Test-case number 29a: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part I : in a stagnant liquid Hien Ha-Ngoc, Institute of Mechanics, 264 Doi Can, Hanoi, Vietnam. E-Mail: [email protected] Jean Fabre, Institut de M´ecanique des Fluides de Toulouse, All´ee du Pr. Camille Soula, 31400 Toulouse, France. E-Mail: [email protected]

1

Practical significance and interest of the test-case

The motion of a long bubble in a channel or in a tube has a practical interest for modeling slug flow that occurs in various industrial applications. This motion is a result of two mechanisms (i) the action of gravity, (ii) transport by the liquid movement. In fully-developed flow, the motion of a long bubble is uniform and independent on its length. This means that the bubble moves with a constant velocity that does not depend on the types of wakes that are formed behind the bubble: they are slave of the bubble motion. The influence of the liquid movement can be therefore characterized by a velocity profile imposed far upstream from the bubble head. We provide in two articles (part I and part II) analytical and numerical solutions for the velocity and shape of 2D long bubbles (plane and axis-symmetrical) in stagnant liquid (part I) and in flowing liquid (part II). The solutions are considered for the case when viscous effects on the bubble motion are negligible i. e. when the motion is inertial. In this regime the influence of surface tension will be discussed. According to White & Beardmore (1962) and Wallis (1969), the effect of viscosity is negligible when the inverse viscosity number N f =(gD 3 )1/2 /ν is greater than 300 or according to Zukoski (1966) when the bubble Reynolds number Re=VD/ ν is greater than 100. Neglecting the viscosity is justified by observing that the growth rate of the boundary layer created at the wall and at the bubble surface is slow. Moreover the acceleration of the liquid around the bubble nose prevents the boundary layer to grow. These results suggest that in the case of large enough Reynolds number potential flow theory can be applied. Such an inertia-controlled regime is often met in the industrial applications (H´eraud, 2002). In a moving frame attached to the bubble, the flow is steady. For an inviscid fluid and a two-dimensional flow, the liquid motion is described by a Poisson equation for the stream function with a source term depending on the vorticity. This vorticity is determined by the velocity profile given ahead of the bubble and remains constant along a streamline. As a result, a rotational flow can be described by only one equation for the stream function (Batchelor, 1967, pp. 507-593)(Lamb, 1932, pp. 244-245). This equation can be solved with corresponding boundary conditions. The bubble shape is determined from an equation obtained by combining the Bernoulli equation and the jump conditions at the interface. An algorithm based on a boundary element method (BEM) solving simultaneously the Poisson equation and the equation at the interface has been proposed by Ha-Ngoc (2003). It allows determining accurately the flow characteristics and the bubble shape. The results have been compared to different theoretical, experimental and numerical results (Zukoski, 1966, Collins et al. , 1978, Vanden-Bro¨eck, 1984a,b, Bendiksen, 1984, Cou¨et & Strumulo, 1987).

2/15

2

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Definitions and model description

We consider the motion of a long gas bubble in a two-dimensional channel or in a vertical tube filled-up with a liquid. The gas density is relatively small compared to that of the liquid so that the gas motion can be neglected and the pressure in the bubble, considered as constant. Indeed the pressure difference in the gas due to its motion should be of the order of magnitude of ρG V 2 /2, considerably less than that in the liquid. The reference calculations to be proposed have been carried out in dimensionless form. The reference scales are: for the length, the half-width, a, of a channel or the radius R of a tube, for the velocity, (gD)1/2 , where D=2a. In stagnant liquid, the flow may be characterized by two dimensionless parameters: • The bubble Reynolds number, Re: VD . ν

(1)

ρgD2 , σ

(2)

Re = • The E¨ otv¨os number, Eo: Eo =

where V is the bubble velocity, ν, the kinematic viscosity of the liquid, ρ, its density, σ, the surface tension and g, the gravity. The dimensionless bubble velocity is represented by the bubble Froude number, Fr, defined as : V (3) Fr = √ . gD Experiences have shown that there exists a critical bubble Reynolds number Re ∗ over which the bubble velocity does not depend on the viscosity. Zukoski (1966) has shown that for a vertical tube Re ∗ is about 100. In this regime, only inertia, gravity and possibly surface tension balance. Therefore, the liquid can be considered as inviscid and the bubble motion is characterized by only one dimensionless parameter, the E¨ otv¨os number, Eo. It can considered that the solutions of present study corresponds to: Re > 100.

(4)

For an inclined channel, two cases must be distinguished: (i) a case when the bubble touches the upper wall and (ii) a case when the bubble has no contact with a wall. For the case (i), a relation between the bubble velocity and the contact angle θ0 of the liquid with the wall should be given to complete the problem formulation. A sensitivity analysis on the influence of the contact angle on the bubble velocity can be done. In fact, the contact angle may vary between 0 and π, but for small surface tension values, Eo>100, it can be shown that the bubble velocity is not sensitive to θ0 . In further calculations the contact angle θ0 = π/2 will be always used. For the case (ii), the interface is a smooth line and there exists a stagnation point at the interface in the moving coordinates. The condition of smoothness at this point can be used for determining the bubble velocity. For vertical flow, when the symmetry is imposed, the problem can be treated as the case (i) with given interface slope angle at symmetry axis θ0 = π/2. It can be shown that there exist multiple solutions for this problem and the criterion of maximum velocity suggested by Garabedian (1957) is used for selecting the physically observable solution.

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

3

3/15

Motion in horizontal channel

Benjamin (1968) obtained an exact analytical solution for the drift velocity and the bubble shape for the horizontal case with negligible surface tension (σ=0 i. e. Eo → ∞). For a bubble in a channel, the velocity is given by: p V = 0.5 gD (5) The coordinates of bubble shape are given in Table 1 (see Appendix) and the shape is presented in Figure 1 in comparison with a numerical solution by Ha-Ngoc (2003).

Figure 1: Shape of a long bubble moving in a horizontal channel at Eo → ∞ (numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

The numerical solution is obtained by an iterative procedure. The calculation begins with initial bubble shape and bubble velocity V 0 . In the (x, y) coordinates, where the x -axis coincides with the axis of symmetry and the y-axis passes by the stagnation point (0, -1), the following initial bubble shape x=X (y) is used:   y+1 2 X(y) = −1 − A ln (1 − ( ) ) , 0 < d < 2, −1 6 y 6 d (6) d where A and d are the arbitrary parameters. For the numerical solution, the conditions at infinity upstream the bubble and in the receding film are taken on lines perpendicular to the channel walls. The distances of these lines to the stagnation point have been chosen so that they do not affect the solution. The interface is discretized and approximated by a polygon. A non-uniform discretization was used with the mesh size changing from ds min near the stagnation point to ds max far downstream. The sensitivity analysis of the bubble length, l, and the discretization for Eo=40÷4000 has shown that the converged solution could be obtained with ds min 660.025 and l >5. The solution becomes more sensitive to the mesh size at small Eo. The numerical results presented here have been carried out with ds min =0.0125, ds max =0.2 and l =8. For fixed Eo, different converged solutions can be obtained from different initial conditions (bubble shape and bubble velocity V 0 ). For each Eo, there exists a solution whose velocity is the greatest: it is considered as the physical solution. This solution is not sensitive to the initial conditions. For example, for Eo=100, the solution with the greatest velocity, V = 0.44, can be obtained for a rather large range of the parameters A, d and V 0 : {A=1.5, d =1.2, V 0 =0.5}, {A=1.5, d =1.0, V 0 =0.7}, {A=2.5, d =1.5, V 0 =0.30} etc.

4/15

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

The results of bubble velocity are presented in Figure 2 for E¨ otv¨os number in the range [10, 1000]. The solution calculated by utilizing the boundary element method is in good agreement with the results of Cou¨et & Strumulo (1987) who used a method based on the conformal mapping technique. The calculated bubble shapes are presented in Figure 3 and in Table 2 (Appendix) for different E¨otv¨ os numbers (Eo=10, 100, 1000).

Figure 2: Dimensionless bubble velocity versus E¨ otv¨ os number for horizontal channel (numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

Figure 3: Horizontal bubble shapes for different E¨ otv¨ os number (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

4

Motion in inclined channel

According to the experimental results of Maneri & Zuber (1974), three different shape regimes may be observed when the inclination angle varies: • The first regime corresponds to inclination angles from α =0◦ (horizontal position) to about 60◦ . The bubble touches the upper wall and the liquid drains in the film

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

5/15

under the bubble. • In the second regime the bubble is vertical or slightly deviated from the vertical (80◦ 6α690◦ ). The liquid drains in two films around the bubble and the bubble velocity is nearly as in vertical channel. • For inclination between 60◦ and 80◦ , there is a change in both velocity and shape. This is a transition region that is generally unknown and may be characterized by the instability of the liquid film. It must be pointed out that the simulation converges well for 60◦ 6α680◦ , but it gives two different solutions according to the case under investigation: (i) or (ii). None of these two solutions seems to be a physical one. We provide here the numerical solutions for the first flow regime i.e. for 0◦ 6α660◦ . For the simulation, the initial bubble shape of the form (6) and interface discretization with ds min =0.0125, ds max =0.2 and l =8 have been used. The bubble velocity is presented in Figure 4 as a function of inclination for different surface tensions (Eo=10, 100, 1000). It should be noted that the numerical results are comparable with the experimental results of Maneri & Zuber (1974). The velocity increases with inclination from the horizontal position, reaches its maximum value at about α=30◦ ÷45◦ and then decreases.

Figure 4: Dimensionless bubble velocity: influence of channel inclination (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

The calculated bubble shapes are presented in Figure 5 and in Table 3 for different inclination angle (α=0o , 15o , 45o ) at Eo=100.

5

Motion in vertical channel and in tube

The vertical case has been the most extensively investigated, theoretically, experimentally and numerically. For the purely inertial regime, the dimensionless bubble velocity is about 0.345 for an axis-symmetrical bubble and 0.23 for a plane bubble (Dumitrescu, 1943, Davies & Taylor, 1950, Vanden-Bro¨eck, 1984b, Collins, 1965, Zukoski, 1966, Bendiksen,

6/15

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Figure 5: Bubble shapes for different inclinations at Eo=100 (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8 ).

1984, 1985, Mao & Dukler, 1990, 1991). The influence of surface tension was studied by Vanden-Bro¨eck, Cou¨et and Strumolo for plane case and by Zukoski and Bendiksen for axis-symmetrical case. Vanden-Bro¨eck (1984b) has show that the case of zero surface tension is a mathematically degenerate case. The solution for this case can be obtained by letting σ→0 but not by setting σ=0. By studying the multiple solution problem, Cou¨et and Strumolo have noted that the solutions could densely cover the whole segment [0, 0.23]. For the boundary element method used here, the solution converges well for Eo4. For the axis-symmetrical case, it requires also a domain discretization to take the surface integration on the flow domain. A simple triangulation procedure has been used (Ha-Ngoc, 2003). The bubble velocity is presented in the Figure 6 for plane bubbles and in Figure 7 for axis-symmetrical bubbles as a function of surface tension for Eo∈[5, 4000]. Some examples of bubble shapes are presented in the Figures 8 and 9. Bubble shape coordinates are given in Tables 4 and 5 (Appendix). The accuracy of the simulation can be appreciated by the fact that the bubble velocity and shape are in very good agreement with the experimental results of Davies & Taylor (1950), Zukoski (1966) and numerical results of Vanden-Bro¨eck (1984b), Mao & Dukler (1990) for small surface tension cases (Ha-Ngoc, 2003).

6

Acknowledgements

This work was partly sponsored by the CNRS France in the frame of a Program of Scientific Cooperation with NCST Vietnam. The authors wish to thank Dr H. Lemonnier for his helpful comments and suggestions.

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

7/15

Figure 6: Velocity of a plane bubble in vertical channel in function of surface tension (numerical results by BEM with ds min =0.025, ds max =0.1 and l =4).

Figure 7: Velocity of an axis-symmetrical bubble in vertical tube in function of surface tension (numerical results by BEM with ds min =0.05, ds max =0.1 and l =4).

8/15

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Figure 8: Bubble shapes in vertical flow for different surface tensions: plane bubble (present numerical results by BEM with ds min =0.025, ds max =0.1 and l =4).

Figure 9: Bubble shapes in vertical flow for different surface tensions: axis-symmetrical bubble (present numerical results by BEM with ds min =0.05, ds max =0.1 and l =4).

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

7

9/15

Appendix

Theoretical solution of Benjamin (1968) x/a y/a 0.0136 -0.978 0.0582 -0.9162 0.1362 -0.8264 0.2428 -0.7258 0.3662 -0.6296 0.4948 -0.5454 0.6202 -0.4754 0.7386 -0.4182 0.8488 -0.3714 0.9504 -0.3330 1.1310 -0.2744 1.3562 -0.2158 1.6480 -0.1584 1.8732 -0.1248 2.5632 -0.0602 4.1254 -0.0116 4.7906 -0.0058 -

Numerical solution of Ha-Ngoc (2003) x/a 0 0.0021 0.0068 0.0125 0.0186 0.0259 0.0342 0.0438 0.0548 0.0673 0.0815 0.0977 0.1160 0.1368 0.1604 0.1872 0.2174 0.2517 0.2905 0.3344 0.3841

y/a -1 -0.9877 -0.9761 -0.965 -0.9541 -0.9423 -0.9296 -0.9158 -0.9008 -0.8845 -0.8669 -0.8479 -0.8272 -0.805 -0.7810 -0.7552 -0.7276 -0.6981 -0.6666 -0.6331 -0.5978

x/a 0.4402 0.5035 0.5749 0.6554 0.7459 0.8477 0.9619 1.0897 1.2326 1.3920 1.5695 1.7667 1.9636 2.1611 2.3590 2.5571 2.7555 2.9539 3.1524 3.3509 3.5495

y/a -0.5607 -0.5219 -0.4817 -0.4403 -0.3980 -0.3554 -0.3129 -0.2712 -0.2309 -0.1926 -0.1571 -0.1248 -0.0989 -0.0780 -0.0611 -0.0477 -0.0369 -0.0284 -0.0214 -0.0161 -0.0117

x/a 3.7481 3.9467 4.1453 4.3439 4.5425 4.7411 4.9397 5.1383 5.3369 5.5355 5.7341 5.9327 6.1313 6.3300 6.5286 6.7272 6.9258 7.1244 7.3230 7.5216 7.7202

y/a -0.0083 -0.0056 -0.0034 -0.0018 -0.0004 0.0006 0.0015 0.0021 0.0027 0.0031 0.0034 0.0036 0.0038 0.0040 0.0039 0.0041 0.0040 0.0040 0.0038 0.0035 0.0039

Table 1: Interface coordinates of a long bubble moving in a horizontal channel at Eo → ∞ (numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

10/15

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Eo = 10 x/a y/a 0 -1 0.0001 -0.9875 0.0006 -0.9750 0.0013 -0.9625 0.0023 -0.9501 0.0037 -0.9363 0.0057 -0.9212 0.0083 -0.9046 0.0117 -0.8864 0.0161 -0.8663 0.0218 -0.8444 0.0289 -0.8204 0.0378 -0.7943 0.0488 -0.7659 0.0623 -0.7351 0.0788 -0.7019 0.0989 -0.6661 0.1231 -0.6278 0.1521 -0.5872 0.1867 -0.5441 0.2276 -0.4990 0.2758 -0.4521 0.3321 -0.4038 0.3976 -0.3545 0.4733 -0.3049 0.5603 -0.2556 0.6595 -0.2073 0.7720 -0.1607 0.8991 -0.1166 1.0418 -0.0755 1.2013 -0.0380 1.3791 -0.0042 1.5767 0.0257 1.7739 0.0494 1.9716 0.0686 2.1696 0.0843 2.3677 0.0974 2.5660 0.1085 2.7644 0.1181 2.9628 0.1266 3.1613 0.1343 3.3598 0.1414 3.5583 0.1482 3.7568 0.1547 3.9553 0.1611 4.1538 0.1674 4.3523 0.1736 4.5508 0.1798 4.7494 0.1860 4.9479 0.1920 5.1464 0.1979 5.3449 0.2034 5.5435 0.2085 5.7420 0.2129 5.9406 0.2167 6.1392 0.2195

Eo = 100 x/a y/a 0 -1 0.0003 -0.9875 0.0011 -0.9750 0.0025 -0.9626 0.0044 -0.9502 0.0069 -0.9367 0.0103 -0.9218 0.0147 -0.9056 0.0202 -0.8879 0.0271 -0.8686 0.0356 -0.8476 0.0460 -0.8248 0.0585 -0.8002 0.0734 -0.7737 0.0912 -0.7451 0.1122 -0.7145 0.1368 -0.6817 0.1656 -0.6468 0.1992 -0.6097 0.2381 -0.5706 0.2830 -0.5295 0.3347 -0.4865 0.3942 -0.4420 0.4623 -0.3964 0.5400 -0.3499 0.6282 -0.3030 0.7281 -0.2562 0.8411 -0.2106 0.9681 -0.1666 1.1106 -0.1247 1.2699 -0.0862 1.4475 -0.0513 1.6449 -0.0203 1.8421 0.0033 2.0398 0.0221 2.2379 0.0367 2.4362 0.0477 2.6346 0.0566 2.8331 0.0633 3.0317 0.0683 3.2302 0.0726 3.4288 0.0755 3.6274 0.0778 3.8260 0.0798 4.0246 0.0810 4.2232 0.0822 4.4219 0.0831 4.6205 0.0835 4.8191 0.0842 5.0177 0.0844 5.2163 0.0846 5.4149 0.0851 5.6135 0.0850 5.8121 0.0852 6.0107 0.0854 6.2094 0.0851

Eo = 1000 x/a y/a 0 -1 0.0007 -0.9875 0.0026 -0.9752 0.0055 -0.9630 0.0091 -0.9510 0.0137 -0.9380 0.0196 -0.9240 0.0267 -0.9087 0.0352 -0.8922 0.0453 -0.8744 0.0572 -0.8551 0.0711 -0.8344 0.0873 -0.8120 0.1060 -0.7879 0.1276 -0.7621 0.1524 -0.7345 0.1808 -0.7050 0.2133 -0.6735 0.2505 -0.6401 0.2929 -0.6048 0.3412 -0.5676 0.3960 -0.5287 0.4583 -0.4882 0.5288 -0.4464 0.6085 -0.4035 0.6985 -0.3600 0.7998 -0.3165 0.9137 -0.2732 1.0415 -0.2312 1.1843 -0.1907 1.3438 -0.1528 1.5214 -0.1178 1.7187 -0.0864 1.9157 -0.0615 2.1133 -0.0415 2.3113 -0.0258 2.5095 -0.0131 2.7079 -0.0033 2.9064 0.0045 3.1049 0.0108 3.3034 0.0155 3.5020 0.0194 3.7006 0.0223 3.8992 0.0248 4.0978 0.0265 4.2964 0.0280 4.4950 0.0292 4.6936 0.0300 4.8922 0.0308 5.0908 0.0312 5.2894 0.0317 5.4881 0.0319 5.6867 0.0323 5.8853 0.0324 6.0839 0.0325 6.2825 0.0327

Table 2: Interface coordinates of long bubbles in horizontal channel for different E¨ otv¨ os numbers (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

x/a 0 0.0003 0.0011 0.0023 0.0041 0.0064 0.0096 0.0137 0.0189 0.0253 0.0332 0.0429 0.0545 0.0684 0.0850 0.1046 0.1276 0.1546 0.1860 0.2227 0.2651 0.3141 0.3705 0.4353 0.5095 0.5942 0.6905 0.7998 0.9234 1.0626

α = 15˚ y/a x/a -1 1.2188 -0.9875 1.3937 -0.9750 1.5887 -0.9626 1.7840 -0.9502 1.9802 -0.9366 2.1771 -0.9217 2.3744 -0.9054 2.5721 -0.8876 2.7699 -0.8681 2.9679 -0.8469 3.1660 -0.8238 3.3642 -0.7988 3.5624 -0.7717 3.7607 -0.7424 3.9591 -0.7109 4.1574 -0.6769 4.3558 -0.6406 4.5542 -0.6018 4.7527 -0.5605 4.9511 -0.5168 5.1496 -0.4707 5.3481 -0.4224 5.5466 -0.3722 5.7450 -0.3204 5.9436 -0.2673 6.1421 -0.2136 6.3406 -0.1598 6.5391 -0.1068 6.7377 -0.0550 6.9362

y/a -0.0055 0.0413 0.0848 0.1210 0.1517 0.1777 0.2003 0.2201 0.2374 0.2532 0.2672 0.2802 0.2922 0.3033 0.3138 0.3236 0.3329 0.3417 0.3501 0.3582 0.3658 0.3732 0.3802 0.3870 0.3935 0.3997 0.4057 0.4113 0.4166 0.4214

x/a 0 0.0002 0.0009 0.0020 0.0034 0.0054 0.0081 0.0116 0.0160 0.0216 0.0285 0.0369 0.0470 0.0593 0.0740 0.0914 0.1120 0.1364 0.1649 0.1983 0.2373 0.2826 0.3350 0.3958 0.4659 0.5464 0.6386 0.7441 0.8639 0.9999

11/15

α = 45˚ y/a x/a -1 1.1533 -0.9875 1.3258 -0.9750 1.5190 -0.9626 1.7129 -0.9502 1.9081 -0.9365 2.1043 -0.9215 2.3011 -0.9050 2.4983 -0.8870 2.6958 -0.8673 2.8936 -0.8457 3.0915 -0.8222 3.2896 -0.7965 3.4877 -0.7686 3.6859 -0.7383 3.8841 -0.7056 4.0824 -0.6701 4.2808 -0.6320 4.4791 -0.5909 4.6775 -0.5470 4.8760 -0.5002 5.0744 -0.4505 5.2729 -0.3979 5.4713 -0.3429 5.6698 -0.2856 5.8683 -0.2264 6.0668 -0.1659 6.2653 -0.1049 6.4639 -0.0439 6.6624 0.0157 6.8609

y/a 0.0735 0.1284 0.1795 0.2225 0.2587 0.2898 0.3166 0.3399 0.3607 0.3790 0.3957 0.4108 0.4247 0.4375 0.4493 0.4605 0.4708 0.4806 0.4897 0.4985 0.5067 0.5145 0.5220 0.5290 0.5359 0.5422 0.5485 0.5543 0.5599 0.5650

Table 3: Interface coordinates of long bubbles in inclined channel for different inclination angles at Eo=100 (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

12/15

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Eo = 10 x/a y/a 0 0 0.0005 0.0250 0.0020 0.0499 0.0045 0.0748 0.0080 0.0996 0.0129 0.1261 0.0193 0.1543 0.0278 0.1843 0.0384 0.2162 0.0517 0.2498 0.0681 0.2851 0.0880 0.3220 0.1119 0.3603 0.1402 0.3997 0.1736 0.4399 0.2125 0.4806 0.2575 0.5212 0.3090 0.5613 0.3674 0.6003 0.4332 0.6376 0.5066 0.6728 0.5879 0.7055 0.6774 0.7353 0.7748 0.7619 0.8733 0.7841 0.9726 0.8025 1.0723 0.8181 1.1724 0.8313 1.2727 0.8427 1.3732 0.8525 1.4738 0.8609 1.5745 0.8681 1.6753 0.8741 1.7762 0.8790 1.8770 0.8833 1.9779 0.8874 2.0788 0.8915 2.1797 0.8955 2.2806 0.8992 2.3815 0.9021 2.4824 0.9043 2.5834 0.9063 2.6843 0.9086 2.7852 0.9113 2.8862 0.9138 2.9871 0.9154 3.0881 0.9166 3.1890 0.9181 3.2899 0.9202 3.3909 0.9221 3.4919 0.9231 3.5928 0.9236

Eo = 100 x/a y/a 0 0 0.0005 0.0250 0.0020 0.0500 0.0044 0.0748 0.0079 0.0996 0.0127 0.1261 0.0191 0.1543 0.0274 0.1844 0.0380 0.2163 0.0513 0.2499 0.0677 0.2852 0.0876 0.3221 0.1117 0.3602 0.1404 0.3994 0.1744 0.4391 0.2141 0.4790 0.2602 0.5183 0.3131 0.5565 0.3731 0.5931 0.4405 0.6273 0.5156 0.6588 0.5986 0.6869 0.6896 0.7118 0.7882 0.7336 0.8875 0.7514 0.9875 0.7659 1.0876 0.7784 1.1881 0.7889 1.2886 0.7979 1.3893 0.8061 1.4900 0.8131 1.5907 0.8194 1.6915 0.8253 1.7924 0.8304 1.8932 0.8352 1.9941 0.8396 2.0950 0.8436 2.1958 0.8475 2.2967 0.8509 2.3977 0.8542 2.4986 0.8573 2.5995 0.8601 2.7004 0.8629 2.8013 0.8654 2.9023 0.8678 3.0032 0.8701 3.1042 0.8722 3.2051 0.8744 3.3060 0.8763 3.4070 0.8782 3.5079 0.8797 3.6089 0.8804

Eo = 1000 x/a y/a 0 0 0.0005 0.0250 0.0020 0.0500 0.0044 0.0748 0.0078 0.0996 0.0126 0.1261 0.0189 0.1544 0.0272 0.1844 0.0376 0.2163 0.0507 0.2500 0.0669 0.2855 0.0865 0.3225 0.1103 0.3608 0.1387 0.4002 0.1722 0.4403 0.2116 0.4806 0.2572 0.5204 0.3097 0.5592 0.3694 0.5962 0.4367 0.6308 0.5117 0.6623 0.5947 0.6906 0.6857 0.7155 0.7843 0.7369 0.8838 0.7543 0.9837 0.7686 1.0840 0.7806 1.1844 0.7907 1.2850 0.7995 1.3857 0.8072 1.4864 0.8141 1.5872 0.8202 1.6880 0.8258 1.7888 0.8308 1.8897 0.8355 1.9906 0.8398 2.0914 0.8437 2.1923 0.8474 2.2932 0.8508 2.3941 0.8540 2.4951 0.8570 2.5960 0.8599 2.6969 0.8625 2.7978 0.8650 2.8988 0.8674 2.9997 0.8697 3.1007 0.8718 3.2016 0.8739 3.3025 0.8758 3.4035 0.8776 3.5044 0.8791 3.6054 0.8798

Table 4: Interface coordinates of plane bubbles in vertical channel for different E¨ otv¨ os numbers (present numerical results by BEM with ds min =0.025, ds max =0.1 and l =4).

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Eo = 10 x/R r/R 0 0 0.0019 0.0500 0.0075 0.0996 0.0169 0.1488 0.0299 0.1970 0.0474 0.2467 0.0698 0.2974 0.0976 0.3487 0.1313 0.4001 0.1711 0.4511 0.2175 0.5010 0.2705 0.5493 0.3304 0.5954 0.3971 0.6386 0.4707 0.6786 0.5510 0.7149 0.6380 0.7474 0.7314 0.7761 0.8265 0.8001 0.9226 0.8200 1.0192 0.8367 1.1163 0.8509 1.2136 0.8630 1.3111 0.8735 1.4087 0.8826 1.5065 0.8903 1.6044 0.8966 1.7023 0.9016 1.8003 0.9054 1.8984 0.9086 1.9964 0.9118 2.0944 0.9154 2.1924 0.9193 2.2904 0.9229 2.3885 0.9253 2.4865 0.9267 2.5846 0.9278 2.6827 0.9297 2.7807 0.9325 2.8787 0.9349 2.9768 0.9360 3.0749 0.9363 3.1730 0.9374 3.2710 0.9396 3.3691 0.9415 3.4672 0.9421 3.5652 0.9416

Eo = 100 x/R r/R 0 0 0.0018 0.0500 0.0071 0.0997 0.0160 0.1489 0.0284 0.1973 0.0451 0.2472 0.0668 0.2983 0.0939 0.3500 0.1269 0.4018 0.1663 0.4531 0.2125 0.5032 0.2658 0.5513 0.3263 0.5964 0.3943 0.6377 0.4697 0.6741 0.5520 0.7058 0.6406 0.7335 0.7354 0.7573 0.8317 0.7757 0.9286 0.7908 1.0258 0.8039 1.1233 0.8144 1.2210 0.8232 1.3187 0.8315 1.4166 0.8382 1.5145 0.8442 1.6124 0.8500 1.7103 0.8547 1.8083 0.8592 1.9063 0.8636 2.0043 0.8670 2.1023 0.8708 2.2003 0.8740 2.2984 0.8768 2.3964 0.8799 2.4944 0.8823 2.5925 0.8848 2.6905 0.8873 2.7886 0.8892 2.8866 0.8915 2.9847 0.8933 3.0828 0.8951 3.1808 0.8971 3.2789 0.8985 3.3769 0.9003 3.4750 0.9015 3.5731 0.9013

13/15

Eo = 1000 x/R r/R 0 0 0.0018 0.0500 0.0072 0.0997 0.0160 0.1489 0.0279 0.1974 0.0438 0.2476 0.0646 0.2991 0.0910 0.3511 0.1233 0.4034 0.1617 0.4554 0.2073 0.5061 0.2600 0.5547 0.3201 0.6005 0.3878 0.6421 0.4627 0.6797 0.5447 0.7121 0.6333 0.7398 0.7282 0.7631 0.8246 0.7814 0.9215 0.7965 1.0188 0.8085 1.1163 0.8188 1.2140 0.8274 1.3118 0.8349 1.4097 0.8416 1.5076 0.8472 1.6055 0.8527 1.7035 0.8572 1.8015 0.8617 1.8995 0.8656 1.9975 0.8692 2.0955 0.8726 2.1935 0.8756 2.2916 0.8786 2.3896 0.8812 2.4876 0.8839 2.5857 0.8862 2.6837 0.8884 2.7818 0.8906 2.8799 0.8925 2.9779 0.8945 3.0760 0.8962 3.1740 0.8981 3.2721 0.8996 3.3702 0.9012 3.4682 0.9024 3.5663 0.9021

Table 5: Interface coordinates of axis-symmetrical bubbles in vertical tube for different E¨ otv¨ os numbers (present numerical results by BEM with ds min =0.05, ds max =0.1 and l =4).

14/15

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

References Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge University Press. Bendiksen, K. H. 1984. An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphase Flow, 10, 467–483. Bendiksen, K. H. 1985. On the motion of long bubbles in vertical tubes. Int. J. Multiphase Flow, 11, 797–812. Benjamin, T. B. 1968. Gravity currents and related phenomena. J. Fluid Mech., 31, 209–248. Collins, R. 1965. A simple model of the plane gas bubble in a finite liquid. J. Fluid. Mech., 22, 763–771. Collins, R., De Moraes, F. F., Davidson, J. F., & Harrison, D. 1978. The motion of a large gas bubble rising though liquid flowing in a tube. J. Fluid. Mech., 89, 497–514. Cou¨et, B., & Strumulo, G. S. 1987. The effects of surface tension and tube inclination on a two-dimensional rising bubble. J. Fluids Mech., 184, 1–14. Davies, R. M., & Taylor, G. 1950. The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. Roy. Soc. Ser. A, 200, 375–390. Dumitrescu, D. T. 1943. Str¨ omung an einer Luftblase im Senkrechten Rohr. Z. Angew. Math. Mech., 23, 139. Garabedian, P. R. 1957. On steady-state bubbles generated by Taylor instability. Proc. Roy. Soc. Ser. A., 241, 423–431. Ha-Ngoc, H. 2003. Etude th´eorique et num´erique du mouvement de poches de gaz en canal et en tube. Ph.D. thesis, Institut National Polytechnique de Toulouse, France. H´eraud, P. 2002. Etude exp´erimentale de dynamique de bulles. Ph.D. thesis, Universit´e de Provence, France. Lamb, H. 1932. Hydrodynamics. Cambridge University Press. Maneri, C., & Zuber, N. 1974. An experimental study of plane bubbles rising at inclination. Int. J. Multiphase Flow, 1, 623–645. Mao, Z. S., & Dukler, A. E. 1990. The motion of Taylor bubbles in vertical tubes I. A numerical simulation for the shape and rise velocity of Taylor bubbles in stagnant and flowing liquids. J. Comp. Phys., 91, 132–160. Mao, Z. S., & Dukler, A. E. 1991. The motion of Taylor bubbles in vertical tubes II. Experimental data and simulations for laminar and turbulent flow. Chem. Eng. Sci., 46, 2055–2064. Vanden-Bro¨eck, J. M. 1984a. Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids, 27, 1090–1093. Vanden-Bro¨eck, J. M. 1984b. Rising bubble in two-dimensional tube with surface tension. Phys. Fluids, 27, 2604–2607. Wallis, G. B. 1969. One Dimensional Two-phase Flow. McGraw-Hill, New York.

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

15/15

White, E. T., & Beardmore, R. H. 1962. The velocity of rise of single cylindrical bubbles through liquids contained in vertical tubes. Chem. Eng. Sci., 17, 351–361. Zukoski, E. E. 1966. Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J.Fluid Mech., 25, 821–837.