# (247e) Nonlinear Dynamics of Bubble Collapse

- Conference: AIChE Annual Meeting
- Year: 2018
- Proceeding: 2018 AIChE Annual Meeting
- Group: Nuclear Engineering Division - See also ICE
- Session:
- Time:
Monday, October 29, 2018 - 5:39pm-6:00pm

**Nonlinear
Dynamics of Bubble Collapse**

Jyoti

Bhati^{1} , Swapan Paruya^{1},^{ }S.

Pushpavanam^{2}

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.

** **

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:

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/m^{2}-K). In Fig .2 we see that_{}the

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

** **