Table 3 Notation

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)