Turbulent Flow of Nonnewtonian Fluids in Pipes

For turbulent flow of newtonian fluids in pipes, the experimental pressure-gradient data are represented by a friction factor-Reynolds number plot (Fig. 6.10). It seems logical to do the same for nonnewtonian fluids, but in so doing we must redefine the Reynolds number. 

For newtonian fluids the viscosity is independent of the shear stress, so there is no ambiguity as to which value of the viscosity to use in the Reynolds number. However, for a nonnewtonian fluid the viscosity is a strong function of the shear stress and from Eq. 6,3 we see that the shear stress decreases linearly with the distance from the wall, becoming zero at the tube center. Thus, there is no obvious choice of the correct viscosity to use in calculating the Reynolds number. Numerous theories and methods have been proposed for determining the critierion for laminar-turbulent transition and the proper Reynolds number to use in Fig. 6.10. The following method is the simplest and the most widely used, 

If we accept this as the definition of the laminar-flow Reynolds number, then for any constitutive equation which can be integrated twice to give the nonnewtonian equiyalent of the Poiseuille equation, Eq. 15.9 cian be used to define a working Reynolds number. For example, for power-law fluids (Eq. 15.7) this leads (Prob. 15.7) to 

The decrease in the friction factor for polymerj solutions compared with newtonian fluids can be quite startling. Dissolving as little as 5 ppm of some polymers in water produces a solution with only 60 percent of the friction factor of water at high Reynolds numbers [1, p. j88]. Such pressure-loss-reduction additives are in current large-scale industrial use [8, 9]. 

In this brief chapter we have not discussed elastic effects in fluid flow. 

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