I've noticed a confusing difference between the theory manual for the newest version of Fluent (18.2) and previous papers on/versions of Fluent regarding volume fraction in the governing equations for granular flow. The new theory manual seems to use volume fraction *squared *in all of the granular viscosity terms (shear, bulk, frictional and collisional), whereas the old manual uses a linear dependence (i.e. raised to the power of one). I've attached an example below:

From page 550 of the 18.2 ANSYS Fluent Theory Guide (see double appearance of the alpha_s term):

From Fluent 6 and Wachem paper "Derivation, Simulation and Validation of a Cohesive Particle Flow CFD Model" (first line of equation 28 below, epsilon is volume fraction here):

https://www.sharcnet.ca/Software/Fluent6/html/ug/node907.htm

This alpha_s versus alpha_s^2 difference exists for all terms in the "new" expressions for granular viscosity. Am I missing something?