberg-Rabinowitsch equation [10,11] it facilitates the measurement of the wall shear rate from Q versus Ap data for any fluid that satisfy the assumptions leading up to Equation 8.4. 

Equation 3B.18 is known as the Weissenberg-Rabinowitsch-Mooney (WRM) equation in honor of the three rheologists who have worked on this problem. An alternate equation can be derived for fluids obeying the power law model between shear stress and the pseudo shear rate ... 

Various forms of this equation are used, a common form (often termed the Weissenberg-Rabinowitsch or Rabinowitsch-Mooney equation) being,... 

The shear rate at the die wall is expressed by the Rabinowitsch-Weissenberg [55] equation for steady laminar flow of a time-independent fluid as,... 

As discussed above, the flow curves of polymer fluids can be obtained by Equations 8.18 and 8.38 (or 8.39), and the viscosities of the fluids can be calculated by Equation 8.41. While deriving these equations, one of the assumptions is that the flow pattern is constant along the pipe. However, in a real capillary flow, the polymer fluid exhibits different flow patterns in the entrance and exit regions of the pipe. For example, the pressure drops at the die entrance and exit regions are different from AP/Z. Therefore, corrections, e.g., Bagley correction, are needed to address the entrance and exit effects. Another assumption is that there is no slip at the wall. However, in a real flow, polymer fluid may slip at the wall and this reduces the shear rate near the wall. The Mooney analysis can be used to address the effect of the wall slip. In addition, the velocity profile shown in Figure 8.13 is a parabolic flow. However, the tme flow in the die orifice is not necessarily a simple parabolic flow, and hence Weissenberg-Rabinowitsch correction often is used to correct the shear rate at the wall for the non-parabohc velocity profile. 

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