Shear flow of a suspension of bubbles rising in an inclined channel

Abstract

A weak, laminar shear flow of a monodisperse suspension of high-Reynolds-number, low-Weber-number bubbles is studied in a novel experimental configuration. Nitrogen bubbles are formed through an array of small capillaries at the base of a tall channel with a small inclination from the vertical. The bubbles generate a unidirectional shear flow, in which the denser suspension near the bottom wall falls and the lighter suspension near the top wall rises. Profiles of the bubble and liquid velocities and the bubble volume fraction are obtained using hot-film and dual impedance probes. To our knowledge, measurements of the laminar shear properties of a nearly homogeneous bubble suspension have not previously been reported.A steady shear flow is observed in which the bubble velocity variation across the channel is typically less than 20% of the mean bubble velocity. The velocity and volume fraction gradients increase with channel inclination and exhibit little or no dependence on the mean gas volume fraction. To explain the magnitude of the volume fraction gradients, it is necessary to consider the effects of both the lift force and the effective bubble-phase diffusivity in balancing the segregating tendency of the cross-channel component of the buoyancy force. The bubble velocity gradient can be understood in terms of a balance of the component of the buoyancy force parallel to the channel walls and an effective viscosity associated with the Reynolds stresses produced by bubble-induced liquid velocity fluctuations. Theories for bubbles rising with potential-flow hydrodynamic interactions predict an instability of the homogeneous state due to a negative Maxwell pressure. However, the hydrodynamic diffusivity inferred from our experiments is large enough to mitigate the clustering effects of the Maxwell pressure. Consistent with this, a vigorous instability of the homogeneous state of the bubble suspension is only observed at volume fractions larger than 5%–20% with the critical volume fraction depending on the angle of inclination.

A kinetic theory for particulate systems with bimodal and anisotropic velocity fluctuations

“…Substituting the values of T 1 , 1 , V 2 1 , etc., determined from the particle dynamics simulations into the rhs of the above equation and comparing the resulting P 22 1 with that determined in dynamic simulations show that the above expression provides a good estimate for values of x 2 close to the channel walls where the LV approximation makes a dominant contribution. Near the center of the channel, however, P 22 1 evaluated using the above expression is nearly twice that obtained from the simulations, suggesting that using the dense gas viscosity expressions results in substantial errors.…”

“…Since we shall not be explicitly solving for T 22 p , it is necessary to use an approximate closure for the other pressure component, P 22 p . We shall use…”

“…On the other hand, since we are using closure relations for P 22 p which are only approximate, we need to modify the boundary conditions for V 2 p at the channel walls ͓Eq. ͑75͒ for P 22 1 plus an analogous equation for P 22 2 ͔. The results of particle dynamics show that V 2 1 is much smaller than V 2 2 .…”

“…The ratio V 2 1 / ͱ 2T 22 1 that appears in the volumetric flux condition Eq. ͑72͒ was found to be approximately constant in our particle dynamics simulations.…”

“…[8][9][10]13,15 The potential flow approximation is expected to be valid when the Reynolds number based on the bubble radius and characteristic velocity is large compared with unity but the Weber number, the ratio of inertial to surface tension forces, is small enough so that the bubbles are approximately spherical, and the liquid is free of surface-active impurity. [16][17][18][19][20] This somewhat ideal case can be studied experimentally, 21,22 numerically using large scale simulations which account for the hydrodynamic interactions among bubbles, 15 and analytically using the methods of statistical mechanics 23 and kinetic theory of dense granular materials. [24][25][26][27][28][29][30][31][32][33] A complete set of equations of motion derived using these numerical simulations and analytical techniques for spherical bubbles is given by Spelt and Sangani.…”