Steady laminar flow of non-Newtonian bubbly suspensions in pipes

Abstract

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 ( τ 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 τ w versus 8 V/ D plot (log–log scale) is linear with a slope of unity at low and high values of wall stress ( τ w/( σ/ R) < 0.4 and τ w/( σ/ R) > 1.0, where σ is the interfacial tension and R is the bubble radius). At intermediate values of wall stress (0.4 < τ w/( σ/ R) < 1.0), the slope of τ w versus 8 V/ 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 τ 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 .