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Journal of Non-Newtonian Fluid Mechanics, Vol.147, No.1-2, 129-137, 2007
Steady laminar flow of non-Newtonian bubbly suspensions in pipes
The steady laminar flow of concentrated bubbly suspensions in pipes is modeled using the Pal-Oldroyd rheological constitutive equation for bubbly suspensions [R. Pal, Rheological constitutive equation for bubbly suspensions, Ind. Eng. Chem. Res. 43 (2004) 5372-5379]. Equations are developed to predict the following: (a) wall shear stress (tau(w)) versus apparent wall-shear rate (8 V/D, where V is the average velocity and D is the pipe diameter); (b) velocity profile; and (c) friction factor. The tau(w) versus 8V/D plot (log-log scale) is linear with a slope of unity at low and high values of wall stress (tau(w)/(sigma/R) < 0.4 and tau(w)/(sigma/R) > 1.0, where sigma is the interfacial tension and R is the bubble radius). At intermediate values of wall stress (0.4 < tau(w)/(sigma/R) < 1.0), the slope of tau(w) versus 8V/D plot on a log-log scale is less than unity. The velocity profile is parabolic at low and high values of wall stress; at intermediate values of tau(w) the velocity profile is significantly flatter as compared with the parabolic profile. At low values of Reynolds number (N-Re), the friction factor (f) follows the usual equation for Newtonian fluids (f= 16/N-Re) provided that the Reynolds number is evaluated using the zero shear-rate viscosity of bubbly suspension. At high values of Reynolds number too, the friction factor follows the usual relationship (f= 16/N-Re) provided that the Reynolds number is evaluated using the high shear-rate limiting viscosity of bubbly suspension. At intermediate values of Reynolds number, the friction factor deviates from the standard relationship, f= 16/N-Re (C) 2007 Published by Elsevier B.V.