a | tank or conduit half width (m) |
c | sound velocity (m/s) |
d | diameter (m) |
f | frequency (Hz) |
G | modulus (Pa) |
g | gravitational acceleration (\(\text{ m/s}^2\)) |
K | consistency (\(\text{ Pa } \text{ s}^{n}\)) |
h | film thickness (m) |
L | air column or cavity length (m) |
\(\delta L\) | open-end correction (m) |
l | submerged bubble length (m) |
\(l'\) | effective neck length (m) |
m | power law exponent (\(\beta \propto f_{\mathrm{p}}^{m}\)) |
n | flow index |
\({\mathcal {O}}\) | the order of |
P | pressure (Pa) |
\(\Delta P\) | overpressure (Pa) |
p | bubble buoyancy pressure (Pa) |
R | radius (m) |
r | correlation coefficient |
t | time (s) |
U | ascent velocity (m/s) |
V | bubble volume (m3) |
v | velocity (m/s) |
W | tank or conduit width (m) |
\(\beta\) | decay rate (1/s) |
\({\dot{\gamma }}\) | strain rate (1/s) |
\(\gamma _{\mathrm{h}}\) | ratio of specific heats |
\(\eta\) | viscosity (Pa s) |
\(\rho\) | density (kg/m3) |
\(\Delta \rho\) | density difference (kg/m3) |
\(\sigma\) | surface tension coefficient (N/m) |
\(\phi\) | volumetric packing fraction |
\(\tau\) | shear stress (Pa) |