Numerical Investigation on Influence of Number of Bubbles on Laser-Induced Microjet

Abstract

In this study, the impact of the number of bubbles on the velocity of laser-induced microjet is numerically investigated, focusing on the pressure wave propagation generated by multiple laser-induced bubbles. First, we show that the microjet velocity increases with the increasing impulse of the pressure wave propagating to the meniscus direction. This result indicates that it is possible to study the structure of the pressure field generated from bubbles to investigate the effect on microjet generation. In addition, it is found that the microjet is weakened with the increase in the number of bubbles. Next, we show that the propagation of the pressure waves has two types. The first type is propagating from a bubble to a meniscus. The second type is propagating round trip between nearby bubbles or by the bubble itself. Finally, we explain the reason for the decrease in the microjet velocity with the increasing number of bubbles by an expansion history of the bubbles, which depends on their interaction with the pressure waves. These results could help to design not only laser-induced microjet generation but also devices that use laser-induced bubbles generated in a microchannel.

1. Introduction

Tagawa et al. proposed a device for generating a microjet by focusing pulsed laser on a liquid-filled capillary [1], and Figure 1 shows schematics of the laser-induced microjet. This microjet has a diameter of up to 30 μm and travels at a speed of up to 850 m/s. The jet formation from the meniscus has been studied by several research groups. Antkowiak et al. studied the jet generation from the meniscus of a test tube filled with liquid by impacting the tube bottom on a hard surface [2]. Jet formation by bubble collapse on a free surface has been studied [3,4,5]. However, in these methods, compared with the method proposed by Tagawa et al. [1], the jet scale is larger, and the jet speed is about one order of magnitude smaller. High-speed microjets are very useful for practical engineering applications, such as needle-free drug injection [6,7,8,9], inkjet printing [10], and nano-droplet generation [11]. Therefore, considerable interest has been focused on microjet generation in the field of microfluidic devices.

The generation process of laser-induced microjets is as follows: a high-pressure laser-induced bubble is generated by focusing a pulsed laser in a liquid-filled capillary (Figure 1a). The pressure wave generated by the laser-induced bubble propagates to the meniscus (Figure 1b). When the pressure wave interacts with the meniscus, the focusing effect of the flow occurring at the curved meniscus causes high-speed microjet generation (Figure 1c). Note that the pressure wave propagation is not a simple process. A complex pressure field is formed inside the microtube through multiple reflections on the tube wall and other bubbles, and the microjet velocity also has complex behavior responding to the bubble conditions.

It has frequently been reported that multiple bubbles are often generated by focusing the laser in a liquid-filled capillary. Tagawa et al. experimentally investigated the jet velocity in a case in which multiple bubbles are generated. Additionally, they especially investigated the relationship between the approximated total bubble volume and jet velocity by changing the laser focusing position [1]. In their study, the bubble volume was geometrically approximated from the optical path of the laser, meaning that the volume increment is correlated with the number of bubbles. The experimental results [1] showed that the jet velocity increased with the approximated bubble volume, which indicates that the bubble volume and number of bubbles significantly affect the jet velocity. However, the mechanisms through which these parameters affect the jet velocity are not clear. To efficiently generate a microjet with less laser energy, it is necessary to investigate the effects of the laser-induced bubble volume and the number of bubbles on the jet velocity in detail and clarify the mechanism. In this study, we mainly focused on the effect of the number of bubbles.

It is important to clarify the characteristics of the pressure waves propagating from laser-induced bubbles to understand the microjet formation. Peters et al. numerically simulated laser-induced jets using the boundary integral method and showed that the summation of the pressure impulses decides the microjet velocity [12]. In addition, using a hydrophone and a high-speed camera, Hayasaka et al. experimentally showed that the jet velocity is roughly proportional to the pressure impulse [13].

There have been many studies on the pressure waves and pressure fields generated by laser-induced bubbles [14,15,16,17,18,19,20,21,22,23]. Kodama et al. investigated the pressure waves produced by an argon fluoride excimer laser, ruby laser, and shock tubes [15]. Moreover, Sankin et al. experimentally investigated the interactions between the pressure waves generated by a single laser-induced bubble and lithotripter shock waves [16]. Klaseboor et al. investigated the interaction of a single laser-induced bubble with lithotripter shock waves using the boundary element method [17]. Quinto-Su et al. investigated a cavitation bubble cloud between two laser-induced bubbles [19]. They also investigated the time evolution of the bubbles and the pressure waves in the cavitation cloud through experiments and simulations. Hsiao et al. simulated the dynamics of two bubbles and the pressure field around bubbles in a channel using a boundary element method [20]. Beig simulated two bubbles near a wall and showed the pressure field around two bubbles [22]. Kyriazis et al. numerically investigated the effects of the initial pressure of a single bubble and meniscus shape on microjet velocity [23]. However, few investigations have focused on the structure of the pressure waves propagating in the meniscus direction from multiple bubbles inside a microchannel.

In this study, we numerically simulate the microjet generation and the pressure waves propagating from multiple bubbles. First, we confirm a relationship between impulse and microjet velocity by simulating the case with and without the meniscus as in the previous studies [12,13]. In addition, we evaluate the effect of the number of bubbles on the microjet velocity and the impulse. The simulation is conducted for various numbers of bubbles with fixed total initial energy and bubble volume. Next, we investigate the detailed structure of pressure waves propagating in the meniscus direction and the pressure-time variation is explained at a specific point based on the pressure wave propagation path. Finally, we discuss the reason for the changing microjet velocity with the number of bubbles using the expansion history of the bubbles. To reduce the computational cost, we simulate the pressure wave propagation in a two-dimensional microchannel in this study. In a microtube, the focusing effect of the pressure waves occurs because the waves generated at the center of the axis are reflected on the surrounding tube wall and return to the tube axis. In the two-dimensional microfluidic channel, the focusing effect becomes weaker, and the quantitative discrepancy appears in the pressure impulse strength and jet velocity. However, the basic process of pressure field generation such as wave reflection on the wall and bubbles can be taken into account even in the two-dimensional microchannel, and the objective of this study is to make a quantitative discussion and understand the basic phenomena of pressure field generation.

2. Computational Methods

2.1. Governing Equations

The boundary integral method used in the previous study accurately predicts the microjet velocity [12]. However, we consider it unsuitable for our research purpose because pressure wave propagation needs to be investigated. In this study, the underwater pressure wave propagation is numerically simulated using the 5-equation model [24,25], which is one of the compressible multi-phase fluid models. The 5-equation model is widely used for compressive multiphase flow simulations such as bubble-shock interaction, bubble collapses, and underwater explosions [22,26,27,28,29,30]. This model can be written as follows:

$$\frac{\partial {\alpha }_i{\rho }_i}{\partial t}+\nabla · \left({\alpha }_i{\rho }_i\mathit{u}\right)=0,$$

(1)

$$\frac{\partial \rho \mathit{u}}{\partial t}+\nabla · \left(\rho \mathit{u}\mathit{u}+p\mathit{I}\right)=0,$$

(2)

$$\frac{\partial \rho \mathit{u}}{\partial t}+\nabla · \left(\rho \mathit{u}\mathit{u}+p\mathit{I}\right)=0,$$

(3)

$$\frac{\partial {\alpha }_g}{\partial t}+\nabla · \left(\mathit{u}{\alpha }_g\right)=\left({\alpha }_g+K\right)\nabla · \mathit{u} ,$$

(4)

where $\alpha$ is the volume fraction, $\rho$ is the density, $\mathit{u}=\left(u,v,w\right)$ denote the velocity components (Cartesian coordinates), $e$ is the total specific energy, $p$ is the pressure, and the subscript $i \in \left\{g,l\right\}$ indicates gas or liquid. Equation (1) shows that the mass conservation laws are defined both for the gas and liquid phases. Here, $K$ is

$$K={\alpha }_g{\alpha }_l\left(\frac{{\rho }_l{c}_l^2-{\rho }_g{c}_g^2}{{\alpha }_l{\rho }_g{c}_g^2+{\alpha }_g{\rho }_l{c}_l^2}\right)$$

(5)

where $c$ is the speed of sound. A term $K\nabla ·\mathit{u}$ in Equation (4) follows from an asymptotic expansion of the total-disequilibrium model of Baer and Nunziato [31]. The 5-equation model can be corrected to capture bubble collapse accurately by adding term $K\nabla ·\mathit{u}$ proposed by Kapila et al. [24,32,33]. On the other hand, it is also known that the term $K\nabla ·\mathit{u}$ induces numerical instability in strong compression or expansion in mixture regions due to its non-conservative nature [33,34]. The following relationships are held for the physical quantities.

$$1={\alpha }_g+{\alpha }_l,$$

(6)

$$\rho ={\alpha }_g{\rho }_g+{\alpha }_l{\rho }_l,$$

(7)

$$\rho e=\rho ϵ+\frac{1}{2}\rho \left(u^2+v^2+w^2\right),$$

(8)

$$\rho ϵ={\alpha }_g{\rho }_g{ϵ}_g+{\alpha }_l{\rho }_l{ϵ}_l,$$

(9)

$$p=p_g=p_l,$$

(10)

$$\mathit{u}={\mathit{u}}_g={\mathit{u}}_l,$$

(11)

where $ϵ$ is the internal specific energy. The closure of the system is obtained through the specification of an equation of state (EoS). To account for fluid properties, various EoS have been proposed considering molecular effects (See Métayer and Saurel [35]). To close the system, we introduced the Stiffened gas EoS (SG EoS):

$$p_i=\left({\gamma }_i-1\right){\rho }_i{ϵ}_i-{\gamma }_i{p}_{ref,i},$$

(12)

where $\gamma$ is the specific heat ratio, and $p_{ref}$ is the reference pressure. By assuming that the pressures of each phase are equal at the interface, Equation (12) can be replaced by [22,25,36]

$$p=\frac{\rho ϵ-\eta }{\xi },$$

(13)

$$\xi ={\displaystyle \sum }_{i\in \left\{g,l\right\}}\frac{{\alpha }_i}{{\gamma }_i-1},$$

(14)

$$\eta ={\displaystyle \sum }_{i\in \left\{g,l\right\}}\frac{{\alpha }_i{\gamma }_i{p}_{ref,i}}{{\gamma }_i-1}.$$

(15)

All the physical quantities were nondimensionalized as follows:

$$\mathit{x}=\frac{\tilde{\mathit{x}}}{R},$$

(16)

$$\rho =\frac{\tilde{\rho }}{{\rho }_{ref}},$$

(17)

$$\mathit{u}=\frac{\tilde{\mathit{u}}}{c_{ref}},$$

(18)

$$p=\frac{\tilde{p}}{{\rho }_{ref}{c}_{ref}^2},$$

(19)

$$\rho e=\frac{\tilde{\rho }\tilde{e}}{{\rho }_{ref}{c}_{ref}^2},$$

(20)

where $\mathit{x}$ is the position vector. The tilde (~) indicates a dimensional quantity. The reference values are set as $R=35 \left[\mathsf{\mu }\mathrm{m}\right]$, ${\rho }_{ref}=1.204 \left[\mathrm{kg}/{\mathrm{m}}^3\right]$, and $c_{ref}=341.0 \left[\mathrm{m}/\mathrm{s}\right]$.

In this study, we disregard the viscosity, cavitation, and surface tension. The previous study by Peter et al. also ignored cavitation and viscosity, but fortunately, it was reported that the microjet velocity was successfully reproduced accurately compared to the experiment [12]. In addition, Hayasaka et al. showed that the surface tension did not hinder the microjet formation [13]. On the other hand, the surface tension and viscosity may have some quantitative influence on the expansion motion of bubbles, and cavitation bubbles may have some impact on pressure wave propagation. However, in this study, the pressure propagation is analyzed and discussed in the simple condition and settings to obtain basic knowledge.

2.2. Simulation Settings and Conditions

In this study, we numerically simulate the pressure wave propagation from multiple high-pressure bubbles (the number of bubbles $N \in \left\{1,2,3\right\})$ inside a two-dimensional microchannel. To evaluate microjet velocity and impulse, we set up two simulation conditions.

Figure 2 shows the simulation settings of a microchannel with the meniscus (w-meniscus setting) for evaluating the microjet velocity. To capture the bubble interface and meniscus, the grid spacing is set to 0.04 in the range from$x\in \left[-6.2, 30\right]$. In this case, we do not apply the term $K\nabla ·\mathit{u}$due to strong expansion in the meniscus. In some studies simulating an underwater explosion with strong expansion waves, it was reported that the term $K\nabla ·\mathit{u}$ should not be applied for stable computation [27,30].

Figure 3 shows the simulation settings for a microchannel without the meniscus (w/o-meniscus setting) for evaluating impulse. The computational grid of the w/o-meniscus setting is uniform in the range from $x\in \left[-12, 12\right],$and the grid spacing is 0.04. In the regions of $x<-12$ and $x<12$, the computational grid is proportionally stretched in the directions of the left and right boundaries. In this study, the pressure wave impulse is evaluated at the inspection surface. The inspection surface for investigating the pressure impulse is the cross-section of the tube at 12 from the bubble center in the x-direction. The grid dependence is identified in Appendix B.

To measure the time history of the pressure, the pressure probe is placed at (12, 0). In this case, we evaluate the impulse as the following equation.

$$I={{\displaystyle \int }}_0^T{{\displaystyle \int }}_S^{}pdsdt={\displaystyle \sum }_{\Delta t\in \left[0, T\right]}^{}{\displaystyle \sum }_{j\in \left\{\mathrm{cell} \mathrm{indexes}\right\}}^{}{p}_j∆s_j∆t,$$

(21)

where $∆t$ is the time interval, $j$ is the index of the cell on the inspection surface, $∆s$ is the area of the cell face.

The bubble radius is set to 1 when the bubble number is 1. The parameters of the SG EoS is set as in Equation (22) following Shukla et al. [26] in the nondimensional value. The initial conditions are set as in Equation (23).

$$\left({\gamma }_g, {\gamma }_l, p_{ref,g}, p_{ref,l}\right)=\left(1.4, 6.12, 0, 2450\right),$$

(22)

$$\left({\rho }_g,{\rho }_l,u,v,w,{\alpha }_g,p\right)=\left\{\begin{array}{l}\left(83.3, 0, 0, 0, 0, 1, 571.4\right)\hspace{1em}\mathrm{Laser-induced\; bubble}\\ \left(0, 833, 0, 0, 0, 0, 0.7143\right)\hspace{1em}\mathrm{Liquid}\\ \left(1, 0, 0, 0, 0, 1, 0.7143\right)\hspace{1em}\hspace{1em}\mathrm{Air}\end{array}\right..$$

(23)

These conditions are the same for all the computational cases. The bubble pressure is based on the laser energy of 36 [$\mathsf{\mu }$J], which is in the same order of magnitude as that of the pulsed-laser devices frequently used in the experiments [6,7,8,9,13,37]. To keep the total internal energy of the bubbles constant among the three cases, the sum of the bubble volumes is set to be constant. From this restriction, the radius of the bubble $r$ is determined using the following equation:

$$r=\sqrt{\frac{R^2}{N}}.$$

(24)

The bubble radius complies with the typical spot size of a pulsed laser [37]. The bubbles are placed at equidistant intervals, as shown in

Table 1. Bubble positions of each case.

Number of Bubbles	1	2	3
Positions $\left(x,y\right)$ [-]	$\left(0,0\right)$	$\left(0,-3.5\right), \left(0,3.5\right)$	$\left(0,-4.67\right), \left(0,0\right), \left(0,4.67\right)$

.

Usually, a bubble interface is smeared for computational stability [22,28,29,32,33]. The smeared volume fraction is given a hyperbolic tangent function [32].

$${\alpha }_g=\frac{1}{2}\left(1-\tanh \left(\frac{l\left(\mathit{x}\right)-r}{h}\right)\right),$$

(25)

where $\mathit{x}$ is the position of the cell center, and $l\left(\mathit{x}\right)$ is the distance of the bubble center. The variable $h$ is the characteristic length of the corresponding computational cell.

Only the lower side of the microchannel is computed by the implemented symmetric boundary condition. The slip condition is implemented to the microchannel wall, and the outflow boundary condition using zero-order extrapolation is implemented at the left and right sides. The outflow boundary is set far enough away to stretch the grid so that the reflections are not a problem.

The time interval is fixed at $∆t=1.0\times {10}^{-4}$[-] from the Courant–Friedrichs–Lewy condition, and the Courant number was set as 0.2.

2.3. Numerical Schemes

The 5-equation model is discretized using the finite volume method. The numerical fluxes are computed using the Harten–Lax–van Leer contact approximate Riemann solver extended for the 5-equation model [38], and we use the second-order total variation diminishing Runge–Kutta for the time stepping. The following reconstruction methods are used depending on the settings for reasonable computation.

• $\rho -$THINC-MUSCL3

To stabilize the simulation of the w-meniscus setting, we use the third-order monotonic upstream-centered scheme for conservation laws [39] with the minmod limiter [40] (MUSCL3) and$\rho$-THINC [29]. $\rho -$THINC is one of the variations of the tangent of the hyperbola interface capturing method (THINC) [41] to sharpen the volume fraction and density on the interfaces. This coupling reconstruction method ($\rho$-THINC-MUSCL3) captures the interface and stabilizes the simulation.

• MP-WENO5-JS

To accurately capture the pressure waves propagating in the w/o-meniscus setting, we use the five-order monotonicity preserving weighted essentially non-oscillatory schemes with the indicator developed by Jiang and Shu (MP-WENO5-JS) [42,43,44] for a reconstruction.

The validation of the computational solver is described in Appendix A.

3. Result and Discussion

3.1. Microjet Evolution and Time History of Impulse

In this section, we show the relationship between impulse and microjet velocity. In addition, we also discuss the time evolution of microjets and the time history of the impulse. In this case, we used the $\rho$-THINC-MUSCL3 reconstruction to stabilize the simulations.

Figure 4 shows the time evolution of the microjet by the single bubble case. As reported by Tagawa et al. [1], it is confirmed that the microjet is generated from the center of the meniscus. Figure 5 shows the time history of the microjet velocity and microjet formations at the final state. Figure 5a shows that the microjet velocity at the final phase decreases as the number of bubbles increases. In addition, the final position of the microjet head also differs with the difference in the time history of the microjet velocity (Figure 5b–d).

Next, the impulse acting on the inspection surface is discussed. Figure 6 shows the time history of the impulse calculated by Equation (21). Figure 6 shows that the case with the three bubbles has the highest impulse until the time of 3.5. After that, the case with the two bubbles has the highest impulse in the time range from 3.5 to 4.5. After $t=4.5$, the case with the single bubble comes to have the highest impulse. As a summary of the above-mentioned results, the impulse increases with more multiple bubbles in the initial stage. However, after that, the impulse with the single bubble comes to exceed that with more bubbles.

As a summary of the results in this section, the pressure impulse at the final phase decreases as the number of bubbles increases. This tendency is the same as in the microjet velocity. Thus, we conclude that the microjet velocity has a strong correlation with the impulse of pressure propagating from the bubble to the meniscus. These results indicate that it is possible to discuss the effect of bubble conditions on the microjet velocity by investigating the structure of pressure field in the w/o-meniscus setting.

3.2. Pressure Field Evolution around a Single Bubble

In this section, we investigate the structure of the pressure field generated from the single bubble in the w/o-meniscus setting. We used the MP-WENO5-JS reconstruction to accurately capture the pressure waves.

Figure 7 shows the pressure field evolution generated by a single laser-induced bubble. At first, a spherical pressure wave (compression wave) is generated by a high-pressure laser-induced bubble (Figure 7a) and propagates in the liquid (Figure 7b). The compression wave then reaches the microchannel wall and is reflected as a compression wave (Figure 7c). Next, the pressure wave from the wall is reflected at the bubble interface and becomes an expansion wave (Figure 7d). After that, the expansion wave is reflected at the wall (Figure 7e) and is also reflected as a compression wave at the bubble interface (Figure 7f). As a result of this series of wave reflections, a complex pressure field is formed inside the microchannel. In addition, the pressure waves propagating repeatedly between the bubble and wall can be observed.

Figure 8 shows the time history of the pressure at the probe position. As observed in the figure, the time history of the pressure has periodic oscillations with the period $T$. In addition, the pressure peaks are confirmed at a unique timing $t_n$ in the initial phase.

It can be considered that the time-pressure history is generated by the following process: Figure 9 shows the path of the pressure wave from the bubble to the pressure probe. There are two types of dominant pressure wave paths reaching the pressure probe. The first path is by the reflected pressure wave not only on the wall but also on the bubble (Figure 9a). The pressure wave generated by the high-pressure bubble returns to the bubble through the reflection at the wall and then propagates to the pressure probe through the reflection on the bubble. The pressure wave then repeatedly goes back and forth between the bubble and wall. Therefore, the pressure wave propagation to the pressure probe is also periodically repeated. The second path is by the pressure wave propagation from the bubble to the probe through multiple reflections at the wall (Figure 9b). The pressure wave generated by the high-pressure bubble reaches the probe directly or through more than one reflection on the wall. It is considered that the periodical oscillation in the time-pressure history shown in Figure 8 is due to the first pressure path and that the pressure peaks seen in the initial phase come from the second pressure path, from which the pressure wave arrives at the probe through a variety of paths. This results in the delay of the pressure peak depending on the propagation paths.

To verify the above-mentioned pressure oscillation mechanism, first, the periodic time of the pressure oscillation described in Figure 9a is calculated using a simple theory. The period of the pressure peaks $T$is expected to be calculated from the time taken by the pressure wave propagating over the distance $L_p$, which is the round-trip distance between the bubble and wall, as follows:

$$T=\frac{L_p}{c_l}=\frac{D}{c_l}$$

(26)

where $c_l$ is the sound speed of the liquid, and $D=14$ is the width of the channel. In this study, $c_l$ is 4.8 by a dimensionless quantity. Next, the delay time of the pressure peak coming from the second pressure wave path is estimated by considering the length of the pressure path $L_n$, as depicted by the dashed line in Figure 9b. The delay time $t_n$ for the pressure wave to arrive at the probe could be calculated using the following equation:

$$\begin{gathered} t_1=\frac{L_1}{c_l}=\frac{L}{c_l} \\ {t}_2=\frac{L_2}{c_l}=\frac{\sqrt{L^2+D^2}}{c_l} \\ {t}_3=\frac{L_3}{c_l}=\frac{\sqrt{L^2+{\left(2D\right)}^2}}{c_l} \\ \dots \\ {t}_n=\frac{\sqrt{{\left(L^2+(\left(n-1\right)D\right)}^2}}{c_l} \end{gathered}$$

(27)

where $L$is the length between the bubble and the inspection surface. The results of the calculations of Equations (26) and (27) are summarized in

Table 2. Comparison between the simulation and analytical results.

Times [-]	$\mathit{T}$	${\mathit{t}}_1$	${\mathit{t}}_2$	${\mathit{t}}_3$
Simulation Results (Figure 4)	2.9	2.6	3.7	6.2
Results of Equations (26) and (27)	2.9	2.5	3.9	6.4

. The results in Equations (26) and (27) are consistent with the simulation results. However, the equations evaluate a longer time than that of the simulation results. The sound speed in the liquid is changed in the range from 4.8 to 5.0 by the pressure wave propagating from the laser-induced bubble. The pressure wave propagated faster than the speed of sound considered in the equations. In addition, the bubble radius is not considered in Equations (26) and (27). Therefore, the equations evaluated a longer time.

Finally, we summarize the structure of pressure field generated from the single laser-induced bubble in a microchannel. We can see two types of pressure wave propagation. The first type is propagation from the bubble to the meniscus, and the second type is reciprocating propagation around the bubble. These two types of pressure waves generate the structure of the pressure field.

3.3. Pressure Field Evolution around Multi-Bubbles

In this section, the results in the case of multiple bubbles are discussed. The MP-WENO5-JS reconstruction and w/o-meniscus setting are used as in Section 3.2.

Figure 10 shows a comparison of the time evolution of the pressure field among the single bubble, two bubbles and three bubbles. Figure 10a shows that a pressure wave overlap occurred more quickly with increase in the number of bubbles. When the pressure waves overlapped, a strong pressure area is generated. As shown in Figure 10b, a pressure area higher than p = 226 is generated by the overlap of two pressure waves in the case of the two bubbles, and a stronger pressure area over p = 455 is generated by the overlap of three pressure waves in the case of the three bubbles. However, in the case of the single bubble, the pressure waves did not overlap, so the maximum pressure in Figure 10c is about p = 161. In this manner, with the increase in the number of bubbles, multiple pressure waves overlap more quickly, and as a result of the quick pressure-wave overlap, stronger pressure waves arrive at the inspection surface more quickly.

Figure 11 shows the time history of the pressure at the probe. As shown in Figure 11, the pressure oscillation with period T can be seen even in the case of multiple bubbles. Note that period T differs depending on the number of bubbles.

Equations (26) and (27) can be extended to multiple bubbles. Figure 12 shows the pressure wave path to the probe in the case of multiple bubbles. First, the equation for the period $T$ (Equation (26)) is extended. In the case of the multiple bubbles, the round-trip distance of the pressure wave in Equation (26) is turned into the distance between the bubbles or twice the distance between the bubble and the wall. The propagation distance $L_p$ of the pressure wave can be calculated using the following equation:

$$L_p=\frac{D}{N}$$

(28)

and the period $T$ can be calculated using $L_p$ through the following equation:

$$T=\frac{L_p}{c_l}=\frac{D}{N{c}_l}.$$

(29)

Next, the equation for the delay time (Equation (27)) is extended. As in the case of the single bubble, the dashed line in Figure 12b is the path on which the pressure wave propagates from the bubble to the probe. The delay time $t_n$ can be calculated using the following equations:

$$\begin{gathered} t_1=\frac{\sqrt{L^2+s^2}}{c_l} \\ {t}_2=\frac{\sqrt{L^2+{\left(D-s\right)}^2}}{c_l} \\ {t}_3=\frac{\sqrt{L^2+{\left(D+s\right)}^2}}{c_l} \\ \dots \\ {t}_n=\left\{\begin{array}{c}\frac{\sqrt{L^2+s^2}}{c_l} \mathrm{If} n=1\\ \frac{\sqrt{L^2+{\left(\left(n-1\right)D-s\right)}^2}}{c_l} \mathrm{If} n \mathrm{is} \mathrm{an} \mathrm{even} \mathrm{number},\\ \frac{\sqrt{L^2+{\left(\left(n-2\right)D+s\right)}^2}}{c_l} \mathrm{If} n \mathrm{is} \mathrm{an} \mathrm{odd} \mathrm{number}\end{array},\right. \end{gathered}$$

(30)

where s is the distance from the channel center to the bubble. The results of the calculations of Equation (29) and (30) for multiple bubbles are shown in

Table 3. Comparison between the simulation and analytical results for the case of the two bubbles.

Times [-]	$\mathit{T}$	${\mathit{t}}_1$	${\mathit{t}}_2$	${\mathit{t}}_3$
Simulation Results (Figure 4)	1.4	2.6	3.2	4.3
Results of Equation (26) and Equation (27)	1.5	2.6	3.3	4.5

and

Table 4. Comparison between the simulation and analytical results for the case of the three bubbles.

Times [-]	$\mathit{T}$	${\mathit{t}}_1$	${\mathit{t}}_2$	${\mathit{t}}_3$
Simulation Results (Figure 4)	0.9	2.4	2.6	3.0
Results of Equation (26) and Equation (27)	1.0	2.5	2.6	3.1

. For the delay time, Equation (30) is calculated for each bubble, and the three least times are shown. There is a little discrepancy between the analytical and simulation results due to not considering the changing sound speed and the bubble radius. However, the error is less than 10%, and the analytical results are consistent with the simulation results. These results indicate that we can see two types of pressure wave propagation (propagating from the bubble to the meniscus and reciprocating around a bubble) even with multiple bubbles.

Table 5. Comparison of the maximum pressure values and maximum pressure arrival times.

Number of Bubbles	1	$2$	$3$
Maximum Pressure [-]	2.9	2.6	3.7
Time at the Maximum Pressure [-]	2.9	2.5	3.9

shows a comparison of the maximum pressure values and the arrival times of the maximum pressure peak at the probe. The maximum pressure increases with the increase in the number of bubbles. Moreover, the maximum pressure arrival time becomes faster with the increase in the number of bubbles. This result is caused by the larger number of overlapping pressure waves when the number of bubbles increases, as confirmed in Figure 10. In Figure 6, we can see that the impulse increases with increasing bubbles in the initial stage. The increased impulse with a larger number of bubbles at the initial stage is concluded to be due to the pressure wave overlap effect.

3.4. Discussion

All figures and tables should be cited in the main text as Figure 1, Table 1, etc.

The force applied to the surrounding fluid by the expansion of the bubbles is expected to turn into an impulse. In this section, based on the time history of the total bubble volume, the discussion focuses on the reason why the impulse increased with the decrease in the number of bubbles (Figure 13).

Figure 13 shows that the case with the three bubbles had the largest total volume at $t\in \left[0.0, 1.2\right]$ and that the case with the two bubbles then had the highest total volume at $t\in \left[1.2, 2.4\right]$. However, the volume in the case with the single bubble became the largest eventually. The time history of the bubble volume shows the similar trend to the impulse-time history shown in Figure 6. Figure 13 b shows a zoom-up view of the timing with the change in the expansion gradient of the bubble. The timing of the change in the bubble expansion gradient coincides with the timing at which the bubble first interacted with the compression pressure wave, which inhibited the expansion of the bubble.

Table 6. Times when the pressure waves first contacted the bubble surfaces.

Number of Bubbles	1	2	3
Time [-]	2.5	1.0	0.6

summarizes the times when the bubble first contacted the compression pressure wave. In the setup of this study, the interval between the bubbles become shorter with the increase in the number of bubbles. Therefore, as the number of bubbles increases, the interference between the pressure waves and bubbles took place more quickly, which resulted in a smaller bubble expansion, i.e., the impulse decreased.

Tagawa et al. conducted experiments varying the objective lens magnification used to focus the laser with constant laser pulse energy and confirmed that the number of bubbles generated initially reduces with higher lens magnification, which leads to higher pressure impulse [45]. The results of this experiment can be interpreted as the generation of a larger impulse with smaller number of bubbles, and it is consistent with the results obtained in this study. This study clearly shows that the larger pressure impulse with the smaller number of bubbles can be explained to be due to the inhibition of the bubble expansion in the larger number of bubbles, as shown in Figure 13.

4. Conclusions

In this study, the propagation of the pressure waves generated by multiple laser-induced bubbles inside a microchannel filled with liquid was numerically simulated using the 5-equation model. First, we investigated a proportional relationship between impulse and microjet velocity. The result showed that the final pressure impulse and microjet velocity decreased with increasing the number of bubbles. It was confirmed that the pressure impulse has a strong correlation with the jet velocity; therefore, it was concluded that the effect of bubble conditions on the microjet velocity could be addressed by investigating the structure of the pressure field even without a meniscus. Next, we investigated the time evolution of the pressure field and considered the path of the pressure wave in the case with a single bubble. The results showed that the propagation of the pressure waves has two types: from a bubble to a meniscus and reciprocating around a bubble. These two types of propagation were observed also in the case of multiple bubbles. Furthermore, by focusing on the pressure peak observed at the probe position, the maximum pressure increased with the increase in the number of bubbles, and the time of reaching the peak pressure decreased. Finally, the time history of bubble volume was investigated. The reason for the decrease in the impulse with the increase in the number of bubbles can be explained by the expansion history of the bubbles. The increase in the number of bubbles inhibited the early expansion of the bubbles and consequently led to the impulse reduction. It was suggested that the interaction between the bubbles and the pressure wave caused by the placement of multiple bubbles negatively affected the impulse at the inspection surface.