Simulates a bubble rising in a quiescent liquid. The density and viscosity ratios are given, respectively, by ρ_{f}/ρ_{b}=1000 and µ_{f}/µ_{b}=100. The bubble velocity is, in general, a function of the Archimedes number, defined as:
Ar := |
| , (1) |
and the bubble shape is a function of both the Archimedes number and the Bond number (also known as the Eötvos number), defined as:
Bo = |
| . (2) |
We take Ar = 1 and Bo = 5. For these values, the bubble’s shape remains approximately spherical. The terminal velocity can then be determined analytically from the Hadamard-Rybczynski equation:
| = |
| ⎛ ⎜ ⎜ ⎝ | 1− |
| ⎞ ⎟ ⎟ ⎠ | ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ | (3) |
The characteristic velocity is U_{c} = √g D and the characteristic time is t_{c} = D/U_{c}.