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 (τw) versus apparent wall-shear rate (8V/D, where V is the average velocity and D is the pipe diameter); (b) velocity profile; and (c) friction factor. The τw versus 8V/D plot (log–log scale) is linear with a slope of unity at low and high values of wall stress (τw/(σ/R) < 0.4 and τw/(σ/R) > 1.0, where σ is the interfacial tension and R is the bubble radius). At intermediate values of wall stress (0.4 < τw/(σ/R) < 1.0), the slope of τw 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 τw, the velocity profile is significantly flatter as compared with the parabolic profile. At low values of Reynolds number (NRe), the friction factor (f) follows the usual equation for Newtonian fluids (f = 16/NRe) 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/NRe) 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/NRe.