(247e) Nonlinear Dynamics of Bubble Collapse

Nonlinear
Dynamics of Bubble Collapse

Jyoti
Bhati¹ , Swapan Paruya¹,  S.
Pushpavanam²

National
Institute of Technology Durgapur, West Bengal, India -713209

Indian
Institute of Technology Madras , Chennai,Tamil Nadu,  India -600036

Extended
Abstract

Boiling has many applications in the field of
nuclear energy, refrigeration, microelectronic and aerospace. In the boiling,
study of bubble dynamics is very important due to its complexity and its direct
effect on the design of heat transfer equipment.  Bubble collapse is an important
and very complex phenomenon of the bubble dynamics. Bubble collapse occurs only
in subcooling boiling, in which bubbles are surrounded by subcooled water. The
degree of subcooling significantly affects the rate of bubble collapse .When a
bubble collapses, it release high thermal energy from bubble to liquid. This
particular aspect of bubble collapse is also utilized in cavitation reactors,
which requires a very temperature. However, this energy transfer during bubble
collapse creates a number of instability phenomena in the boiling channel and affects
the interfacial heat transfer coefficient. It is noted
that literature lacks steady state analysis of oscillating bubble collapse. In this
paper, the authors report the steady-state analysis and the bifurcation
analysis of bubble collapse at a pressure of 1.0 atm.

Model Equations

We have
used Plesset-Zwick equation (Eq. (1)) and the energy balance equation (Eq. (3))
at bubble wall for the steady sate analysis. Plesset-Zwick equation (modified
Rayleigh-Plesset equation) describes the growth or collapse of a spherical
vapor-bubble.  (eq. (2)) is the
growth rate of bubble that produces liquid inertia force.  is the vapor pressure
at liquid temperature ;  is the ambient
pressure;  is the surface
tension.; h is interfacial heat transfer coefficient.

                                                            
                                  (1)

                                       
                                                                                             (2)

                                                                                           (3)

The energy required for evaporating the liquid is supplied by
thermal diffusion through the liquid. For the spherical bubble, energy balance
for moving liquid at vapor-liquid interface is given by Eq. (3):.

We linearized the above three equation to obtain the following
form:

                                                                                                                           
(4)

where, x= [R,dR/dt, T_v]^T

We
have performed the steady state analysis by arc-length continuation method. And
we also determined the eigenvalues of Jacobian matrix (A) for studying bifurcation
phenomena during bubble collapse. Using bifurcation theory, we study how a
solution branch breaks up into two, giving rise to another class of solutions,
as we vary a parameter. The steady sate of a system is determined by a large
number of parameter which occur in the model. The dependency of a steady sate
on a parameter is generated by varying it and keeping all other constant. This
enables us to study how a steady state solution change. We have used arc – length
continuation method to trace the steady state branches. We have determined set
of eigenvalues on the steady-state branches. From the eigenvalue, we can
clearly understand the stability or oscillating nature of bubble collapse.

Numerical
Results

In
the present study on the stability analysis of the bubble collapse, we have T_∞
(subcooled water temperature) and R (initial bubble radius) to be the
bifurcation parameters. We obtained the variation of T_v (vapour
temperature) vs. R (by arc -length continuation method) and eigenvalues for
every T_v and R. From the results presented in Fig.1, we see that the
vapour temperature increases as R is increased if interfacial heat transfer
coefficient is constant (h=1000W/m²-K). In Fig .2 we see thatthe
vapour temperature decreases as initial bubble radius is increased, if
interfacial heat transfer coefficient varies with R (Nu=hD/k_l=2 for
spherical bubble collapse). The eigenvalues indicate that unstable bubble
oscillations take place during bubble collapse. Smaller bubbles create
relatively more unstable phenomena during collapse. As initial bubble radius is
increased, we obtain unstable nodes – the unstable decrease of radius without
oscillation.

Fig 1. Steady -state of T_v and R by Arc
length continuations method where h=1000W/ (m^2-K) at 0K

 Fig.2
Steady –state analysis of bubble collapse by using arc-length continuation
method where (Nu=2) at K

Keywords: Bubble,
Bubble collapse, Stability analysis, Bifurcations, Arc- length continuation