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For most non-Newtonian fluids the constitutive relation between the deviatoric stress tensor T and the applied strain rates E can be described by a time independent scalar function ?=?(??), such that T=2?(??)E.
Non-Newtonian fluids occur commonly in our world. Power law model is applicable for time independent non-Newtonian fluids and can be written as ? = K ?n. where, ? is shear stress, ? is shear rate, K is called flow consistency index and n is called flow behavior index.
In the Power Law model, a value of n=1 corresponds to a Newtonian fluid, while 0 < n < 1 is a Shear Thinning (pseudoplastic) fluid and n > 1 represents a Shear Thickening (dilatant) fluid. To employ this model, users must have values for K and n, or have rheological test data.
For a Newtonian Fluid, the stress is proportional to the rate of deformation (the change in velocity in the directions of the stress). In other words, ?ij = µ ( ?ui ?xj + ?uj ?xi ) .
In the lower shear rate range, the fluid is Newtonian (?a=?o) and in the higher shear rate range the fluid acts as a power law fluid (?=K(??)n).