Laminar pipe flow, nonnewtonian fluids

Earlier investigators studying the drag-reducing phenomenon in viscoelastic fluids often used Re and Reeff. The former is generally valid only for dilute polymer solutions, in which case the solution viscosity is quite close to that of the solvent. The use of Reeff seems inappropriate in the study of the drag coefficient because it does not represent any physical property of nonnewtonian fluids, although it produces a unique reference line for experimental friction data in laminar pipe flow ... 

FIGURE 10.5 Velocity profile in fully developed laminar pipe flow for nonnewtonian power-law fluids. 

Experimental measurements of pressure drop for purely viscous nonnewtonian fluids flowing through a circular tube in the fully developed laminar flow region confirm this prediction. In fact, this relationship also applies to fully established flow of viscoelastic fluids through circular tubes as demonstrated by Tung et al. [38], The reason for this is that there is no mechanism for elasticity to play a role under fully established pipe flow conditions. Equation 10.32 is recommended for the prediction of pressure drop for nonnewtonian fluids, both purely viscous and viscoelastic, in fully established laminar pipe flow. 

An exception to the generally observed drag reduction in turbulent channel flow of aqueous polymer solutions occurs in the case of aqueous solutions of polyacrylic acid (Carbopol, from B.F. Goodrich Co.). Rheological measurements taken on an oscillatory viscometer clearly demonstrate that such solutions are viscoelastic. This is also supported by the laminar flow behavior shown in Fig. 10.20. Nevertheless, the pressure drop and heat transfer behavior of neutralized aqueous Carbopol solutions in turbulent pipe flow reveals little reduction in either of these quantities. Rather, these solutions behave like clay slurries and they have been often identified as purely viscous nonnewtonian fluids. The measured dimensionless friction factors for the turbulent channel flow of aqueous Carbopol solutions are in agreement with the values found for clay slurries and may be correlated by Eq. 10.65 or 10.66. The turbulent flow heat transfer behavior of Carbopol solutions is also found to be in good agreement with the results found for clay slurries and may be calculated from Eq. 10.67 or 10.68. 

For laminar flow of nonnewtonian fluids in circular pipes, we can readily calculate the behavior from pipe flow data in pipes of other sizes or from data from any kind of viscometer. 

Most fluids with pronounced nonnewtonian behavior have such high viscosities that their flow is laminar in most industrially interesting situations. We saw in Sec. 6.3 that for any fluid the shear stress at any point in a horizontal circular pipe is given by... 

The laminar flow of various kinds of fluids in circular pipes can be easily compared by plotting (DI4) —dPldx) versus 32QI irD ) = SV JD, as shown in Fig, 15.3. This plot (or its equivalent on logarithmic paper) is very widely used in nonnewtonian flow calculations and publications. Its merit can be seen by rewriting Poiseuille s equation (Eq. 6.8) in the form... 

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