RABINOWITSCH-MOONEY equation

The solution to the problem of determining the wall shear rate for a non-Newtonian fluid in laminar flow in a tube relies on equation 2.6. 

The volumetric flow rate through an annular element of area perpendicular to the flow and of width Sr is given by 

Provided there is no slip at the tube wall, the first term in equation 3.14 vanishes. 

the velocity gradient is equal to the shear rate y so equation 3.14 can be written as 

Just as the variation of vx with r was unknown, so is the variation of y. However, if the fluid is time-independent and homogeneous, the shear stress is a function of shear rate only. The inverse is that the shear rate y is a function of shear stress rrx only and the variation of rrx with r is known  

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In order to determine the true shear rate at the wall it is necessary to use the Rabinowitsch-Mooney equation ... 

This material is seen to be shear thinning. It is possible that it may exhibit a yield stress but confirmation of this would require measurements at lower shear rates. Note that the Rabinowitsch-Mooney equation is still valid when a non-zero yield stress occurs. 

When data are available in the form of the flow rate-pressure gradient relationship obtained in a small diameter tube, direct scale-up for flow in larger pipes can be done. It is not necessary to determine the r-y curve with the true value of y calculated from the Rabinowitsch-Mooney equation (equation 3.20). 

Equation 3.29 is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, n = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to < 1 and consequently the correction factor has values greater than unity, showing that the wall shear rate yw is of greater magnitude than the value for Newtonian flow. Similarly, for shear thickening behaviour, yw is of a... 

When trying to determine the flow behaviour of a material suspected of exhibiting wall slip, the procedure is first to establish whether slip occurs and how significant it is. The magnitude of slip is then determined and by subtracting the flow due to slip from the measured flow rate, the genuine flow rate can be determined. The standard Rabinowitsch-Mooney equation can then be used with the corrected flow rates to determine the tw-jw curve. Alternatively, the results can be presented as a plot of tw against the corrected flow characteristic, where the latter is calculated from the corrected value of the flow rate. 

This must be done for each of a range of values of the wall shear stress tw. The standard Rabinowitsch-Mooney equation can then be used with the corrected values of uc ... 

The analogous expressions for AP and yw for non-Newtonian fluids depend on the rheological model describing the fluid. The shear rate at the wall 7W is given by the Rabinowitsch-Mooney equation (86, 87), which in its general form is independent of the rheology of the fluid ... 

The Rabinowitsch-Mooney equation is important because it gives a general relationship between AP and flow rate (or vm)... 

Metzner and Reed (89) and Dodge and Metzner (90) derived a generalized Reynolds number from the Rabinowitsch-Mooney equation... 

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

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