Introduction

Bubble dynamics in liquids subjected to external acoustic fields play crucial roles in various industrial and biomedical applications. A single bubble immersed in a liquid under an acoustic field behaves as a nonlinear oscillator1, and understanding this behavior is essential for applications such as polymer devolatilization, composite processing, and ultrasonic cleaning devices2. While extensive research has been conducted on bubble dynamics in Newtonian fluids3,4,5,6,7,8, the behavior of bubbles in viscoelastic or non-Newtonian fluids remains less well understood and requires further investigation9.

The study of bubble dynamics began with an examination of cavitation erosion caused by ship propellers, starting with Rayleigh’s pioneering work in 1917 on cavitation bubble collapse10. Despite its potentially damaging effects, cavitation has been applied in various fields, including protein folding11, degradation of harmful water pollutants12, and biomedical treatments13. For instance, the stress fields generated by cavitation can lead to the fragmentation of kidney stones14 and the removal of bacterial biofilms from hard surfaces such as teeth15. Ultrasound-induced cavitation is also used for safe and efficient drug and gene delivery16,17,18,19,20 and, consequently, for chemotherapy treatments21,22. They examined the industrial viability of cavitation technologies, focusing on their unique ability to release thermal, mechanical, and chemical energy. The study investigated the distinctive characteristics of these technologies, explored ways to apply them more effectively and efficiently, and evaluated their potential applicability in combination with other technologies23.

In biomedical applications, gas bubbles and encapsulated gas bubbles serve as valuable contrast agents for medical ultrasound diagnosis24,25,26,27,28 and tumor identification29. S. Ilke Kaykanat and A. Kerem Uguz30 investigated the dynamics of an encapsulated bubble immersed in a spherical liquid cavity modeled using the Carreau–Yasuda constitutive model and surrounded by an infinite elastic solid. The bubble shell was modeled as a nonlinear neo-Hookean hyperelastic material. The rheological properties of both the shell and the solid either attenuated or amplified the bubble oscillations depending on the frequency. Moreover, the effect of the power-law index was influenced by other parameters such as the driving frequency and cavity size.

Joydip Mondal et al.31 simulated shape-mode oscillations both experimentally and numerically as a result of interactions with ultrasound for biological applications. However, under certain conditions, particularly at high-pressure amplitudes, bubble oscillations can become chaotic32. This inertial cavitation—characterized by bubbles of small radii expanding significantly before rapidly collapsing—may result in tissue damage due to potential cavitation bioeffect33,34,35. Because most biological tissues and fluids, such as blood, exhibit viscoelastic properties36,37, understanding bubble dynamics in non-Newtonian media is crucial for the safe and effective application of these technologies.

The addition of polymers to host fluids has been shown to effectively reduce the drag of turbulent flows38,39 and prevent cavitation. Early studies by Ellis and Hoyt40 and later by Chahine and Fruman41 demonstrated that bubble oscillations in non-Newtonian fluids composed of polymer solutions are attenuated compared with those in Newtonian fluids. This attenuation indicates that viscoelastic properties can influence bubble dynamics, potentially mitigating unwanted cavitation effects.

Previous theoretical treatments for bubble dynamics in non-Newtonian fluids have employed various rheological models. The initial motivation for studying this subject stemmed from observations of cavitation suppression in polymer solutions40,41. Early work used the linear Maxwell model to describe a spherical bubble in a viscoelastic fluid42,43. Subsequently, other viscoelastic models have been employed, including the three-constant Oldroyd fluid44,45, the Jeffreys model13,46,47, and the upper-convected Maxwell (UCM) model investigated by Kim48. Levitskii and Listrov49 examined the influence of rheological parameters on bubble oscillations using the Oldroyd model. Also, oscillations of a single bubble have been investigated in a Giesekus fluid50,51 Hao Tang et al.52 modeled the oscillation of a bubble near the interface of a two-phase fluid system, in which one of the fluids exhibits non-Newtonian behavior described by the Herschel–Bulkley model. Shima et al.45 further analyzed the effects of the relaxation and retardation times on frequency-response curves and the relationship between the initial bubble radius and the maximum pressure at the bubble. Brujan53 extended these studies by considering the compressibility effects of viscoelastic liquids.

More recent studies have explored the nonlinear dynamics of oscillating bubbles in viscoelastic fluids. Jiménez-Fernández and Crespo54 found that viscoelastic parameters can modulate bubble oscillations, with the bubble radius growing more than in the Newtonian case. Naude and Méndez55 considered thermal damping effects and showed that for high Deborah numbers, oscillations become chaotic regardless of thermal damping. Additionally, nonlinear oscillations and cavitation regimes for encapsulated bubbles have been widely researched. Khismatullin and Nadim26 modeled contrast agents as bubbles encapsulated by a thin elastic solid membrane, while Sarkar et al.56 developed a viscoelastic model of microbubbles composed of a flexible surfactant layer. Zhang and Li57 studied the mass transfer during the radial oscillations of such bubbles.

Despite these advancements, the dynamical response of spherical oscillating bubbles immersed in non-Newtonian fluids still requires further study. The behavior of bubbles in viscoelastic fluids is complex due to the interplay between fluid elasticity, viscosity, and compressibility. A multiscale analysis for the linearized version of the governing equation for the Maxwell model was performed, which was only valid for small amplitudes of acoustic forced oscillations58. In recent years, researchers have considered additional factors influencing bubble oscillation and dynamics. A-Man Zhang et al.8 considered various aspects such as fluid compressibility, bubble interactions, bubble migration, and other effects to present a comprehensive study of bubble dynamics in water, highlighting their influence on bubble behavior and interface evolution. Additionally, a numerical study investigated acoustic multicavitation dynamics in a generalized Newtonian fluid under the influence of external fields such as shear stress, electrical conductivity, surface tension-induced pressure, and magnetic fields59.

In this study, we address the complexities of bubble dynamics in viscoelastic fluids by examining the behavior of a single acoustically driven bubble, modeled using the Oldroyd-B constitutive equation. Unlike most previous studies that assume an adiabatic process for bubble oscillations, such as13,46,47,54, our study considers an isothermal process, which influences the pressure dynamics within the bubble and is more relevant to certain biomedical applications and low-frequency acoustic conditions. Furthermore, this study distinguishes between the solvent and polymeric viscosities in the governing equations and determines their ratio to ensure that the relaxation time is larger than the retardation time. This enables the model to predict tension-thickening elongational viscosity, which can significantly affect bubble dynamics. We introduce a semi-analytical approach to solve the governing equations, significantly enhancing both the accuracy and efficiency of our solutions. We employed Backward Differentiation Formula (BDF) with Newton’s method as a time stepping scheme and implicit finite element method, our numerical method demonstrates excellent stability, allowing us to reach relatively high Deborah and Weber numbers that are typically challenging to achieve. We conducted numerous sensitivity analyses to identify suitable parameter values within the context of this study. By varying parameters such as the Deborah number, we account for changes in the material’s relaxation time, allowing us to simulate bubble oscillations in fluids with different viscoelastic properties and we conduct a comprehensive investigation into the effects of various and wide range of flow parameters and non-Newtonian characteristics, providing deeper insights into the complex bubble dynamics governing viscoelastic fluid flows. Furthermore, we validate our results against experimental data. We also examine the simultaneous variation of two parameters and their impacts on bubble oscillation, such as acoustic pressure and Deborah number, as well as Elasticity number and Weber number separately. Notably, our analysis covers not just the usual Reynolds number ranges examined in earlier works but also extremely low Reynolds numbers (around Re = 0.01) representing creeping flow conditions. This results in elasticity numbers reaching ten or more, a range that has not yet been explored in the current literature.

The results of this study contribute to the development of safer and more effective applications in biomedical ultrasound and other technologies where bubble dynamics in viscoelastic media play a critical role.

The structure of the paper is organized as follows: First, we derive the governing equations for our study, which includes developing a modified version of the Rayleigh-Plesset equation tailored for the Oldroyd-B model and converting these equations into ordinary differential equations (ODEs). We then reduce the equations to develop formulations for the upper-convected Maxwell (UCM) model. Next, we present the oscillation behaviors for both Newtonian and viscoelastic cases, validating our results for the UCM model. Finally, we investigate the effects of individual parameter variations on the oscillations, as well as the combined effects of modifying two parameters simultaneously.

Bubble oscillation in Oldroyd-B fluids

To investigate the oscillation of a bubble in an incompressible viscoelastic fluid, the continuity and Cauchy momentum equations must be simplified based on certain assumptions. The problem assumes that the acoustic wave source is in close proximity to the biofluid in which the bubble oscillates. Under these conditions, energy attenuation of the acoustic wave is considered negligible, as the short distance minimizes energy loss. Additionally, the use of lower frequencies and a homogeneous medium further contributes to this negligible attenuation. The bubble is assumed to be spherically symmetric, with velocity components only in the r-direction. Additionally, if the stress tensor is assumed to be traceless and symmetric in the φ- and θ-directions, it follows that \(\:{\tau\:}_{rr}\:=\:-2{\tau\:}_{\theta\:\theta\:}\:=\:-2{\tau\:}_{\phi\:\phi\:}\), while the other components of the stress tensor are zero. Under these assumptions, the continuity equation in the r-direction is simplified to Eq. (1).

$$\:\frac{1}{{r}^{2}}\frac{\partial\:}{\partial\:r}\left({r}^{2}{v}_{r}\right)=0$$
(1)

Since at \(\:r=R\), the velocity in \(\:r\)-direction is \(\:\dot{R}\), then \(\:{v}_{r}\) will be given by:

$$\:{v}_{r}=\frac{\dot{R}{R}^{2}}{{r}^{2}}$$
(2)

where \(\:R\) is the radius of bubble, \(\:\dot{R}\) is the radial velocity of bubble and \(\:r\) is radial distance from bubble center.

By substituting \(\:{v}_{r}\) from Eq. (2) into the momentum equation in \(\:r\)-direction, we obtain Eq. (3).

$$\begin{aligned}&{}\rho \left(\frac{\partial\:{v}_{r}}{\partial\:t}+{v}_{r}\frac{\partial\:{v}_{r}}{\partial\:r}+\frac{\:{v}_{\theta\:}}{r}\frac{\partial\:{v}_{r}}{\partial\:\theta\:}+\frac{{v}_{\phi\:}}{r{sin}\theta\:}\frac{\partial\:{v}_{r}}{\partial\:\phi\:}-\frac{{{v}_{\theta\:}}^{2}+{{v}_{\phi\:}}^{2}}{r}\right)\\&=-\left[\frac{1}{{r}^{2}}\frac{\partial\:}{\partial\:r}\left({r}^{2}{\tau\:}_{rr}\right)+\frac{1}{r{sin\theta\:}\:}\frac{\partial\:}{\partial\:\theta\:}({\tau\:}_{\theta\:r\:}{sin}\theta\:)\right.\\&\left.+\frac{1}{r{sin}\theta\:}\frac{\partial\:}{\partial\:\phi\:}\left({\tau\:}_{\phi\:r\:}\right)-\frac{{\tau\:}_{\theta\:\theta\:\:}+{\tau\:}_{\phi\:\phi\:\:}}{r}\right]+\frac{\partial\:p}{\partial\:r}+\rho g\end{aligned}$$
(3)

In Eq. (3), \(\:\rho\:\) is density of fluid surrounding the bubble and \(\:r\) is a radial distance from the center of bubble, \(\:{v}_{r}\), \(\:{v}_{\theta\:}\:\)and \(\:{v}_{\phi\:}\) are the component of velocity vector in the spherical coordinate system, \(\:{\tau\:}_{\phi\:r}\:a{nd\:\tau\:}_{\theta\:r\:}\) are the non-diagonal terms of the stress tensor. By considering the preceding assumptions and integrating over the fluid domain, we derived the general format of the Rayleigh-Plesset equation, which is applicable to both Newtonian and non-Newtonian fluids60:

$$\:R\ddot{R}+\frac{3}{2}\:{\dot{R}}^{2}=\frac{{P}_{l}-{P}_{\infty\:}\left(t\right)}{\rho\:}-\frac{1}{\rho\:}{\tau\:}_{rr}\left(R\right)+\frac{2}{\rho\:}\underset{R}{\overset{\infty\:}{\int\:}}\frac{{\tau\:}_{rr}-{\tau\:}_{\theta\:\theta\:}}{r}dr$$
(4)

Taking into account that the tension tensor is traceless and symmetric as stated previously, Eq. (4) can be rewritten as Eq. (5).

$$\:R\ddot{R}+\frac{3}{2}\:{\dot{R}}^{2}=\frac{{P}_{l}-{P}_{\infty\:}\left(t\right)}{\rho\:}-\frac{1}{\rho\:}{\tau\:}_{rr}\left(R\right)+\frac{3}{\rho\:}\underset{R}{\overset{\infty\:}{\int\:}}\frac{{\tau\:}_{rr}}{r}dr$$
(5)

where \(\:{P}_{\infty\:}\left(t\right)\) is the transient ambient pressure that stimulates the oscillation, \(\:{P}_{l}\) is the liquid pressure. The difference between the outer and inner radial tractions of the bubble, \(\:{T}_{rr,l}\) and \(\:{T}_{rr,b}\), equals to \(\:2\sigma\:/R\), where \(\:\sigma\:\) is the surface tension. Hence:

$$\:{T}_{rr,l}-{T}_{rr,b}=\frac{2\sigma\:}{R}$$
(6)

According to Dalton’s law of partial pressure61, the pressure within the bubble,\(\:\:{P}_{b},\:\)is equal to the summation of the liquid vapor pressure, \(\:{P}_{v},\) and the gas pressure, \(\:{P}_{g}\left(t\right),\) which is assumed to undergo a polytropic process (\(\:{P}_{g}{V}^{n}=cte)\):

$$\:{P}_{b}\left(t\right)={P}_{v}+{P}_{g}\left(t\right)$$
(7)

When the bubble is at rest and in equilibrium, where \(\:r={R}_{0}\), the equilibrium pressure of the gas can be calculated from Eq. (8).

$$\:{P}_{g}={P}_{0}-{P}_{v}+\frac{2\sigma\:}{{R}_{0}}$$
(8)

where \(\:{P}_{0}\) is the initial equilibrium pressure and considering the behavior of a gas as an ideal gas undergoing a polytropic process. The state equation of the gas pressure is given by Eq. (9).

$$\:{P}_{g}\left(t\right)=\left({P}_{0}-{P}_{v}+\frac{2\sigma\:}{{R}_{0}}\right){\left(\frac{{R}_{0}}{R}\right)}^{3n}$$
(9)

Consequently, the traction value at the surface is expressed by Eq. (10).

$$\:{T}_{rr,b}=-{P}_{b}\left(t\right)=-{P}_{v}-\left({P}_{0}-{P}_{v}+\frac{2\sigma\:}{{R}_{0}}\right){\left(\frac{{R}_{0}}{R}\right)}^{3n}$$
(10)

\(\:{T}_{rr,l}\), is the radial component of the traction on the suspension side of the bubble, as defined by the following equation:

$$\:{T}_{rr,l}=-{P}_{l}-4{\eta\:}_{s}\frac{\dot{R}}{R}+{\tau\:}_{p,rr}\left(R\right)$$
(11)

where \(\:{P}_{l}\) is the hydrostatic pressure, \(\:{\eta\:}_{s}\) is solvent viscosity and \(\:{\tau\:}_{p,rr}\left(R\right)\) is polymeric tension or extra tension. By considering the polytropic process as an isothermal process, the value of \(\:n\) would be equal to 1, and by substituting equations (10)and (11) in Eq. (6), \(\:{P}_{l}\) will be derived as follows:

$$\:{P}_{l}=-4{\eta\:}_{s}\frac{\dot{R}}{R}-\frac{2\sigma\:}{{R}_{0}}+{P}_{v}+\left(\varDelta\:p+\frac{2\sigma\:}{{R}_{0}}\right){\left(\frac{{R}_{0}}{R}\right)}^{3}+{\tau\:}_{p,rr}\left(R\right)$$
(12)

where \(\:\varDelta\:p={P}_{0}-{P}_{v}\).

By inserting \(\:{P}_{l}\:\) from Eq. (12), and considering \(\:{P}_{\infty\:}\left(t\right)={P}_{0}(1+\epsilon\:\text{sin}\left(\omega\:t\right))\), in which \(\:{P}_{0}\:\)is the initial equilibrium pressure and \(\:\epsilon\:\) is the coefficient of the amplitude of the acoustic pressure, and \(\:\omega\:\) or \(\:f\:\)is the angular frequency of the acoustic pressure, the Rayleigh-Plesset equation will yield to:

$$\:\ddot{R}+\frac{3}{2}\:{\dot{R}}^{2}=-\varDelta\:p+\left(\varDelta\:p+\frac{2\sigma\:}{{R}_{0}}\right){\left(\frac{{R}_{0}}{R}\right)}^{3}-4{\eta\:}_{s}\frac{\dot{R}}{R}-{P}_{0}\epsilon\:{sin}\left(\omega\:t\right))-\frac{2\sigma\:}{R}\:\:+\frac{3}{\rho\:}\underset{R}{\overset{\infty\:}{\int\:}}\frac{{\tau\:}_{p,rr}}{r}dr$$
(13)

The term \(\:4{\eta\:}_{s}\dot{R}/R\) in Eq. (13) is the result of calculating the integral term in Rayleigh-Plesset equation, considering the solvent viscosity, which is a Newtonian fluid; thus, if the fluid is pure Newtonian the Rayleigh-Plesset equation be simplified to Eq. (14).

$$\:R\ddot{R}+\frac{3}{2}\:{\dot{R}}^{2}=-\varDelta\:p+\left(\varDelta\:p+\frac{2\sigma\:}{{R}_{0}}\right){\left(\frac{{R}_{0}}{R}\right)}^{3}-4{\eta\:}_{s}\frac{\dot{R}}{R}-{P}_{0}\epsilon\:{sin}\left(\omega\:t\right))-\frac{2\sigma\:}{R}\:\:$$
(14)

In order to take into account, the viscoelastic properties of the fluid, the Oldroyd-B constitutive equation, Eq. (16)62, is used to calculate the integral on the right-hand side of Eq. (13).

$$\:\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}=\begin{array}{c}{\tau\:}_{p}\\\:=\end{array}+\begin{array}{c}{\tau\:}_{s}\\\:=\end{array}$$
(15)
$$\:\begin{array}{c}{\tau\:}_{p}\\\:=\end{array}+{\lambda\:}_{\:}\left\{\left(\frac{\partial\:}{\partial\:t}\right(\begin{array}{c}{\tau\:}_{p}\\\:=\end{array})+\:\:\varvec{v}.\:\nabla\:\begin{array}{c}{\tau\:}_{p}\\\:=\end{array}-{\left(\nabla\:v\right)}^{T}.\:\begin{array}{c}{\tau\:}_{p}\\\:=\end{array}-\begin{array}{c}{\tau\:}_{p}\\\:=\end{array}.\:\:(\nabla\:v)\right\}={2\eta\:}_{p}\begin{array}{c}D\\\:=\end{array}$$
(16)

where \(\:\varvec{v}\) is the velocity vector, and \(\:\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}\) is the total stress tensor consisting of the Newtonian stress tensor (\(\:\begin{array}{c}{\tau\:}_{s}\\\:=\end{array})\) and polymeric stress tensor \(\:\begin{array}{c}{\tau\:}_{p}\\\:=\end{array}\). Also, \(\:{\lambda\:}_{\:}\) is relaxion time of fluid and the deformation tensor, \(\:\begin{array}{c}D\\\:=\end{array}\), is defined as:

$$\:\begin{array}{c}D\\\:=\end{array}=\frac{\nabla\:\varvec{v}+{\left(\nabla\:\varvec{v}\right)}^{T}}{2}$$
(17)

where \(\:\nabla\:\varvec{v}\:and\:{\left(\nabla\:\varvec{v}\right)}^{T}\) are the gradient of velocity and its transpose respectively. Also, \(\:{{\lambda\:}_{\:}}^{*}\) is the retardation time of the fluid, which is given by \(\:{{\lambda\:}_{\:}}^{*}=\frac{{\eta\:}_{s}}{{\eta\:}_{s}+{\eta\:}_{p}}\lambda\:\) when slip parameter is zero63.

By considering that the tension tensor is traceless and the only velocity component is in the r-direction, Eq. (16) yields to:

$$\:{\dot{\tau\:}}_{rr,p}+\left(\frac{1}{\lambda\:}+\frac{4{R}^{2}\dot{R}}{{r}^{3}}\right){\:\tau\:}_{rr,p}=-\frac{4{\eta\:}_{p}{R}^{2}\dot{R}}{\lambda\:{r}^{3}}$$
(18)

Here, we take \(\:B\left(R\right)\:\)to represent the integral in Eq. (13). To calculate \(\:B\left(R\right),\:\)we implement the Lagrangian coordinates47, introduced as \(\:y={r}^{3}-{R\left(t\right)}^{3}\), with derivative of \(\:\frac{dy}{dr}=3{r}^{2}.\) This transformation gives us:

$$\:B\left(R\right)={\int\:}_{R}^{\infty\:}\frac{{\tau\:}_{rr,p}(y,t)}{r}dr={\int\:}_{0}^{\infty\:}\frac{{\tau\:}_{rr,p}(y,t)}{y+{R}^{3}}dy$$
(19)

By making derivatives with respect to time and using Leibniz’s formula, we would have:

$$\:\frac{d}{dt}{\int\:}_{0}^{\infty\:}\frac{{\tau\:}_{rr,p}(y,t)}{y+{R}^{3}}dy={\int\:}_{0}^{\infty\:}\frac{d}{dt}\left[\frac{{\tau\:}_{rr,p}(y,t)}{y+{R}^{3}}\right]dy$$
(20)

Therefore, the time derivative of \(\:B\) is obtained as follows:

$$\:3\dot{B}={\int\:}_{0}^{\infty\:}\frac{{\dot{\tau\:}}_{rr,p}(y,t)}{y+{R}^{3}}dy$$
(21)

Multiplying Eq. (18) by \(\:dy/(y+{R}^{3})\) and integrating over the fluid domain with respect to \(\:dy\) yields to Eq. (22).

$$\:3{\lambda\:}_{\:}\dot{B}+3B+4{\lambda\:}_{\:}{R}^{2}\dot{R}{\int\:}_{0}^{\infty\:}\frac{{\dot{\tau\:}}_{rr,p}(y,t)}{{\left(y+{R}^{3}\right)}^{2}}dy=-{4\eta\:}_{p}\frac{\dot{R}}{R}$$
(22)

To solve the integral in Eq. (22), we use the definition of integration by parts: \(\:\int\:xdz=xz-\int\:zdx\) with \(\:x={\left(y+{R}^{3}\right)}^{-1}\) and \(\:dz={\dot{\tau\:}}_{rr}(y,t)dy\). By this substitution, we have:

$$\:dx=-\frac{dy}{{\left(y+{R}^{3}\right)}^{2}},\:\:\:\:\:\:\:\:\:z=\int\:{\dot{\tau\:}}_{rr,p}\left(y,t\right)dy=-3{R}^{2}\dot{R}{\tau\:}_{rr,p}$$
(23)

At this stage, by evaluating the integral in Eq. (22), it can be expressed as:

$$\:{\int\:}_{0}^{\infty\:}\frac{{\dot{\tau\:}}_{rr,p}(y,t)}{y+{R}^{3}}dy=-{\left(\frac{3{R}^{2}\dot{R}{\tau\:}_{rr,p}}{y+{R}^{3}}\right)}_{0}^{\infty\:}-3{R}^{2}\dot{R}{\int\:}_{0}^{\infty\:}\frac{{\tau\:}_{rr,p}(y,t)}{{\left(y+{R}^{3}\right)}^{2}}dy$$
(24)

By substituting Eq. (24) in Eq. (21), one obtains:

$$\:{\int\:}_{0}^{\infty\:}\frac{{\tau\:}_{rr,p}(y,t)}{{\left(y+{R}^{3}\right)}^{2}}dy=\frac{{\tau\:}_{rr,p}\left(R\right)}{{R}^{3}}-\frac{\dot{B}}{{R}^{2}\dot{R}}$$
(25)

Then, by considering the calculation of the integral in Eq. (25), Eq. (22) will turn into Eq. (26).

$$\:-{\lambda\:}_{\:}\dot{B}+3B+4\left({\lambda\:}_{\:}{\tau\:}_{rr,p}\left(R\right)+{\eta\:}_{p}\right)\frac{\dot{R}}{R}=0$$
(26)

We introduce dimensionless variables marked with asterisks as follows:

$$\:{R}^{*}=\frac{R}{{R}_{0}},\:\:\:\:\:{t}^{*}=\frac{t}{{t}_{c}},\:\:\:\:\:U=\sqrt{\frac{{P}_{0}-{P}_{v}}{\rho\:}},\:\:\:\:\:\:{t}_{c}=\frac{{R}_{0}}{U},\:\:\:\:\:{\epsilon\:}^{*}=\epsilon\:\left(1-\frac{{P}_{v}}{{P}_{0}}\right),\:\:\:\:\:{\tau\:}_{rr,p}^{*}=\frac{{\tau\:}_{rr,p}}{\rho\:{U}^{2}},\:\:\:\:{\omega\:}^{*}=\omega\:{t}_{c}$$

As a result, the following physical dimensionless parameters emerge in the equation:

$$\:Re=\frac{\rho\:U{R}_{0}}{{\eta\:}_{t}}\:,\:\:\:\:\:We=\frac{\rho\:{U}^{2}{R}_{0}}{\sigma\:},\:\:\:\:\:De=\frac{\lambda\:}{{t}_{c}},\:\:\:\:\:\:{R}_{\mu\:}=\frac{{\eta\:}_{s}}{{\eta\:}_{s}+{\eta\:}_{p}}$$

where \(\:Re\) corresponds to the Reynolds number (ratio of inertia forces to viscous forces) in which \(\:{\eta\:}_{t}\) is the total viscosity and is the sum of\(\:\:{\eta\:}_{s}\) (solvent viscosity) and \(\:{\eta\:}_{p}\) (polymeric viscosity) i.e., \(\:{\eta\:}_{t}=\:{\eta\:}_{s}\) + \(\:{\eta\:}_{p}\). Weber number (\(\:We)\:\)represents the ratio of inertia forces to surface tension forces. The Deborah number (\(\:De\)) denotes the ratio of the relaxation time of the viscoelastic fluid to the characteristic time (\(\:{t}_{c}\)) of bubble motion. Consequently, the equations form a system of ordinary differential equations, equations (27) to (30), which must be solved to obtain a solution for this system of equations. For convenience, we eliminate the asterisks for dimensionless variables.

$$\:V=\dot{R}$$
(27)
$$\:\dot{V}=\frac{1}{R}\left\{\frac{2}{We}\left(\frac{1}{{R}^{3}}-\frac{1}{R}\right)-1-\epsilon\:\text{sin}\left(\omega\:t\right)+\frac{1}{{R}^{3}}-\frac{3}{2}{V}^{2}+3B-\frac{4{R}_{\mu\:}}{Re}\frac{V}{R}\right\}$$
(28)
$$\:\dot{B}=\frac{3}{De}B+4{\tau\:}_{rr,p}\frac{V}{R}+\frac{4(1-{R}_{\mu\:})}{De.Re}\frac{V}{R}$$
(29)
$$\:{\dot{\tau\:}}_{rr,p}=-\frac{1}{De}{\tau\:}_{rr,p}-4{\tau\:}_{rr,p}\frac{V}{R}-\frac{4(1-{R}_{\mu\:})}{De.Re}\frac{V}{R}$$
(30)

The initial conditions for the four ordinary differential equations are:

$$\:{R}_{0}=1,\:{v}_{0}=0,\:{B}_{0}=0,\:{\tau\:}_{0}=0$$

In the case of Newtonian fluids, the system of equations can be reduced to two ordinary differential equations, (31(and (32(.

$$\:V=\dot{R}$$
(31)
$$\:\dot{V}=\frac{1}{R}\left\{\frac{2}{We}\left(\frac{1}{{R}^{3}}-\frac{1}{R}\right)-1-\epsilon\:\text{sin}\left(\omega\:t\right)+\frac{1}{{R}^{3}}-\frac{3}{2}{V}^{2}-\frac{4}{Re}\frac{V}{R}\right\}$$
(32)

Upper convected Maxwell model

In order to validate the results obtained for the oscillation of bubbles in an Oldroyd-B fluid, the oscillation of bubble in upper-convected Maxwell (UCM) fluid is compared to one of the results of a previous study.

The constitutive equation of the UCM model is defined as63:

$$\:\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}+{\lambda\:}_{\:}\left\{\left(\frac{\partial\:}{\partial\:t}\right(\begin{array}{c}{\tau\:}_{t}\\\:=\end{array})+\:\:\varvec{v}.\:\nabla\:\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}-{\left(\nabla\:v\right)}^{T}.\:\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}-\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}.\:\:(\nabla\:v)\right\}={2\eta\:}_{0}\begin{array}{c}D\\\:=\end{array}$$

where \(\:{\eta\:}_{0}\) is the zero-shear viscosity and \(\:\begin{array}{c}{\tau\:}_{t}\\\:=\end{array}\) represents the total stress tensor.

In order to facilitate comparison with the Oldroyd-B model, the same dimensionless variables will be employed for UCM model. The dimensionless numbers, Reynolds number, Weber number, and Deborah number will be as follows:

$$\:Re=\frac{\rho\:U{R}_{0}}{{\eta\:}_{0}},\:\:\:\:\:We=\frac{\rho\:{U}^{2}{R}_{0}}{\sigma\:},\:\:\:\:\:De=\frac{\lambda\:}{{t}_{c}}\:\:\:\:\:\:$$

Therefore, the final dimensionless equations for a bubble oscillating in UCM fluid are given in equations (33) to (36).

$$\:V=\dot{R}$$
(33)
$$\:\dot{V}=\frac{1}{R}\left\{\frac{2}{We}\left(\frac{1}{{R}^{3}}-\frac{1}{R}\right)-1-\epsilon\:\text{sin}\left(\omega\:t\right)+\frac{1}{{R}^{3}}-\frac{3}{2}{V}^{2}+3B\right\}$$
(34)
$$\:\dot{B}=\frac{3}{De}B+4{\tau\:}_{rr,t}\frac{V}{R}+\frac{4}{De.Re}\frac{V}{R}$$
(35)
$$\:{\dot{\tau\:}}_{rr,t}=-\frac{1}{De}{\tau\:}_{rr,t}-4{\tau\:}_{rr,t}\frac{V}{R}-\frac{4}{De.Re}\frac{V}{R}$$
(36)

In addition, this system of ordinary differential equations can be derived through considering \(\:{R}_{\mu\:}\) equals to zero64, and the oscillation of bubbles in a UCM fluid is considered to validate the result of study relating to Oldroyd-B model.

In order to investigate how elastic forces and inertial forces contribute to bubble oscillation, the Elasticity number is considered. In rheology, Elasticity number (El) is a dimensionless number that quantifies the ratio of elastic forces to inertial forces as: \(\:El=\frac{De}{Re}\) or \(\:El=\frac{\lambda\:\eta\:}{\rho\:{R}_{0}^{2}}\)66. Nevertheless, for the Newtonian case this parameter is negligible.

Results and discussion

We employ a combination of numerical methods to solve the nonlinear system of ordinary differential equations governing bubble dynamics in both viscoelastic and Newtonian fluids. Specifically, we utilize Newton’s method to address non-linearities and the Backward Differentiation Formula (BDF) for stable time integration. Initially, we present results for a Newtonian fluid to provide a baseline comparison with viscoelastic fluids. Subsequently, we validate our numerical approach using the Oldroyd-B model by comparing it to the Upper Convected Maxwell (UCM) fluid model, where we set \(\:{R}_{\mu\:}=0\) in the Oldroyd-B model. Consequently, we examine the dynamics of bubble oscillation in the Oldroyd-B fluid, with particular emphasis on the effects of different non-dimensional numbers, i.e. De, We, Re, \(\:\epsilon\:\), and \(\:{\upomega\:}\).

Newtonian case

When a bubble oscillates within a Newtonian fluid, the tension tensor is defined by a Newtonian constitutive equation. Consequently, equations (31) and (32) are solved numerically to model this behavior. Notably, in the Newtonian case, the absence of elastic effects prevents bubble oscillation at low values of non-dimensional parameters, such as Re = 0.01, We = 0.5, ε = 0.1, and ω = 0.4, which are the base case for bubble oscillation in a viscoelastic fluid. This difference underscores the stabilizing influence of elastic effects on bubble oscillation. The bubble oscillations in the Newtonian fluid under specific conditions are shown in Fig. 1.

Fig. 1
figure 1

Bubble oscillation in a Newtonian fluid, \(\:e\:=0.01\:\:We\:=0.5\:\:\epsilon\:\:=\:0.1\:\omega\:\:=0.\:4\).

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Oscillation of bubble in viscoelastic fluid: validation

To validate this mathematical procedure, the results of oscillating bubble in a UCM fluid, specifically when \(\:{R}_{\mu\:}=0\:\)are compared with those from a previous study. The physical properties of bubble oscillation in medical applications are reported in55:

$$\:{\eta\:}_{p}=30cP,\rho\:={10}^{3}kg/{m}^{3},\:f=3MHz\:,\:{R}_{0}=0.73\mu\:m,\:{p}_{A}=0.4MPa,\:\sigma\:=0.072N/m$$

Figure 2 illustrates the oscillation of a bubble in a UCM fluid for the equivalent values for dimensionless parameters: \(\:De=0.48,Re=0.28\), \(\:We\) = 1.2, \(\:\epsilon\:=0.8,\) \(\:\omega\:=0.11\), and \(\:{R}_{\mu\:}=0.1\). The oscillation process is isothermal in this study and the referenced paper. For these dimensionless numbers, the elasticity number is approximately 1.71. As when the Reynolds number increases and the elasticity effect diminishes, the amplitude of bubble oscillation increases.

The bubble oscillation is plotted over 11 cycles, with time as the ratio of dimensionless time to the acoustic period. The results align well with those of previous study. In the referred study, the oscillating a bubble through an isothermal process in a UCM fluid was examined, yielding the maximum and minimum bubble radii of approximately 1.16 and 0.89, respectively. These values closely match the maximum and minimum bubble radii shown in Fig. 2.

Fig. 2
figure 2

Bubble oscillation in a UCM fluid, \(\:De=0.48,Re=0.28\), \(\:We\) = 1.2, \(\:\epsilon\:=0.8,\) \(\:\omega\:=0.11\), and \(\:{R}_{\mu\:}=0\).

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Fig. 3
figure 3

The change of bubble radius in proportion to initial radius, \(\:De=0.532,Re=6.49\), \(\:We=245\), \(\:\epsilon\:=0.01,\) and \(\:{R}_{\mu\:}=0.1\).

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Additionally, the proportional change in bubble radius relative to the initial radius was investigated across frequencies ranging from \(\:10\) to \(\:50\:kHz\). These results were compared with an experimental study that modeled the material using a linear viscoelastic Kelvin-Voigt66. In that study, the initial bubble radius was \(\:171\:{\upmu\:}\text{m}\), and the agarose gel (2% concentration) exhibited a shear modulus of \(\:256\pm\:29\:\text{k}\text{P}\text{a}\). For this concentration, the shear viscosity (\(\:{\eta\:}_{t}\)) and surface tension were measured as \(\:0.280\pm\:0\)0.03 \(\:\text{P}\text{a}.\text{s}\) and \(\:0.0676\:\text{N}.{m}^{-1}\), respectively. Using these parameter values, we computed the dimensionless numbers and numerically solved the governing system of equations. The blue triangular markers in Fig. 3 represent the bubble radius change ratio across the tested frequency range. These results demonstrate strong agreement with both the analytical (solid red line) and theoretical (red triangular markers) predictions from the aforementioned study, as shown in Fig. 3.

Oscillation of bubbles in Oldroyd-B fluid

The coupled ordinary differential equations (27) to (30) are solved numerically for a variety of cases. Some parameter values, such as frequency and surface tension, are determined based on previous studies, including those by Jiménez-Fernández54 and J. Naude, F. Méndez55, in which typical values were assumed and subsequently used to calculate dimensionless numbers. Other parameters, such as the fluid relaxation time, bubble initial radius, and polymeric viscosity, exhibit a wide range of values across different studies8,54,55,67. In this work, various sensitivity analyses are conducted to select parameter values appropriate for the context of our study, while ensuring that the aforementioned ranges are covered within our chosen parameter spectrum. The initial case is defined using the following parameters: \(\:De\:=\:0.1,\:\:Re\:=\:0.01,\:\:We\:=\:0.5,\:\:\epsilon\:\:=\:0.1,\:\:\omega\:\:=\:0.4,\:\:{R}_{\mu\:}=0.1\). In each step, only a single parameter is modified, while keeping the others constant. For these non-dimensional numbers, the amplitude of the oscillations is approximately 1.0006, which is attributed to high elasticity number (\(\:El=10)\). To avoid numerical problems, such values are chosen for the initial case. The periodic oscillation of a bubble with a low-amplitude of bubble oscillation is depicted in Fig. 4. Additionally, for all cases involving the Oldroyd-B fluid, \(\:{R}_{\mu\:}=0.1\) is maintained and since \(\:{{\lambda\:}_{\:}}^{*}={R}_{\mu\:}\lambda\:\),it can be deduced that \(\:{{\lambda\:}_{\:}}^{*}<\lambda\:\),enabling the model to predict tension-thickening elongational viscosity, which impacts bubble oscillations62.

Fig. 4
figure 4

Bubble oscillation in a Oldroyd-B fluid with\(\:\:De\:=\:0.1,\:\:Re\:=\:0.01,\:\:We\:=\:0.5,\:\:\epsilon\:\:=\:0.1,\:\:\omega\:\:=\:0.4,\:\:{R}_{\mu\:}=0.1,f\)=\(\:3MHz\), \(\:=0.072N/m\).

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Impact of the Deborah number on bubble oscillation

First, we solve the system of equations numerically by increasing the Deborah number. The oscillation of the bubble radius was computed for various Deborah numbers. In Fig. 5, the oscillation of the bubble is illustrated for different Deborah numbers. It is obvious that an increase in Deborah number cannot significantly affect the bubble oscillation for the values considered for other parameters. As Deborah number increases, the oscillation amplitude slightly increases. It can be concluded that when the elasticity number is in the order of ~ 10 or more, increasing the relaxation time does not significantly affect the configuration of the bubble oscillation. Since the Reynolds is 0.01, the viscous effect is dominant and increasing the elastic force is damped by viscous force and it cannot considerably influence the dynamics of bubble oscillation. In this study, the maximum Deborah number achieved is \(\:De=2.4\). Beyond this value, specifically at \(\:De>2.41\), the numerical simulation encounters limitations, resulting in singularities that limit further analysis.

Fig. 5
figure 5

Impact of the Deborah number on bubble oscillation (\(\:Re=0.01\), \(\:We\)=0.5, \(\:\epsilon\:=0.1,\omega\:=0.4,\:{R}_{\mu\:}=0.1,\:f\)=\(\:3MHz\), \(\:=0.072N/m\)), a \(\:De=1\), b \(\:De=2\), and c \(\:De=2.4\).

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Impact of the Reynolds number on bubble oscillation

An increase in the Reynolds number results in an increase in the amplitude of bubble oscillations, and this parameter can have a significant impact on the amplitude of oscillations. As illustrated in Fig. 6, the nondimensional radius increases from 1.003 to approximately 1.01 as the Re number rises from 0.05 to 0.65. Beyond a certain Reynolds number, specifically between \(\:Re=0.45\) and \(\:Re=0.65\), oscillations are still periodic; however, the peaks of the oscillation pattern diverge, exhibiting two and three local maxima per cycle at these respective Reynolds numbers, respectively. When the Reynolds number reaches approximately 1.1 or higher, bubble oscillations transit from periodic to chaotic, as shown in Fig. 7. In this chaotic regime, the oscillation amplitude varies irregularly across cycles, lacking discernible pattern. This behavior indicates that as the Reynolds number increases, leading to lower Elasticity number (El), elastic forces are increasingly dominated by inertial forces, resulting in an amplified oscillation.

Fig. 6
figure 6

Impact of the Reynolds number on bubble oscillations (\(\:De=0.1,We\)=0.5, \(\:=0.1,\:\:\omega\:=0.4,\:{R}_{\mu\:}=0.1\), \(\:f\)=\(\:3MHz\), \(\:\sigma\:=0.072N/m\)),\(\:\:a)\:Re=0.05,\:\:b)\:Re=0.1,\:\:c)\:Re=0.45,\:and\:d)\:Re=0.65\).

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Fig. 7
figure 7

Oscillation of bubble, \(\:De=0.1,Re=1.1\), \(\:We\)=0.5, \(\:\epsilon\:=0.1,\:\omega\:=0.4,\:{R}_{\mu\:}=0.1,\:\:f=3MHz\:and\:\sigma\:=0.072N/m\:.\).

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Figure 8 illustrates that as the Reynolds number increases, the maximum bubble radius initially exhibits a steep growth rate. However, as the Reynolds number approaches approximately \(\:0.25\), the slope of the graph begins to decrease, indicating a reduced rate of increase in the bubble radius. An increasing in the Reynolds number results in lower elasticity numbers, thereby diminishing the elastic effect.

Fig. 8
figure 8

The maximum radius with respect to Reynolds number, \(\:De=0.1\), \(\:We\)=0.5, \(\:\epsilon\:=0.1,\) \(\:\omega\:=0.4,\:{R}_{\mu\:}=0.1,\:f\)=\(\:3MHz\:\)and \(\:\:\sigma\:=0.072N/m\).

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Impact of the Weber number on bubble oscillation

As shown in Fig. 9, an increase in the Weber number results in a longer duration before the oscillation stabilizes. However, once the Weber number reaches approximately 0.1, the amplitude of the oscillation becomes nearly constant. This indicates that further increases in the Weber number beyond 0.1 have little to no effect on the oscillation amplitude for the given dimensionless parameters. In these scenarios, with an elasticity number of 10 and very low inertial forces, a decrease in surface tension—resulting in higher Weber numbers—does not significantly alter the bubble oscillation amplitude. Nevertheless, higher Weber numbers do prolong the time required for the bubble oscillation to stabilize.

Fig. 9
figure 9

Impact of Weber number on bubble oscillation, (\(\:De=0.1,Re=0.01\), \(\:\epsilon\:=0.1,\:\omega\:=0.4,\:\:{R}_{\mu\:}=0.1,\:f\)=\(\:3MHz,\:\sigma\:=0.072N/m\:\), a \(\:We\)=0.01, b \(\:We\)=0.1, c \(\:We\)=0.5, and d \(\:We\)=2.

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Fig. 10
figure 10

Maximum radius with respect to Weber number, \(\:De=0.1,Re=0.01\), \(\:\epsilon\:=0.1,\) \(\:\omega\:=0.4,\:{R}_{\mu\:}=0.1,f\)=\(\:3MHz\).

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Figure 10 shows how the maximum radius changes as the Weber number increases. The diagram indicates that when \(\:We\ll\:1\) and increases slightly, the maximum radius rises sharply; however, for higher values of the Weber number, the maximum bubble radius does not change with an increment of the Weber number. The Elasticity number (El) is approximately \(\:10\) in this case, and the other dimensionless numbers remain consistent with the initial case. Consequently, when the Weber number reaches a sufficiently high level, the bubble’s behavior under parameters values does not significantly affect its oscillation.

Impact of frequency on bubble Oscillation

As shown in Fig. 11, increasing the frequency results in a decrease in oscillation amplitude. For instance, when the angular frequency is increased from 0.1 to 2, the amplitude of oscillation diminishes to 0.99 of its initial value. The maximum frequency is observed to be \(\:5.3\), beyond which the numerical solution reaches a singularity.

In contrast to Newtonian cases, it is evident that elastic effects can stabilize bubble oscillation; however, in the absence of sufficient elasticity, the bubble may fail to oscillate periodically, exhibiting chaotic behavior or collapsing shortly after displaying non-periodic dynamics.

Fig. 11
figure 11

Impact of frequency on bubble oscillation \(\:De=0.1,Re=0.01\), \(\:We\)=0.5, \(\:\epsilon\:=0.1,{R}_{\mu\:}=0.1\) and \(\:\sigma\:=0.072N/m\). a \(\:\omega\:=0.1\), \(\:f\)=\(\:3MHz\) b \(\:\omega\:=0.3\), \(\:f\)=\(\:8.18MHz\:\)and c) \(\:\omega\:=2\), \(\:f\)=\(\:54.54MHz\).

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Figure 12 illustrates that the maximum bubble radius decreases as the frequency increases. At a frequency of approximately \(\:1.6\), the rate of change in the radius stabilizes. Conversely, at a frequency of approximately \(\:5.3\), the bubble collapses.

Fig. 12
figure 12

The effect of angular frequency on maximum radius, \(\:De=0.1,Re=0.01\), \(\:We\)=0.5, \(\:\epsilon\:=0.1,\:{\:R}_{\mu\:}=0.1\:and\:\sigma\:=0.072N/m\).

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Impact of acoustic pressure on bubble Oscillation

The value of the \(\:\epsilon\:\), which represents the magnitude of the non-dimensional ratio of acoustic pressure to ambient pressure, has a significant effect on the amplitude of the isothermal bubble oscillations. As the amplitude of the acoustic pressure increases, the input oscillation amplitude also rises, causing the bubble to oscillate with a larger radius, as shown in Fig. 13. However, when the acoustic pressure reaches a critical high level (\(\:\epsilon\:=1.35)\), the bubble collapses due to the high acoustic pressure, which exerts high strain on the bubble.

Fig. 13
figure 13

Impact of acoustic pressure on bubble dynamics, \(\:De=0.1,Re=0.01\), \(\:We=0.5\), \(\:\omega\:=0.4,\:f=3MHz\:and\:\:\sigma\:=0.072N/m\:,\:{R}_{\mu\:}=0.1\), a \(\:\epsilon\:=0.5\), b \(\:\epsilon\:=1.3,\) and c \(\:\epsilon\:=1.35\).

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Figure 14 illustrates how increasing acoustic pressure affects both the maximum and minimum bubble radii. As the acoustic pressure rises, the maximum radius increases while the minimum radius decreases, leading to significant increase in the amplitude of bubble oscillation. The relationship between the maximum or minimum bubble radii versus acoustic pressure is roughly linear.

Fig. 14
figure 14

The effect of acoustic pressure on (a) maximum and (b) minimum radii, \(\:De=0.1,Re=0.01\), \(\:We\)=0.5, \(\:\omega\:=0.4,\:\:{R}_{\mu\:}=0.1,\:f=3MHz\:and\:\:\sigma\:=0.072N/m\).

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As demonstrated in Figs. 15 and 16, an increase in the Deborah number and acoustic pressure or the Reynolds number and acoustic pressure results in a corresponding rise in the maximum bubble radius. Additionally, each figure presents the numerical results from two different perspectives. However, the simultaneous increases in the Reynolds and acoustic pressure have a more significant effect on the evolution of the maximum bubble radius. In Fig. 15, The maximum bubble radius reaches approximately 1.007 for \(\:\epsilon\:=0.525\), whereas it reaches 1.065 for the same value of \(\:\epsilon\:\) in Fig. 16. Thus, changes in the Reynolds number appear to exert a greater influence on bubble oscillation than modifications to the Deborah number. From a computational perspective, increasing the Reynolds number, which leads to a smaller elasticity number, results in fewer numerical problems and it is possible to obtain the result for a wider range of Reynolds number values comparing to Deborah number. To avoid numerical instabilities, since these two parameters increased, we halved the angular frequency to 0.2.

Fig. 15
figure 15

The effect of simultaneous increase of Deborah number and acoustic pressure, \(\:Re=0.01\), \(\:We\)=0.5\(\:,\) \(\:\omega\:=0.2,\:{R}_{\mu\:}=0.1\:,\:f=3MHz\:and\:\:\sigma\:=0.072N/m\).

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Fig. 16
figure 16

The effect of the simultaneous increase of Reynolds number and acoustic pressure, \(\:De=0.01\), \(\:We\)=0.5\(\:,\) \(\:\omega\:=0.2,\:{R}_{\mu\:}=0.1\:,\:f=3MHz\:and\:\:\sigma\:=0.072N/m\).

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Interaction between the Weber number and Deborah number

Changing the Elasticity number (El) influences bubble dynamics through its interaction with other dimensionless numbers. When the Deborah number (De) increases while the Reynolds number (Re) remains constant, El also increases. Simultaneous increases in the Weber number (We) can result in new oscillation patterns, including changes in the maximum radius achieved by the bubble. In some specific parameter combinations, the bubble exhibits multiple radius peaks during one oscillation period. Figure 17 presents the bubble oscillation results for varying El and We. The oscillation amplitude exhibits a nonlinear trend as El and We increase. Beyond De = 1.7 and We = 3.8, further increases in either parameter led to bubble collapse. To better isolate the effect of increasing El with increasing De, the Reynolds number is set to unity.

Fig. 17
figure 17

The effect of the simultaneous increase of Weber number and Deborahnumber\(\:.Re\:=\:1,\:\:\epsilon\:\:=\:0.1,\:\:\omega\:\:=\:0.2,\:\:{R}_{\mu\:}=0.1,f=3MHz,\:\sigma\:=0.072N/m\:\).

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Conclusion

In this paper, we investigated the oscillation of a bubble containing isothermal gas in an Oldroyd-B fluid. The governing equations of this study are the constitutive equation for the Oldroyd-B fluid and the modified Rayleigh-Plesset equation. By applying the Leibniz transformation, we converted these two equations—one of which is an integro-differential equation—into four ordinary differential equations. We then solved these equations using an implicit, damped version of Newton’s method with an automatically determined damping factor. For improved stability, we employed the backward differentiation formula as our time-stepping method. We obtained results for modified values of dimensionless numbers that are significant to the problem. Our findings illustrate that adding the viscoelastic effect can stabilize the oscillation of the bubble in Newtonian fluid specifically, when viscous force overcomes the elastic force. In viscoelastic cases in which \(\:{R}_{\mu\:}\) equals to 0.1, when the elasticity number (El) is sufficiently high (on the order of 10 or more), increasing the Deborah number does not affect the bubble’s oscillation because the viscous effect dampens the higher elastic effect. In such cases, increasing the Weber number does not increase the maximum bubble radius but prolongs the time required for the bubble to reach stable oscillation. Conversely, if El is approximately 1, increasing the Weber number can lead to a higher maximum bubble radius. Furthermore, regardless of whether the elasticity number is low or high, a high Reynolds number can result in a higher amplitude of bubble oscillation. In addition, increases of acoustic pressure leads to rising the oscillation amplitude. In our study, when the frequency ω equals to 0.1, the bubble oscillates with higher amplitude compared to higher frequencies, as these cases are far from the resonance condition. These results demonstrate that the elasticity number (El) plays a crucial role in the dynamics of bubble oscillation. Although the OldroydB model captures several viscoelastic characteristics, it falls short in representing the pronounced shearthinning typical of many biofluids. Adopting more sophisticated constitutive laws such as the Giesekus, or PhanThien–Tanner models would extend the rheological scope and enable a more accurate portrayal of diverse biologicalfluid behaviors.