5.2. Interfacial momentum transfer

Hereafter, we omit \(\overline{( ~~ )}\) for simplicity.

5.2.1. Drag

See also

• Drag force acting on spherical body

• Deformed bubbles

The drag force acting on a bubble in liquid is given by

(5.24)\[\mathbf{F}_{D} = - C_{D} \frac{\rho_{L}}{2} \left| \mathbf{v}_{G} - \mathbf{v}_{L} \right| \left( \mathbf{v}_{G} - \mathbf{v}_{L} \right) \frac{\pi d^{2}}{4}\]

where \(d\) is the sphere-volume equivalent bubble diameter, and \(C_{D}\) is the drag coefficient. Suppose that \(N\) bubbles are in the averaging volume \(V\). The total momentum transfer from \(L\) to \(G\) due to drag is therefore written as

(5.25)\[\mathbf{M}_{D}^{l \rightarrow g} = \frac{N \mathbf{F}_{D}}{V} = - \frac{3}{4} \alpha_{G} C_{D} \rho_{L} \left| \mathbf{v}_{G} - \mathbf{v}_{L} \right| \left( \mathbf{v}_{G} - \mathbf{v}_{L} \right) \]

The reaction:

(5.26)\[\mathbf{M}_{D}^{l \rightarrow g} = - \mathbf{M}_{D}^{g \rightarrow l}\]

For poly-dispersed bubbly flows, the interface area concentration and Sauter mean diameter are more suitable for considering the momentum transfer through bubble surfaces. Let \(P(d)\) be the probability density of bubbles within \(d(d)\) (a small fraction of diameter range). The mean diameter is defined by

(5.27)\[\overline{d} = \int_{0}^{\infty} P(d) d d(d)\]

while Sauter mean diameter is defined by

(5.28)\[d_{S} = \frac{\int_{0}^{\infty} P(d) d^{3} d(d)}{\int_{0}^{\infty} P(d) d^{2} d(d) }\]

The interface area concentration, \(a_{\text{int}}\), is given by

(5.29)\[a_{\text{int}} = \frac{6 \alpha_{G}}{d_{S}}\]

Thus,

(5.30)\[\mathbf{M}_{D}^{l \rightarrow g} = - \frac{a_{\text{int}}}{8} C_{D} \rho_{L} \left| \mathbf{v}_{G} - \mathbf{v}_{L} \right| \left( \mathbf{v}_{G} - \mathbf{v}_{L} \right)\]

5.2.2. Lift

See also

• Lift acting on spherical body in inviscid flow with weak shear

• Lift reversal

The lift force acting on a bubble is given by

(5.31)\[\mathbf{F}_{L} = - C_{L} \rho_{L} \frac{\pi d^{3}}{6} \left| \mathbf{v}_{G} - \mathbf{v}_{L} \right| \times \nabla \times \mathbf{v}_{L}\]

Being similar to the drag the momentum transfer is given by

(5.32)\[\mathbf{M}_{L}^{l \rightarrow g} = - \alpha _{G} C_{L} \rho_{L} \left| \mathbf{v}_{G} - \mathbf{v}_{L} \right| \times \nabla \times \mathbf{v}_{L}\]

5.2.3. Virtual mass

See also

Virtual mass

(5.33)\[\mathbf{M}_{VM}^{l \rightarrow g} = \alpha_{G} C_{VM} \rho_{L} \left( \frac{D \mathbf{v}_{L}}{Dt} - \frac{D \mathbf{v}_{G}}{Dt} \right) \]

5.2.4. Another closures

• Turbulent dispersion force

• Wall lubrication force