Table 2 Dimensionless numbers for a rising bubble : the Eötvös \(Eo = \Delta \rho g d_{\mathrm{be}}^2 / \sigma\) (bubble size, Eq. (4)), the Morton \(Mo = g \eta ^4 / \rho \sigma ^3\) (fluid property), and the Reynolds \(Re = \rho U d_{\mathrm{be}} / \eta\) (bubble velocity) numbers (see Tables 3, 4 for notations). For magma, \(\eta = 230 - 1.7 \times 10^4\) Pas (Additional file 7: Material), \(d_{\mathrm{be}} = 1 - 10\) m are assumed and the Re is calculated from the Eo, Mo using an empirical function (Clift et al. 1978)
Case | Eo | Mo | Re | flow type |
|---|---|---|---|---|
Expt. (\(\phi = 0.4\)) | \(20 - 520\) | \(3 \times 10^3 - 1 \times 10^8\) | \(8 \times 10^{-4} - 5\) | viscous |
Expt. (all) | \(20 - 520\) | \(0.1 - 1 \times 10^8\) | \(8 \times 10^{-4} - 1 \times 10^2\) | viscous - transitional |
Basaltic magma | \(7 \times 10^4 - 7 \times 10^6\) | \(2 \times 10^8 - 7 \times 10^{15}\) | \(0.02 - 800\) | viscous - inertial |