1 Introduction

The oscillations of bubbles driven by an external sound field can be essential in biomedical applications, including targeted drug/gene delivery and use as contrast agents in medical imaging [1,2,3,4]. Microbubbles are valuable for drug delivery by enhancing cell and tissue permeability. In targeted drug delivery, microbubbles are injected through the vasculature. Once they reach the target area, a focused ultrasound induces bubble break-up, releasing the drug within the bubble into the target tissue [5,6,7,8,9,10].

Shape stability is essential for the targeted drug delivery to control the microbubble break-up [11] or to estimate the bubble collapse pressure in tissue ablation [12]. The onset of bubble break-up is directly related to the loss of spherical shape due to the growing shape instabilities at the interface. Also, in ultrasound imaging, the stable shape oscillations of the bubbles are used to increase blood tissue contrast, thus improving the image quality of blood vessels in ultrasound examination [13]. In this case, stable bubbles in the blood live longer than unstable bubbles, extending the imaging time.

Plesset [14] and Prosperetti [15] used perturbation methods to study the shape stability analysis of a microbubble [14,15,16,17,18,19,20]. Plesset [14] applied spherical harmonics to represent non-spherical bubble surfaces. Two coupled ordinary differential equations (ODEs) are obtained: an equation for the mean bubble radius and an equation for the non-spherical mode amplitudes [14]. Prosperetti [15] extended this inviscid model to account for rotational flow by adding two corrections: a viscous correction in the dynamic boundary condition and a rotational correction in the velocity field. Based on this model, the shape stability of a bubble is investigated [16,17,18,19, 21,22,23,24]. Lu et al. [21] examined the stability of the spherical oscillations of thick-shelled microbubbles using perturbation theory with harmonic non-spherical small perturbations and solving an eigenvalue problem, assuming that the surrounding liquid is Newtonian. Liu et al. [25] investigated the shape stability of a bubble in a liquid fully covered by an elastic solid. Klapcsik et al. [26] investigated the effect of dissipation on the shape stability of a harmonically excited bubble. The results show that the viscosity of the liquid domain prevents the bubbles from losing their spherical shape. Gaudron et al. [27] studied the shape stability of a bubble surrounded by an elastic medium using classical perturbation theory [14]. Two distinct constitutive models are considered for the surrounding medium: linear elasticity and Neo-Hookean hyperelasticity. They concluded that surface tension and elasticity have stabilizing effects, while inertia can be either stabilizing or destabilizing, depending on the interfacial velocity and acceleration. Murakami et al. [28] analytically investigated the shape stability of a bubble in a Kelvin-Voigt viscoelastic medium. They observed that the viscosity increases the damping rate, thus suppressing the shape instability. Yang et al. [29] studied the onset of non-spherical shapes of a bubble in a soft material by explicitly accounting for nonlinear interactions between the cavitation bubble and the solid surroundings.

Most of the above-cited references considered Newtonian liquids, even though non-Newtonian liquids are prevalent in biomedical applications. Yang et al. [30] studied the growth (collapse) of a spherical bubble in an unbounded generalized Newtonian fluid numerically. They observed that the energy dissipation rate decreases with increasing power–law index. Shima and Tsujino [31] numerically investigated the spherical bubble dynamics in unbounded polymer solutions obeying a power–law model. They concluded that the effect of damping on the bubble oscillations is more significant for a small initial bubble radius and a large polymer solution concentration. Arefmanesh et al. [32] investigated the effect of shear-thinning liquid on a confined gas bubble. They showed that the oscillation amplitude decreases faster for a higher power–law index, and the bubble reaches its equilibrium radius faster.

This study investigates the dynamics of a bubble surrounded by a power–law liquid under the effect of an acoustic field. Here, the surrounding liquid is assumed to be non-Newtonian to simulate the blood viscosity [33, 34]. Most analyses to date investigated spherical bubble dynamics in a Newtonian liquid. However, the bubble dynamics may deviate from a spherical behavior when the bubble is under a continuous force [16,17,18,19, 21,22,23].

The paper is organized as follows. In Sect. 2, the details of the physical model and the assumptions are given. Section 2 also includes the base and the perturbed state equations. In Sect. 3, the effects of the power–law index, the apparent viscosity, and the driven frequency are studied. Section 4 presents the conclusions.

2 Physical model

The physical system consists of a bubble in an incompressible and non-Newtonian liquid described in spherical coordinates by a radial distance r, and a polar angle \(\theta \), as shown in Fig. 1. Here, \(p_\infty (t)=p_{atm}+p_a \sin (\omega _d t)\). \(p_a\) is the amplitude of the acoustic field pressure, and \(\omega _d\) is the frequency of the acoustic field. \(p_{atm}\) is the reference radius and equals to 1 atm. R is the time-dependent mean bubble radius, \(\textbf{u}\) is the velocity vector, \(p_b(t)\) is the pressure inside the bubble, t is the time, \(\rho , k\), and \(\mu \) are the density, power–law index, and apparent viscosity of the surrounding liquid. The interface between the gas and the surroundings is assumed to be infinitesimally thin. It is assumed that the gas is considered ideal with negligible heat exchange between the bubble and the liquid.

Fig. 1
figure 1

Non-spherical bubble geometry in spherical coordinates. The solid and dashed lines show the spherical and non-spherical oscillations

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With the assumption of axisymmetric deformation, the bubble surface function, \(F(r, \theta , t)\), can be expressed by the Legendre polynomials,

$$\begin{aligned} F(r, \theta , t)=r-R(t)-\sum ^\infty _{n=2} a_n(t) P_n(\cos \theta )=0, \end{aligned}$$
(1)

where \(a_n\) is the perturbation amplitude, and \(\left| a_n(t)\right| \ll R(t)\). \(P_n \) is a Legendre polynomial of degree \(n \geqslant 2\). The summation is from \(n = 2\) since \(n = 1\) is related to the translation of the non-deformed bubble and is not considered here.

The continuity and the momentum equations for the surrounding liquid are written as,

$$\begin{aligned} \nabla \cdot \textbf{u}=0, \end{aligned}$$
(2)

and

$$\begin{aligned} \rho \left( \frac{\partial \textbf{u}}{\partial t}+ \textbf{u} \cdot \nabla \textbf{u}\right) =-\nabla p+[\nabla \cdot \pmb {\tau }], \end{aligned}$$
(3)

respectively. Here, \(\textbf{u}\), p, and \(\pmb {\tau }\) are the velocity, pressure, and stress fields of the surrounding liquid, respectively. The non-Newtonian surrounding liquid obeys a power–law constitutive equation. The relation between the stress and the deformation-rate tensor is given as [32],

$$\begin{aligned} \pmb {\tau }=\mu {\dot{ \pmb {\gamma }}}, \end{aligned}$$
(4)

where \(\pmb {{\dot{\gamma }}}\) is the strain-rate tensor. The apparent viscosity, \(\mu \) is given by

$$\begin{aligned} \mu =m{{\dot{\gamma }}}^{k-1}. \end{aligned}$$
(5)

Here, m is the consistency index and \({\dot{\gamma }}\) is the the second invariant of the strain-rate tensor, \(\pmb {{\dot{\gamma }}}\). At the bubble surface, kinematic condition, no-slip, and stress balance are given as

$$\begin{aligned}{} & {} \frac{\partial F}{\partial t}+({\textbf{u}} \cdot \nabla ) F=0, \end{aligned}$$
(6)
$$\begin{aligned}{} & {} {\textbf{u}}\cdot \textbf{t}={\textbf{u}}_{g} \cdot \textbf{t}, \end{aligned}$$
(7)

and

$$\begin{aligned} {\textbf{n}} \cdot {\textbf{T}} -{\textbf{n}}\cdot {\textbf{T}}_g =\sigma {\textbf{n}} \nabla \cdot {\textbf{n}}, \end{aligned}$$
(8)

where the subscript g represents the gas phase. \(\sigma \) is the surface tension and \(\textbf{T}\) is the stress tensor, and \({\textbf{n}}=\frac{\nabla F}{\Vert \nabla F\Vert }\) is the outward unit normal vector [15]. For low acoustic field pressure amplitude and high frequency, the bubble dynamics are qualitatively independent of the selection of the gas phase equations of state [35]. Therefore, it is assumed that the internal bubble pressure obeys the ideal gas assumption and can be determined by the polytropic law as,

$$\begin{aligned} p_b=p_v+p_{b,eq}\left( \frac{R_0}{R}\right) ^{3 \lambda }. \end{aligned}$$
(9)

Here, \(p_v\) and \(p_{b,eq}\) are the vapor pressure inside the bubble and the equilibrium pressures. \(\lambda \) is the polytropic exponent and \(R_0\) is the initial bubble radius.

2.1 Field decomposition

Since velocity changes due to non-spherical perturbations are much smaller than those due to radial motion, the velocity field is decomposed as follows [15],

$$\begin{aligned} {\textbf{u}}={\textbf{u}}_0+\epsilon {\textbf{u}}_1, \end{aligned}$$
(10)

subscripts 0 and 1 denote radial motion, base state, and non-spherical corrections, respectively. The pressure field is decomposed accordingly,

$$\begin{aligned} p=p_0+\epsilon p_1, \end{aligned}$$
(11)

where \(\epsilon \) is of order \(\left| a_n\right| / R\).

2.1.1 Spherical oscillations

The base state is a spherical gas bubble oscillating under an acoustic field. The initial liquid pressure is larger than the initial gas pressure. The r-component of continuity and momentum equations for the base state are written in spherical coordinates as,

$$\begin{aligned} \frac{1}{r^2} \frac{\partial \left( r^2 u_{0,r}\right) }{\partial r}=0, \end{aligned}$$
(12)

and

$$\begin{aligned} \rho \left( \frac{\partial u_{0,r}}{\partial t}+u_{0,r} \frac{\partial u_{0,r}}{\partial r}\right) =-\frac{\partial p_0}{\partial r}+[\nabla \cdot \pmb {\tau _0}]_r, \end{aligned}$$
(13)

respectively. \(u_{0,r}\) is the r-component of the base velocity. Here, \(\pmb {\tau _0}\) is traceless.

The solution of Eq. 12 with \(u_{0,r}(r=R)={\dot{R}}\), gives

$$\begin{aligned} u_{0,r}=R^2 \frac{{\dot{R}}}{r^2}, \end{aligned}$$
(14)

where \({\dot{R}}\) is the rate of change of the bubble radius with respect to time. For the velocity profile given by Eq. 14, Eq. 4 yields the following expression for \(\tau _{0,r r}\):

$$\begin{aligned} \tau _{0,r r}=-2\mu _{0} \frac{R^2 {\dot{R}}}{r^3} \end{aligned}$$
(15)

where \(\mu _{0}=M\left| \frac{R^2 {\dot{R}}}{r^3}\right| ^{k-1}\) and \(M=2\,m(2 \sqrt{3})^{k-1}\). The normal stress balance at the bubble-liquid interface is given as,

$$\begin{aligned} p_0(R)-\tau _{0,r r}\left( R\right) +\frac{2 \sigma }{R}=p_b, \end{aligned}$$
(16)

where \(p_0(R)\) is the pressure in the liquid at the bubble interface. The expression for \(p_b\) is given by the polytropic expression (Eq. 9). Equations 14, 15 and 16 are inserted into Eq. 13 and integrated from R to infinity to yield the following differential equation for the change of the bubble radius with respect to time as,

$$\begin{aligned} \rho \left( {\ddot{R}} R+1.5 {\dot{R}}^2\right) =p_{b,eq}\left( \frac{R_0}{R}\right) ^{3\kappa }-p_\infty (t)-\frac{2 \sigma }{R}-\frac{2M}{k}|{\dot{R}}(t)|^{k-1} \frac{{\dot{R}}}{R^k}, \end{aligned}$$
(17)

where \({\ddot{R}}\) is the second order time derivative and \(p_{b,eq}=p_{atm}+2\sigma /R_0-p_v\). Equation 17 represents an ordinary differential equation governing the bubble radius, derived from the equation of motion (Eq. 3) under an approximation valid to the order of \({\dot{R}}/c_\infty \), where \({\dot{R}}\) denotes the speed of the bubble wall and \(c_\infty =1500\) m/s represents the sound speed in the liquid. In this study, the maximum velocity of the bubble wall is approximately 50 m/s. Hence, the Mach number, \({\dot{R}}/c_\infty \approx 0.04\), is significantly less than unity. For the details of the spherical oscillations of a bubble in a power–law liquid, the reader is referred to [32].

2.1.2 Non-spherical oscillations

The continuity equation and the irrotational equation of motion are given for the non-spherical oscillations as,

$$\begin{aligned} \nabla \cdot {\varvec{u}}_1=0, \end{aligned}$$
(18)

and

$$\begin{aligned} \rho \frac{\partial {\varvec{u}}_1}{\partial t}+\rho \left( {\varvec{u}}_0 \cdot \nabla \right) {\varvec{u}}_1+\rho \left( {\varvec{u}}_1 \cdot \nabla \right) {\varvec{u}}_0+\nabla p_1=0 \end{aligned}$$
(19)

respectively. The boundary conditions at infinity are,

$$\begin{aligned} {\varvec{u}}_1 \rightarrow 0, \quad p_1 \rightarrow 0 \quad \text{ as } r \rightarrow \infty . \end{aligned}$$
(20)

The velocity at the interface in the radial direction is given by [14, 15, 24],

$$\begin{aligned} {\varvec{u}}_1 \cdot {\varvec{n}}_w={\dot{a}}_n P_n(\cos \theta ), \end{aligned}$$
(21)

where w denotes the variable evaluated at the non-spherical bubble wall. The velocity profile \({\varvec{u}}_1\), upon satisfying the conditions 20, and  21, is given by [24]

$$\begin{aligned} {\varvec{u}}_1=\frac{\left( {\dot{a}}_n+\frac{2 {\dot{R}} a_n}{R}\right) R^{n+2}}{(n+1) r^{n+2}}[(n+1) P_n (\cos \theta ){\varvec{e}}_r- P_n^\prime (\cos \theta ) {\varvec{e}}_\theta ] \end{aligned}$$
(22)

Here, \(P_n^{\prime }=\frac{d P_n(\cos \theta )}{d \theta }\). The perturbed stress equation is

$$\begin{aligned} \tau _{1,rr}=\mu _{0}\left( \varvec{u_{0}}\right) {{{\dot{\gamma }}}}_{1,rr}\left( \varvec{u_1}\right) +{\mu }_{1}\left( {{\varvec{u}}}_1\right) {{{\dot{\gamma }}}}_{0,rr}\left( \varvec{u_{0}}\right) , \end{aligned}$$
(23)

where

$$\begin{aligned} {\mu }_{1}=\frac{\partial \mu \left( {{\varvec{u}}}_0\right) }{\partial {\dot{\gamma }}_{rr}} {\dot{\gamma }}_{1,rr}={M(k-1)\left| \frac{\partial u_{0,r}}{\partial r}\right| ^{(k-3)}\left( \frac{\partial u_{0,r}}{\partial r}\right) \left( \frac{\partial u_{1,r}}{\partial r}\right) }. \end{aligned}$$
(24)

The details of Eq. 23 can be found in [36,37,38,39,40].

For an incompressible and irrotational liquid, the contribution of viscosity is from the normal stress balance, given in order \(\epsilon \) as,

$$\begin{aligned} p_1-\tau _{1,rr} +\sigma (n+2)(n-1) \frac{a_n P_{n}\cos (\theta )}{R^2}=0 \end{aligned}$$
(25)

The perturbed pressure, \(p_1\), is obtained after substituting the perturbed stress component, \(\tau _{1,rr}\), into Eq. 25. The governing equation for the perturbation amplitude, \(a_n(t)\) is obtained by integrating the equation of motion, Eq 19, from the non-spherical bubble surface to infinity and substituting \(p_1\) from Eq. 25. To order \(\epsilon \), a linear ODE for perturbation amplitude of non-spherical mode \(a_n(t)\) is obtained as

$$\begin{aligned} \ddot{a}_n(t)+A_n(t) {\dot{a}}_n(t)+B_n(t) a_n(t)=0, \end{aligned}$$
(26)

where

$$\begin{aligned} A_n=3 \frac{{\dot{R}}}{R}+\frac{M(n+2)(n+1)}{\rho R^2}\bigg (\left| \frac{\partial u_{0,r}}{\partial r}\right| ^{k-1}+4(k-1)\left| \frac{\partial u_{0,r}}{\partial r}\right| ^{k-3} \frac{{\dot{R}}^2}{R}\bigg ), \end{aligned}$$
(27)

and

$$\begin{aligned}{} & {} B_n=-(n-1) \frac{{\ddot{R}}}{R}+(n-1)(n+1)(n+2) \frac{\sigma }{\rho R^3}+\nonumber \\{} & {} \quad (n+1) \frac{2M {\dot{R}}}{\rho R^3}\bigg ((n-1)\left| \frac{\partial u_{0,r}}{\partial r}\right| ^{k-1}+4(k-1)(n+2)\frac{{\dot{R}}^2}{R^2}\left| \frac{\partial u_{0,r}}{\partial r}\right| ^{k-3}\bigg ), \end{aligned}$$
(28)

respectively. When \(k=1\), the non-spherical model reduces to the Newtonian model of Prosperetti [15].

3 Results and discussion

In this section, the results of the spherical oscillations (Sect. 3.1) and the non-spherical oscillations (Sect. 3.2) are presented. The evolution of the non-spherical bubble surface in power–law liquid is determined by solving two coupled ordinary differential equations (ODEs): one for the mean bubble radius (Eq. 17), and the other one is for the perturbation amplitude of the non-spherical mode (Eq. 26). These coupled ODEs are solved numerically via MATLAB. The initial conditions are \(R(t=0)=R_0, {\dot{R}}(t=0)=0, a_n(t=0)=0.01 R_0, {\dot{a}}_n(t=0)=0\).

For therapeutic applications, microbubble radii range from 1 to 10 microns. This range of radii for microbubbles makes them about the size of a red blood cell [41]. In these applications, the highest U.S. Food and Drug Administration (FDA)-approved mechanical index (MI) is 1.9 to prevent tissue damage [42]. This index is a measure of acoustic power. Consequently, \(p_a\) and f range from 0.25 to 15 atm and 1 to 8 MHz in this work, which yields a maximum value of 0.5 for MI. Furthermore, following Liu et al. [25] and Kaykanat and Uguz [5] to represent typical conditions for therapeutic applications of microbubbles, the default values of the parameters are chosen as \(M=0.002\) Pa\(\cdot \)s\(^\text {k}\), \(p_a=0.25\) atm, \(R_0=5\) \(\mu \)m, \(\sigma =0.0729\) N m\(^{-1}\), \(\rho =1000\) kg m\(^{-3}\), and \(\kappa =1.4\). The pressure amplitude is chosen to be below Blake’s threshold, around 1 atm, representing a bubble’s explosive growth [43,44,45].

The model and the code are validated for a system of a bubble in an unbounded Newtonian liquid \((k = 1)\), subject to an acoustic field, and the results of Liu et al. [25] are recovered.

3.1 Spherical oscillations

First, the model is employed to investigate the spherical oscillations of the bubble in a power–law liquid (Figs. 2, 3, 4, 5). A parametric study is conducted, and the effects of the consistency index, m, power–law index, k, driving frequency, \(\omega _d\), and the initial bubble radius, \(R_0\), are investigated.

Fig. 2
figure 2

R(t) vs \(t/(2\pi /\omega _d)\) for (a) \(\omega _d=1\text { MHz}<\omega _0\), (b) \(\omega _d=2\text { MHz}\approx \omega _0/2\), (c) \(\omega _d=4\text { MHz}\approx \omega _0\) (radial resonance), and (d) \(\omega _d=8\text { MHz}>\omega _0\). Here, the blue (\(k = 0.75\)), red (\(k=1\)), and black (\(k=1.25\)) represent the shear-thinning, Newtonian, and shear-thickening liquids, respectively

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Fig. 3
figure 3

a Maximum value of bubble wall velocity max(\({\dot{R}}\)), vs power–law index, k and b maximum viscosity, \(\mu _{0,max}\) vs power–law index, k after 15 cycle

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Fig. 4
figure 4

Contour plot (M vs power–law index, k) for \(R_{max}/R_0\): a \(\omega _d=1\) MHz, b \(\omega _d=2\) MHz and c \(\omega _d=4\) MHz\(\approx \omega _0\) after 15 cycle

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Fig. 5
figure 5

Contour plot (power–law index, k vs initial bubble radius, \(R_0\)) for \(R_{max}/R_0\) after 15 cycle, \(f=2\) MHz

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The natural frequency of the radial mode, \(\omega _0\), is determined to be approximately 4 MHz [43,44,45]. The dynamic behavior of the radial modes is shown for \(\omega _d=1\text { MHz}<\omega _0\) (Fig. 2a), \(\omega _d=2\text { MHz}\approx \omega _0/2\) (Fig. 2b), \(\omega _d=4\text { MHz}\approx \omega _0\) (Fig. 2c), and \(\omega _d=8\text { MHz}\approx 2\omega _0\) (Fig. 2d) for different k values. The dark blue, red, and black lines show the shear-thinning (\(k=0.75\)), the Newtonian (\(k=1\)), and the shear-thickening liquids (\(k=1.25\)), respectively.

For the case of radial resonance, as shown in Fig. 2c, the amplitude of the radial oscillation is larger than that of the non-resonant ones (Fig.  2a, b, d). Furthermore, the amplitude of the radial oscillation decreases with the power–law index, k, as shown in Fig. 2b–d. For higher k, the strain rate decreases while the dissipation rate increases. Consequently, the oscillations are damped.

As seen in Fig. 2, the effect of k on the dynamic bubble radius depends on \(\omega _d\). To understand the effect of \(\omega _d\) on the viscosity, first the maximum bubble wall velocity (Fig. 3a), then, the maximum viscosity \(\mu _{0,max}\) (Fig. 3b), \(max\left( \left| {\dot{R}}/R\right| \right) ^{(k-1)}\) are obtained for different k values after 15 cycles of the acoustic field. For the case of radial resonance, bubble wall velocity, which also affects the behavior of the power–law liquid viscosity, is dramatically larger than that of the non-resonant cases, as shown in Fig. 3a. For the Newtonian liquid case, even though the bubble wall velocity changes with \(\omega _d\), the viscosity is, of course, constant, showing the effect of the power–law liquid.

The effect of M, which is related to the consistency index (\(M=2m(2\sqrt{3})^{k-1}\)), and the power–law index, k, on the maximum bubble radius, \(R_{max}/R_0\), are obtained in Fig. 4 for different \(\omega _d\) values, e.g., \(\omega _d=1\) MHz (Fig. 4a), \(\omega _d=2\) MHz (Fig. 4b) and \(\omega _d=4\) MHz (Fig. 4c). In Fig. 4a, the change of \(R_{max}/R_0\) is small with M and k compared to Fig. 4b, c. Therefore, the change of \(R_{max}/R_0\) cannot be observed in Fig. 4a. An inset figure for Fig. 4a is obtained with different color codes to show the effects of M and k on \(R_{max}/R_0\). For all cases, when M is low, \(R_{max}/R_0\) is high and becomes independent of k. As mentioned, M is related to the consistency index. The damping rate is relatively low for lower M values, i.e., small consistency indexes. As a result, lower M values, given in Eq. 15 increase the oscillations of the bubble radius with respect to time. Contrary to this, higher M values intensify the damping due to the increased dissipation rate and reduce the amplitude of the oscillations.

Although this paper focuses on microbubbles due to biomedical applications, in Fig. 5, the effects of larger \(R_0\) and k on \(|R_{max}/R_0|\) are investigated for \(f=2\) MHz. Maximum bubble radius values are obtained after 15 acoustic field cycles. In Fig. 5, the amplitude of the acoustic field is 0.25 atm. The power–law index, k, does not significantly affect the maximum bubble radius for large bubbles. Small bubbles have a larger surface area relative to their volume compared to larger bubbles. Consequently, surface forces become more important for small bubbles. The dependence of the bubble radius on bubble dynamics can also be seen in Eq. 17. As the bubble radius increases, the influence of the power–law index on bubble dynamics decreases with the dissipation term in Eq. 17 (\(\frac{2M}{k}|{\dot{R}}|^{k-1}\frac{{\dot{R}}}{R^k}\)) becomes smaller and dominates for small bubble radii.

3.2 Non-spherical oscillations

Two types of instabilities are observed: Rayleigh–Taylor (RT) and the parametric instabilities [16,17,18,19, 25]. Parametric instability is caused by the parametric resonance between the spherical oscillations and the non-spherical perturbation, which occurs when \(2 \omega _n / \omega _d=j\) [22, 46]. Here, j is most likely 1 or 2 [25, 47, 48]. \(\omega _n\) is the natural frequency of mode n [25, 47, 49]. The parametric instability of mode \(n = 0\) is investigated in Sect. 3.1 through the natural frequency of the spherical mode. If a disturbance grows over many cycles, it is assumed that the bubble will break up under these conditions [50,51,52]. The parametric instability evolves over several bubble oscillations. In contrast, the RT instability occurs during a single violent bubble collapse.

Fig. 6
figure 6

\(a_2(t)/R_0\) vs \(t/(2\pi /\omega _d)\) for (a) \(\omega _d=1\text { MHz}<\omega _0\), (b) \(\omega _d=2\text { MHz}\approx \omega _0/2\), (c) \(\omega _d=4\text { MHz}\approx \omega _0\) (radial resonance), and (d) \(\omega _d=8\text { MHz}>\omega _0\). Here, the blue, red and black lines show the shear-thinning (\(k=0.75\)), Newtonian (\(k=1\)) and shear-thickening (\(k=1.25\)) liquid cases, respectively

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Figure 6 shows the dynamic behavior of the second mode amplitudes for different power–law indexes and frequencies. As observed for the radial oscillations (Fig. 2), the oscillations of the perturbed mode amplitudes damped at a higher power–law index, k, e.g., when \(k=1\) and \(k=1.25\). For the cases out of radial resonance (\(\omega _d\approx \omega _0\)) and higher power–law indexes, e.g., \(k=1\) and \(k=1.25\), the initial disturbances of the shape modes damp and shape instabilities do not take place for this pressure amplitude. Here, the dissipation rate increases with k. Consequently, the amplitude of the oscillation decreases more quickly, and the bubble reaches its equilibrium radius faster.

At radial resonance, the intense radial oscillations lead to shape instability for \(k=0.75\). For lower power–law indexes, e.g., \(k=0.75\), the dissipation rate decreases. The amplitude of the non-spherical modes is small, such that the oscillations are effectively spherical. The time scale of shape oscillation is comparable with the radial oscillation (Fig. 2). In Fig. 6c, mode \(n = 2\) has grown sufficiently after several oscillations due to radial resonance that causes the bubble shape to depart from spherical. At radial resonance (\(\omega _d=4\) MHz), that of the non-spherical oscillations is related to the amplitude of the radial oscillations (Fig. 2c). Thus, non-spherical oscillations develop preferentially when the bubble is at its radial resonance. These are general features of parametric instabilities; surface modes are not directly excited by the applied ultrasound but are driven by radial oscillations.

In Fig. 7, the effect of perturbation mode n on the shape stability is investigated by setting the driving frequencies equal to the natural frequencies of the shape modes instead of radial mode: \(\omega _d=2.65\) MHz\(=\omega _2\) for \(n=2\) and \(\omega _d=9.6\) MHz=\(2\omega _3\) for \(n=3\) [25] for lower consistency index, e.g., \(M=2\times 10^{-4}\) Pa.s\(^k\). By decreasing the consistency index, the damping effect of the power–law index is decreased thus, the parametric instabilities emerge both for shear-thinning (\(k=0.75\)) and Newtonian liquid (\(k=1\)) cases because the driving frequencies are equal or close to a multiple of its natural frequency (2.65 MHz and 9.6 MHz). Unlike the Newtonian liquid, the damping factor for the power–law liquid is

$$\begin{aligned} \frac{M(n+2)(n+1)}{\rho R^2}\left| \frac{{\dot{R}}}{R}\right| ^{k-3}\bigg (\left( \frac{{\dot{R}}}{R}\right) ^2+4(k-1) \frac{{\dot{R}}^2}{R}\bigg ), \end{aligned}$$

which depends on the bubble wall velocity and the power–law index (Eq. 26).

Fig. 7
figure 7

\(a_n(t)/R_0\) vs \(t/(2\pi /\omega _d)\) for \(M=2\times 10^{-4}\) Pa\(\cdot \)s\(^k\) when (a) \(\omega _d=2.65\text { MHz}=\omega _2\), (b) \(\omega _d=9.6\text { MHz}=\omega _3\), (c) \(\omega _d=2.65\text { MHz}=\omega _2\), and (d) \(\omega _d=9.6\text { MHz}=\omega _3\). Here, the blue (\(n = 2\)), red (\(n=3\)), and black (\(n=4\)) represent the perturbation modes

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Fig. 8
figure 8

Bubble wall acceleration, \({\ddot{R}}\), vs \(t/(2\pi /\omega _d)\) for \(\omega _d=8\text { MHz}>\omega _0\), (a) \(p_a=2.5\) atm and \(k=0.75\), (b) \(p_a=2.5\) atm and \(k=1\) and (c) \(p_a=15\) atm and \(k=1.1\). \(a_n(t)/R(t)\) vs \(t/(2\pi /\omega _d)\) for \(\omega _d=8\text { MHz}<\omega _0\), (d) \(p_a=2.5\) atm and \(k=0.75\), (e) \(p_a=2.5\) atm and \(k=1\) and (f) \(p_a=15\) atm and \(k=1.1\)

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RT instability occurs near the minimum bubble radius where the acceleration \({\ddot{R}}\) of the bubble wall is positive (i.e., the gas accelerates into the fluid and vice versa). Solutions of the perturbation amplitudes (\(n = 2\) to \(n = 4\)) normalized by the mean bubble radius are shown in Fig. 8. During the inertial collapse, the bubble wall acceleration reaches the maximum value \(10.5\text {x}10^{9}\) ms\(^{-2}\) (Fig. 8a), \(6.3\text {x}10^{9}\) ms\(^{-2}\) (Fig. 8b) and \(1.9\text {x}10^{10}\) ms\(^{-2}\) (Fig. 8c) and sharply decreases to \(-8\text {x}10^{8}\) ms\(^{-2}\) in 0.020 \(\mu \)s (Fig. 8a), \(-0.49\text {x}10^{8}\) ms\(^{-2}\) in 0.025 \(\mu \)s (Fig. 8b) and \(-1.75\text {x}10^{9}\) ms\(^{-2}\) in in 0.015 \(\mu \)s (Fig. 8c). This instability is caused by the significant acceleration of the power–law liquid into the gas and vice versa. Although the acceleration of the liquid is high, compressible effects on the non-spherical and spherical oscillations are expected to be small because \({\dot{a}}_n/{c_\infty }\) and \({\dot{R}}/{c_\infty }\) are small. The maximum perturbed and unperturbed bubble wall velocities normalized by the sound speed in liquid in Fig. 8 are \({\dot{a}}_{max}/{c_\infty }=0.1\) and \({\dot{R}}_{max}/{c_\infty }=0.04\), where \(c_\infty \approx 1500 m/s\). For this parameter set, the most unstable mode is \(n = 3\). Furthermore, in Fig. 8, the effect of power–law index on the RT-type instability of a bubble is investigated. The RT instability depends only weakly on viscous effects, because the maximum acceleration of the bubble is independent of viscosity when the surrounding liquid is Newtonian. Contrary to the Newtonian liquid case, the power–law index changes the evolution of the mean bubble radius and, thus, the bubble wall acceleration (Fig. 8a–c), which is the most dominant factor to induce the RT-type instability. For higher power–law indexes, the acceleration is reduced, and the perturbation growth is also reduced (red line) (Fig. 8a, b). For higher values of k, e.g., \(k=1.1\), the non-spherical perturbation decays. Thus, to obtain RT-type instability, the amplitude of the acoustic pressure increases, e.g., \(p_a=15\) atm in Fig. 8c.

4 Conclusion

This paper studies the dynamics of spherical and non-spherical oscillations of a bubble in an unbounded power–law liquid. A model for the non-spherical bubble is developed by extending perturbation analysis for a spherical interface between a gas and a power–law liquid. Hence, a better understanding of the surface mode oscillations of bubbles leads to the classification of stable/unstable conditions.

Two types of instabilities are observed in this system, which are Rayleigh-Taylor and the parametric instabilities. Effects of power–law and consistency indexes, the applied acoustic field’s driven frequency, and the amplitude are investigated. The amplitude of non-spherical oscillations is high when shape mode resonance occurs under two conditions: the driving frequency satisfies the radial resonance condition or the Mathieu condition \(\omega _d = 2\omega _n/j\), where j is 1 or 2. For both spherical and non-spherical bubble oscillations, higher power–law indexes reduce the oscillations due to higher damping coefficients. The damping effect of the power–law index can be decreased by decreasing the consistency index. Unlike Newtonian liquids, the driving frequency affects the bubble wall velocity and thus the apparent viscosity. The power–law index changes the evolution of the mean bubble radius and, thus, the bubble wall acceleration, which induces the Rayleigh-Taylor instability, which is not observed for the Newtonian liquid.