Chaotic behavior of gas bubble in non-Newtonian fluid: a numerical study

Abstract

In the present paper, the nonlinear behavior of bubble growth under the excitation of an acoustic pressure pulse in non-Newtonian fluid domain has been investigated. Due to the importance of the bubble in the medical applications such as drug, protein or gene delivery, blood is assumed to be the reference fluid. Effects of viscoelasticity term, Deborah number, amplitude and frequency of the acoustic pulse are studied. We have studied the dynamic behavior of the radial response of bubble using Lyapunov exponent spectra, bifurcation diagrams, time series and phase diagram. A period-doubling bifurcation structure is predicted to occur for certain values of the effects of parameters. The results show that by increasing the elasticity of the fluid, the growth phenomenon will be unstable. On the other hand, when the frequency of the external pulse increases the bubble growth experiences more stable condition. It is shown that the results are in good agreement with the previous studies.

Appendix: Stability analysis

Equations (4) and (5) can be expressed as a system of first-order ordinary differential equations in which the zero point is located on the wall of the spherical bubble:

$$ \left \{ \begin{array}{l} \displaystyle\frac{dR}{dt}=U, \\ \displaystyle\frac{dU}{dt}= \biggl[ -\displaystyle\frac{3}{2}U^2+\frac{p_0}{\rho\omega^2 R_0^2} \biggl((1+\mathit{We})\biggl(\displaystyle\frac{1}{R}\biggr)^{3k}\\ \hphantom{\displaystyle\frac{dU}{dt}=} {}-\mathit{We}\biggl(\displaystyle\frac{1}{R}\biggr)-\bigl(1+\alpha\sin (t)\bigr) \biggr) \biggr]\frac{1}{R}\\ \hphantom{\displaystyle\frac{dU}{dt}=} {}+ \displaystyle\frac{1}{R}\frac{2}{3\mathit{Re}}\biggl(\displaystyle\frac{1}{\omega R_0} \sqrt{\displaystyle\frac {p_0}{\rho}}\biggr)\\ \hphantom{\displaystyle\frac{dU}{dt}=} {}\times\displaystyle\int_0^{\infty} \biggl(\displaystyle\frac{\tau_{rr}(y,t)-\tau_{\theta\theta}(y,t)}{y_i+R^3}\biggr)\,dy,\\ \displaystyle\frac{d\tau_{rr}(y,t)}{dt}= \biggl(\biggl(\displaystyle\frac{-4R^2\dot{R}}{y_i+R^3}\biggr) -\displaystyle\frac{1}{\mathit{De}} \biggr)\tau_{rr}\\ \hphantom{\displaystyle\frac{d\tau_{rr}(y,t)}{dt}=} {}+\displaystyle\frac{4}{\mathit{De}}\biggl(\omega R_0\sqrt{\displaystyle\frac{p_0}{\rho }}\biggr)\biggl(\displaystyle\frac{R^2\dot{R}}{y_i+R^3}\biggr),\\ \displaystyle\frac{d\tau_{\theta\theta}(y,t)}{dt}= \biggl(\biggl(\displaystyle\frac{2R^2\dot {R}}{y_i+R^3}\biggr)-\displaystyle\frac{1}{\mathit{De}} \biggr)\tau_{rr}\\ \hphantom{\displaystyle\frac{d\tau_{\theta\theta}(y,t)}{dt}=} {}-\displaystyle\frac{2}{\mathit{De}}\biggl(\omega R_0\sqrt{\displaystyle\frac{p_0}{\rho}}\biggr) \biggl(\displaystyle\frac{R^2\dot{R}}{y_i+R^3}\biggr). \end{array} \right . $$

(7)

We is the Weber number, defined as

$$ \mathit{We}=\frac{2\sigma}{p_{c}R_{0}}. $$

(8)

Also, in above equation the initial conditions are taken as

$$\begin{aligned} &{R(0) = 1[R_0],} \end{aligned}$$

(9)

$$\begin{aligned} &{\tau_{\theta\theta}(0) =\tau_{rr}(0) = 0,} \end{aligned}$$

(10)

$$\begin{aligned} &{U(0) = 0.} \end{aligned}$$

(11)

This study is conducted for De∼O(1) to avoid numerical difficulties because of the division by this quantity in Eq. (8).