  The Thermodynamic Non-Equilibrium (TNE) effects in the coalescing process of
two initially static bubbles under thermal conditions are investigated by a
Discrete Boltzmann Model (DBM). The spatial distributions of the typical
none-quilibrium quantity, i.e., the Non-Organized Momentum Fluxes (NOMF) during
evolutions are investigated in detail. The density-weighted statistical method
is used to highlight the relationship between the TNE effects and the
morphological or kinetics characteristics of bubble coalescence. It is found
that the $xx$-component and $yy$-component of NOMF are anti-symmetrical; the
$xy$-component changes from an anti-symmetric internal and external double
quadrupole structure to an outer octupole structure during the coalescing
process. More importantly, the evolution of the averaged $xx$-component of NOMF
provides two characteristic instants, which divide the non-equilibrium process
into three stages. The first instant corresponds to the moment when the mean
coalescing speed gets the maximum and at this time the ratio of minor and major
axes is about $1/2$. The second instant corresponds to the moment when the
ratio of minor and major axes gets $1$ for the first time. It is interesting to
find that the three quantities, TNE intensity, acceleration of coalescence and
negative slope of boundary length, show a high degree of correlation and attain
their maxima simultaneously. Surface tension and heat conduction accelerate the
process of bubble coalescence while viscosity delays it. Both surface tension
and viscosity enhance the global non-equilibrium intensity, whereas heat
conduction restrains it. These TNE features and findings present some new
insights into the kinetics of bubble coalescence.
