The effect of weak inertia on the emptying of a tube

Abstract

We present an extension of the classical axisymmetric Bretherton theory giving the thickness of the liquid film left on the walls of a drained tube, treating the case of weak inertia by a regular perturbation method. The results obtained by numerical integration fit Taylor's ͓J. Fluid Mech. 10, 161 ͑1961͔͒ experiments, obtained with viscous fluids ͑glycerine and strong sucrose solutions͒, and Aussillous and Quéré's ͓Phys. Fluids 12, 2367 ͑2000͔͒ experiments with low viscosity liquids ͑hexamethyldisiloxane and water͒ when inertia becomes important. The discrepancies observed between the theory and high Reynolds numbers experiments ͑ReϾ1000͒ are commented on.

CFD modelling of flow and heat transfer in the Taylor flow regime

“…He showed that for a particular Capillary number (Ca), the inertia does not have any significant effect up to a certain value of Reynolds number (Re) and then increases with an increase in Re (see Fig. 9 in Ryck, 2002). The Re value above which inertial effects become important decreases with an increase in the Capillary number (Ca).…”

“…Giavedoni and Saita (1997) and Heil (2001) found that the inertial effects are of importance only for Ca 4 0.05 and the film thickness first decreases for increasing values of Re (Reo280) and then reaches a minimum and finally increases. Ryck (2002) analysed the effect of inertia on the liquid film thickness at low Reynolds numbers (Reo1000) for the front half of the gas bubble using a regular perturbation method and showed that the ratio of Reynolds and Capillary number (Re/Ca) plays an important role in determining the effect of inertia on film thickness. He showed that for a particular Capillary number (Ca), the inertia does not have any significant effect up to a certain value of Reynolds number (Re) and then increases with an increase in Re (see Fig.…”

Confined growth of a vapour bubble in a capillary tube at initially uniform superheat: Experiments and modelling

“…They suggested that this might be because this range lay outside the range of validity of Eq. (20), but the later steady-flow studies in [24,25] Table 2, just as they are with Eqs. (21,24) for the particular experimental conditions of [17].…”

“…Some limited experimental data in [24] for Re B < 10 3 and an extended asymptotic analysis by de Ryck [25] for laminar flows with weak inertia show that the inertial effects in steady flow cause a significant increase in film thickness beyond a critical value of Ca * given by…”

“…The definitions and correlations in [17] have been rewritten here with the gap half-width R as the characteristic length for consistency with the convention used in [24,25]. Noting that the bubble radius Z ∝ t 2 approximately, Moriyama and Inoue calculated the approximately constant accelerationU B from U B /t g and redefined the Bond number as…”