Linear stability analysis of a two-fluid model for air–water bubble columns

Linear stability analysis is performed for the two-dimensional, two-fluid model for gas–liquid flow applied in our previous computational study of bubble columns [Monahan, S.M., Vitankar, V.S., Fox, R.O., 2005. CFD predictions for flow-regime transitions in bubble columns. A.I.Ch.E. Journal 51, 1897–1923]. The growth rate and the velocity of propagation for a small-amplitude disturbance wave are shown to be highly dependent on the wave number, the direction of propagation, and the two-fluid model parameters. Two types of vertical instabilities are identified: one corresponding to the classical analysis of Jackson [1963. The mechanics of fluidized beds. I: the stability of the state of uniform fluidization. Transactions of the Institution of Chemical Engineers 41, 13–21] for the one-dimensional model, and the other due to a second pair of roots to the characteristic equation of the linearized two-dimensional model. Numerical simulations keeping one type or the other of the roots stable (or unstable) show distinctly different dynamics and suggest that large-scale instabilities seen experimentally may be associated with the second type of instability. The latter leads to instability in the horizontal velocities and is associated with a positive lift coefficient in flows without mean shear in the presence of isotropic bubble–bubble interactions (i.e., “bubble pressure”). This instability is thus different than previously reported instabilities due to negative lift or cooperative/hindered rise.