Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

Abstract

The total energy balance of a system consisting of a bubble (radius 0.74 mm) freely rising in viscous liquid under the action of buoyancy was considered. Values of bubble acceleration, local and terminal velocities, and shape deformations calculated by numerically solving the Navier-Stokes equation are compared with experimentally determined values. Additionally, the energies of the system associated with bubble motion (kinetic, potential, and rate of viscous energy dissipation), obtained from simulations, are given. On the basis of energy balances obtained for the system, two independent relations describing the motion of a bubble (Lagrange and Joseph equations) were used to show that, for steady-state conditions, the constant (terminal) velocity of the bubble is a result of balance between buoyancy force and the drag originating from energy dissipation due to fluid viscosity. According to the calculations performed, the discrepancy between the expected and actual results of the Lagrange and Joseph equations for the system considered was $1 %. The estimated added mass coefficient was in close agreement with expected theoretical value for deformed spheres, and increased with decreasing tube radius.

Added mass of rising bubble approaching to solid wall – numerical studies

“…The bubble interface was reproduced and tracked using VOF approach. More details about the discretization parameters, mesh size and discretization algorithms can be found in (Zawala, 2016). Slip boundary conditions were assumed at the cylinder sidewalls, while the top liquid/solid interface was fixed as no-slip.…”

“…The moving body experiences an additional steady drag as a consequence of the work necessary to balance the steady rate of energy dissipation in viscous fluid. For steady-state motion (which is rectilinear) the drag and viscous energy dissipation in unbounded fluid should be constant (Zawala, 2016). When the bubble approaches to the obstacle this situation changes significantly -the added mass increases (Milne-Thomson, 1968;Zawala and Dabros, 2013;Klaseboer et al, 2014).…”