1Graduate School of Science and Technology, Hirosaki University 2Graduate School of Technology, Tokyo University of Agriculture and Technology

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1 (Minori Shirota) 2(Masaham Kameda) 1Graduate School of Science and Technology Hirosaki University 2Graduate School of Technology Tokyo University of Agriculture and Technology 1 ( ) 2 2 $[1][2]$ Batchelor 2 [2] $M \frac{du_{1}}{dt}=-\frac{1}{2}\frac{d}{dt}[m_{0}(u_{1}-u_{2})]+m_{0}\frac{du_{2}}{dt}$ (1) $M$ $M_{0}$ $U_{1}$ U2 Bjerknes $(U\cdot\nabla U=0)$ $du/dt=-\nabla p/\rho$ $M_{0} \frac{du_{2}}{dt}\approx-\nu\nabla p$ (2) $p$ $n$ 2 $0$ 2 Bjerknes - $\nabla$p $>0$ (1) ( ) p $<$

$40 2 Bjerknes 2 Bjerknes $\omega$ $\nu$ $\delta$ $\delta=(2v/\omega)^{1/2}$ $\delta$ R $\delta$ 2 Magnaudet&Legendre[7] (dr/dt) 1$

2 40 2 Bjerknes 2 Bjerknes $\omega$ $\nu$ $\delta$ $\delta=(2v/\omega)^{1/2}$ $\delta$ R $\delta$ 2 Magnaudet&Legendre[7] (dr/dt) 1 $(=2R U_{1}- U2 /v)$ 1 Levich $C_{D}= \frac{48}{re}$ (3) Levich Magnaudet&Legendre Levich $[3\cdot 6]$ [8][9] $\delta$ Magnaudet&Legendre

3 41 Fig 1 Schematic diagram of a levitation cell and camera configuration for image capturing 2 Fig 1 [10] 40 mm 130 mm ( ) (Kulite XCQ A) [11] 2 ( ) ( HPV-I) (1)$345kHz$ $100kPa$ (2) $192kHz$ $25kPa$ ffames $/s$ ( 01 mm ) 50 $mm^{2}/s$ ( KF96-50) 3 $i$ 2 $(i=12)$ Fig 2 (4) (5)

4 $\ddot{r}$ 42 Fig2 Coordinate system used in the present analysis $(1- \frac{\dot{r}_{i}}{c_{l}})r_{j}\ddot{r}_{i}+\frac{3}{2}(1-\frac{\dot{r}_{i}}{3c_{l}})\dot{r}_{i}^{2}$ (4) $= \frac{1}{\rho_{l}}(1+\frac{\dot{r}_{i}}{c_{l}}+\frac{r_{j}d}{c_{l}dt})(p_{g}-\frac{2\sigma}{r_{i}}-\frac{4\mu_{l}\dot{r}_{i}}{r_{i}}p_{l\infty})-\frac{d}{dt}(\frac{r_{3-i}^{2}\dot{r}_{3-i}}{d})$ $\frac{d}{dt}(\frac{1}{2}\rho_{l}v_{j}v_{\dot{n}})=\rho_{l}v_{i}\frac{d}{dt}(\frac{r_{3\dashv}^{2}\dot{r}_{3-i}}{d^{2}})-$ (5) $\dot{r}$ $V=4\pi R^{3}/3$ $D$ 2 $c$ $\sigma$ $\mu$ $l1$ $p_{g}$ pl$\infty$ $v_{j}$ v $v_{ri}=\{\begin{array}{ll}v_{1}+(r_{2}^{2}/d^{2})\dot{r}_{2} (i=1)-v_{2}+(r_{1}^{2}/d^{2})\dot{r}_{1} (i=2)\end{array}$ (6) 2 O((&D)3) (5) $F_{V}$ Magnaudet&Legendre[7] Re $(=2 v_{r} R/\nu)$ 1

5 pl$\infty$ 43 Re $(=2R \dot{r} /\nu_{l})$ 1 $+$ Levich $F_{VL}$ $F_{VL}=( \frac{48}{re})\ovalbox{\tt\small REJECT}_{i}^{2}\frac{1}{2}\rho_{l}v_{ri} v_{\dot{n}} $ (7) $Re$ $Re$ 1 $F_{VM}$ $F_{VM}=( \frac{16}{re})nr_{i}^{2}\frac{1}{2}\rho_{l}v_{\dot{n}} v_{\dot{n}} +8\pi\mu_{l}J\{w(\taut)\frac{d}{dt}[R_{i}(\tau)v_{ri}(\tau)]\}d\tau$ (8) $w$ $v\langle\taut)=\exp[9\nu_{l}\iota\frac{dt }{R_{i}(t )^{2}}]erfc[\sqrt{9\nu_{l}\downarrow\frac{dt }{R_{i}(t^{1})^{2}}}]$ (9) 4 Magnaudet&Legendre[7] (4) (5) 2 (4) p8 $\not\in$ [10] Fig 3 Fig 3(c) (4) Fig 3(c) Fig 3(d) 2 (exp) (F $\omega$ (Fr$M$) Fig 3 Fig 3(e) Re 100 Re Re Magnaudet&Legendre ig 3 $Re$ 1 $F$ $Re$

6 44 Fig 3 Attractive motion of a pair of bubbles under strong acoustic forcing (a)selected frames of images capmred by using 0 $fx0152\bullet 0\{ur0\{u\bullet 01W\bullet 0212\bullet 022l\bullet\bullet \Re l\bullet u\bullet 0\infty 00312\bullet\bullet 032l0y \bullet\bullet 0a\infty 0372r$ high-speed video camera with elapsed time below each image in ms (b) Liquid pressure measured in experiment and used in the calculation Evolution of (c) radius of upper bubble (d) separate distance $D$ between two bubbles and (e) actual Reynolds number $Re$ and Reynolds number based on the radial motion $Re$ O(10) Fig 4 $Re$ 50 Fig 4(b) (FrM) Fig 4 Fig 3

$$\overline{s\subset\in=}6080$ $\hat{\phi}$ $\dot{s_{\infty 0}^{\infty 0}\in c=}4\infty$ $\circ>\cdot$ 100 45 Fig 5 Time evolution of separate distance between two bubbles obtained by numerical$

7 $\overline{s\subset\in=}6080$ $\hat{\phi}$ $\dot{s_{\infty 0}^{\infty 0}\in c=}4\infty$ $\circ>\cdot$ Fig 5 Time evolution of separate distance between two bubbles obtained by numerical calculation: $(a) 50mm^{2}/s$ $P_{a}=130kPa$ $(b)v=5mm^{2}/s$ $P_{a}=50kPa$ $(c)v=5mm^{2}/s$ $P_{a}=5kPa$ (a) (b) $\langle c)$ $-R-\cdot R$ $\dot{s^{100}\in}\epsilon 0=-Ro-\cdot R\cdot$ $c60$ $\Omega 08w4020$ $\frac{b}{g}\omega $ $\propto 0$ $0$ oe $\Phi $0$ $0$ of $fo^{6}\prime dng8$ 10 $0$ 2 $Pe\dot{n}^{4}odofb^{6}Mng8$ 10 2 $p_{e\dot{\mathfrak{n}}odofforcing}^{46}8$ 10 2 $Pe\dot{n}^{4}od$ $\frac{b}{\circ}w40$ 520$ $0$ Fig 6 Reynolds numbers for bubble motions in each case in Fig 5 (a) (b) and (c) Reddy&Szeri ] $Re$ O(l) Reddy&Szeri Reddy&Szeri 1 5 $mm^{2}/s$ lpm Reddy&Szeri 2 ( pa) pl$\infty$ RO (a) (b) (c) 1 2 mm Z)0$=10R_{0}$ $=345kHz$ $R_{0}=009$ Fig 5 Fig 5(a) -50 $mm^{2}/sp_{a}=130kpa$ Fig 3 Fig 5(b) (c) Reddy&Szeri 5 $mm^{2}/s$ (b) $p_{a}=50kpa$ (c) $p_{a}=5kpa$ Fig 5(a) (b) (C) 5 $mm^{2}/s$ (b) Re 500 (Fig 6) (a) (a) Re (C) 5 $mm^{2}/s$ Reddy&Szeri

8 46 $\delta$ $=(2v/0))^{1/2}$ $\delta$/r ( ) 50 $mm^{2}/s$ $f=345khz$ $R_{0}=01$ $\delta/r_{0}\approx 02$ mm $v=5mm^{2}/s$ $\delta$/r$0\approx 007$ $\delta$/r 5 50 $mm^{2}/s$ [7] (1) $O(10)$ (2) Reddy&Szeri ( ) 6 R $\delta/r$ [1] Leighton T G The Acoustic Bubble Academic Press (1994) [2] Batchelor G K An Introduction to Fluid Dynamics Cambridge University Press (1967) [3] Cmm L A Bjerknes forces on bubbles in a stationary sound field J Acoust Soc Am (1975) [4] Barbat T Ashgriz N and Liu C S Dynamics of two interacting bubbles in an acoustic field J Fluid Mech (1999) [5] Harkin A Kaper T J and Nadim A Coupled pulsation and translation of two gas bubbles in a liquid J Fluid Mech (2001) [6] Oguz H N and Prosperetti A A generalization of the impulse and virial theorems with an application to bubble oscillations J Fluid Mech (1990) [7] Magnaudet J and Legendre D The viscous drag force on a spherical bubble with a time-dependent radius Phys Fluids (1998)

9 47 [8] Reddy G A and Szeri A J Coupled dynamics of translation and collapse of acoustically driven microbubbles J Acoust Soc Am (2002) [9] Krefting D Toilliez J O Szeri A J Mettin R and Lauterbom W Translation ofbubbles subjected to weak acoustic forcing and error in decoupling from volume oscillations J Acoust Soc Am 12067(b675(2006) [10] Kameda M and Matsumoto Y Nonlinear oscillation of a spherical gas bubble in acoustic fields J Acoust Soc Am (1999) [11] Shirota M and Kameda M A bubble generator to produce 01 mm-diameter gas bubbles Proceedings ofthe 4th International Conference on Multiphase Flow Paper No 451 (2001)

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