BUBBLES RISING IN AN INCLINED TWO-DIMENSIONAL TUBE AND JETS FALLING ALONG A WALL

J. Autral. Math. Soc. Ser. B 4(999), 332 349 BUBBLES RISING IN AN INCLINED TWO-DIMENSIONAL TUBE AND JETS FALLING ALONG A WALL J. LEE and J.-M. VANDEN-BROECK 2 (Received 22 April 995; revied 23 April 996) Abtract The motion of a two-dimenional bubble riing at a contant velocity U in an inclined tube of width H i conidered. The bubble extend downward without limit, and i bounded on the right by a wall of the tube, and on the left by a free urface. The ame flow configuration decribe alo a jet emerging from a nozzle and falling down along an inclined wall. The acceleration of gravity g and the urface tenion T are included in the free urface condition. The problem i characterized by the Froude number F D U= p gh, the angle þ between the left wall and the horizontal, and the angle between the free urface and the right wall at the eparation point. Numerical olution are obtained via erie truncation for all value of <þ<³. The reult extend previou calculation of Vanden-Broeck [2 4] for þ D ³=2 and of Couët and Strumolo [3] for <þ<³=2. It i found that the behavior of the olution depend on whether <þ<2³=3 or2³=3 þ<³. When T D, it i hown that there i a critical value F c of Froude number for each <þ<2³=3 uch that olution with D ; ³=3and³ þ occur for F > F c, F D F c and F < F c repectively, and that all olution are characterized by D for2³=3 þ<³.whena mall amount of urface tenion T i included in the free urface condition, it i found that for each <þ<³there exit an infinite dicrete et of value of F for which D ³ þ. A particular value F Ł of the Froude number for which T D and D ³ þ i elected by taking the limit a T approache zero. The numerical value of F Ł and the correponding free urface profile are found to be in good agreement with experimental data for bubble riing in an inclined tube when <þ<³=2.. Introduction We conider a bubble riing at a contant peed U in an inclined two-dimenional tube of width H (ee Figure a). The bubble i bounded on the right by a wall of the tube and on the left by a free urface. We aume that the tube i infinitely long and that Department of Mathematic, Kwang Woon Univerity, Wolgye-dong Nowon-Gu, Seoul 39-7, Korea 2 Department of Mathematic and Center for the Mathematical Science Univerity of Wiconin - Madion, Madion, WI 5376 c Autralian Mathematical Society, 999, Serial-fee code 334-27/99 332

[2] Bubble riing in an inclined tube 333 :5 I Ð Λ Λffl U Λ Λffl I Ð :5 I Ð Λ Λffl U Λ Λffl I Ð :5 þ x XyXΛ Λν y S :5 þ x XyXΛ Λν y S :5 :5 J J Xy XX X XXz H :5 :5 J J :5 :5 :5 :5 :5 :5 (a) (b) FIGURE. Sketch of the flow domain and the coordinate. Thi i a computed profile for þ D 7³=2 and F D : and! D. the bubble extend downward without limit. We chooe a frame of reference moving with the bubble. Gravity i acting vertically downward. The effect of the urface tenion T i included in the dynamic boundary condition. The angle between the left wall and the horizontal i denoted by þ and the angle between the negative x-axi and the tangent line at the noe of the bubble by (ee Figure a). The flow configuration of Figure a decribe alo a jet emerging from a nozzle and falling down along a wall. In thi cae the flow i viewed a bounded on the left by an infinite wall and on the right by a emi-infinite wall and a free urface (ee Figure b). The bubble of Figure a ha been tudied experimentally by Maneri [9] for <þ<³=2. No experimental reult have been reported for ³=2 <þ<³, becaue the bubble tend to rie on the upper wall. Variou configuration of jet falling from a nozzle or an aperture have been conidered analytically by Keller and Gee [7] and numerically by Tuck [], Dia and Vanden-Broeck [4], Lee and Vanden- Broeck [8] and other. Thee author included gravity but neglected urface tenion. Our calculation generalize ome of their finding by including urface tenion. We conider olution for the flow of Figure b for all value of <þ<³, although the olution for ³=2 <þ<³are upide down flow with the heavy fluid on top of the light fluid. We have two motivation for conidering uch flow. Firt, upide down flow are commonly oberved. For example, when a liquid i poured from a container, it ometime flow along the underide of the pout of the pouring veel. Thi phenomenon wa tudied experimentally by Reiner [], who called it

334 J. Lee and J.-M. Vanden-Broeck [3] the teapot effect. Analytical and numerical olution of uch pouring flow have been found by Keller [6] and Vanden-Broeck and Keller [5, 6]. Secondly, it i of interet to decribe the local behavior of the flow near the eparation point (that i, the point of interection of the free urface with the right wall) for all <þ<³,andin particular, the dependence of on þ. For each value of þ, the problem i characterized by the Froude number and the Weber number F D U= p gh (.)! D ²U 2 H=T: (.2) Here, ² i the denity of the fluid. Garabedian [5], Birkhoff and Carter [] and Vanden-Broeck [2 4] conidered the problem of a free bubble riing in a two-dimenional tube. Thi configuration can be obtained by reflecting the flow of Figure a with þ D ³=2 into the right wall. After reflection the width of the tube i 2H (ee Figure 2). A we hall ue reult from Vanden-Broeck [2 4] later, we ummarize them here a follow. When! D (that i, T D ), there i a critical value F c ³ :56 of the Froude number F uch that D ³=2 forf < F c ; D ³=3 forf D F c ; D forf > F c : (.3) We note that Vanden-Broeck ([2 4]) ued a Froude number (which we denote here by F r ) baed on the width of the tube. Thu F r i related to F by F r D F= p 2: (.4) The mathematical olution for þ D ³=2 ha three interpretation: a jet falling from a nozzle (ee Figure b), a bubble bounded on the right by a wall and on the left by a free urface (ee Figure a) and a free bubble (ee Figure 2). If we ue the interpretation of Figure b, relation (.3) agree with our everyday experience: there i a jet for each value of F. If we ue the interpretation of Figure 2, we need to require D ³=2 othatthe lope i continuou at the noe of the bubble. Relation (.3) how that there i a olution for each value of < F < F c. Thi i in contradiction with the experiment of Collin [2] which how that there i only one bubble correponding to F ³ :35. Vanden-Broeck [3] howed that the dicrepancy between the experiment and the theory can be removed by introducing urface tenion and taking the limit a the urface tenion approache zero. Vanden-Broeck [3] howed that when! 6D (that i, T 6D ), the angle i a function of! and F. For a fixed value of!, ocillate

[4] Bubble riing in an inclined tube 335 2 :5 :5 U?? 2H 6 x y S -,:5,,:5,2,:5,,:5 :5 :5 FIGURE 2. Computed olution for þ D ³=2andF D F Ł ³ :325. infinitely often around ³=2 a F decreae. There i a countably infinite number of value of F for which D ³=2. Furthermore, a!!, all the olution approach a unique olution characterized by F Ł ³ :325. Therefore an arbitrary mall amount of the urface tenion can be ued to elect a particular olution with T D and D ³=2 (in the abence of urface tenion, all olution for < F < F c are characterized by D ³=2). The value F Ł ³ :325 i in good agreement with the experimental value :35 of Collin [2]. The flow problem with the interpretation of Figure a wa olved numerically by Couët and Strumolo [3]. They retrict their attention to <þ<³=2. Their reult howed that for given value of þ and!, ocillate infinitely often a F approache zero. For each value of <þ<³=2and!, they elected the particular olution correponding to the larget value of F for which D ³=2. In thi paper we extend the calculation of Vanden-Broeck [2 4] and Couët and Strumolo [3] and preent numerical olution of the flow configuration of Figure a andbforvalueof<þ<³. The numerical procedure ue erie truncation. It i imilar to the technique ued by Vanden-Broeck [2 4], Couët and Strumolo [3] and Lee and Vanden-Broeck [8]. It i found that the behavior of the olution depend on whether <þ<2³=3 or 2³=3 þ<³.

336 J. Lee and J.-M. Vanden-Broeck [5] For T D and each <þ<2³=3, there i a critical value F c of the Froude number F, uch that olution with D ; ³=3and³ þ occur for F > F c ; F D F c and F < F c, repectively. Thi include (.3) a a particular cae for þ D ³=2. On the other hand, for T D and each value of 2³=3 þ<³, there i no uch a critical value of F and D forall< F <. When the flow i interpreted a in Figure a, urface tenion can be ued to elect the phyically relevant bubble. We propoe a new election criterion and how that the elected value of the Froude number and the elected profile are in good agreement with the experiment of Maneri [9]. The problem i formulated in Section 2. The numerical procedure i decribed in Section 3 and the reult are preented and dicued in Section 4, 5 and 6. 2. Formulation Let u conider the teady two-dimenional potential flow of an invicid incompreible fluid in a domain bounded on the left by an infinite wall I J andonthe right by a emi-infinite wall IS and a free urface SJ (ee Figure a). A x!, the velocity approache the contant U. We introduce Carteian coordinate with the origin at the eparation point S. Gravity act vertically downward. Then, the gravitational potential i given by G D gx in þ gy co þ: (2.) On the free urface SJ, the Bernoulli equation yield 2 q2 C G.T=²/ K D B; (2.2) where q i the flow peed, K i the curvature of the free urface and B i the Bernoulli contant. We define dimenionle variable by taking U a the unit velocity and H a the unit length. We denote the potential function by and the tream function by. In addition, we introduce the complex velocity by D u iv, and we define the function i by the relation D u iv D e i : (2.3) We denote the complex potential by f D C i. Without lo of generality, we chooe D D atx D y D. It follow from the choice of the dimenionle variable that D on the urface of the bubble SJ and D onthewalli J. The complex potential plane i ketched in Figure 3. We hall eek the function i a an analytic function of f in the trip < <:

[6] Bubble riing in an inclined tube 337 6 I S D - J D I J FIGURE 3. The complex potential plane. In term of the dimenionle variable, (2.2) become e 2 C 2 F.x in þ y co þ/ 2 þ þþþ @ 2! e @þ D B U : (2.4) 2 Here, F and! are the Froude number and the Weber number defined by (.) and (.2), repectively. Differentiating (2.4) with repect to and uing the relation @x @ C i @y @ D u iv D e Ci ; (2.5) we have 2 @ e @ C F 2 e in.þ /! @ @ e @ D onsj: (2.6) @ The kinematic condition on the wall I J and IS yield D ³; D ; (2.7) D ³; D ; <: (2.8) Thi complete the formulation of the problem of determining i. Thi function mut be analytic in the trip < < and atify the condition (2.6) (2.8). 3. Numerical Procedure We olve the problem by following the procedure ued by Vanden-Broeck[2 4], Couët and Strumolo [3] and Lee and Vanden-Broeck [8].

338 J. Lee and J.-M. Vanden-Broeck [7] 6 I I S - J J FIGURE 4. The complex t-plane. We firt map the flow domain on the fourth quadrant of the unit circle in the complex t-plane by the tranformation e ³ 2 f D t C : (3.) 2 t The wall IS goe onto the real interval.; / and the bubble urface SJ onto the circumference (ee Figure 4). Next we define the function.t/ by the relation D e i D [ ln C. C t 2 /] =3. ln C/ =3. t 2 / 2=³ e.t/ : (3.2) Here C i an arbitrary contant between and :5. We chooe C D :2. At the point J and S, ha ingularitie aociated with a thin jet and a flow inide an angle of ³ (ee Figure a). Thee ingularitie are removed in (3.2) by the factor [ ln C. C t 2 /] =3 and. t 2 / 2=³ (ee Birkhoff and Carter [], Vanden-Broeck [2 4] and Lee and Vanden-Broeck [8] for detail). It follow that.t/ can be repreented by a Taylor expanion in power of t. Furthermore, the kinematic condition (2.7) and (2.8) imply that the expanion for.t/ ha real coefficient a n and involve only even power of t. Thu we write.t/ D X a n t 2n : (3.3) nd We decribe point on the free urface SJ by t D e i,where ³=2 <. Uing (3.), we rewrite (2.6) in the form 2 Q d Q ³ cot e d C 2 F 2 e Q in.þ Q/ ³ 2 2! cot d e Q cot d! Q D : (3.4) d d

[8] Bubble riing in an inclined tube 339 Here, Q. / and Q. / denote the value of and on the free urface SJ. We olve the problem by truncating the infinite erie in (3.3) after a finite number of term. We firt conider the cae! D(that i, T D ). The local analyi of Lee and Vanden-Broeck [8] how that the only poible value of are, ³=3 and³ þ. Therefore, we have three different cheme correponding to each of thee value of. In each of them, we truncate the infinite erie in (3.3) after N term and atify the free urface condition (3.4) at the N meh point I D ³ 2N I 2 ; I D ; 2;::: ;N: (3.5) Thi lead a ytem of N nonlinear algebraic equation for the N unknown a n, n D ; 2;::: ;N. We olve thi ytem by Newton method. Once thi ytem i olved for given value of F and! (for each þ), the hape of the bubble i obtained by numerically integrating (2.5). We now conider the cae! 6D (that i, T 6D ). The angle i no longer retricted to the value, ³=3 and³ þ and mut be found a part of the olution. Thu we truncate the infinite erie in (3.3) after N term and atify (3.4) at the N collocation point (3.5). Thi lead to a ytem of N nonlinear algebraic equation for the N unknown a n, n D ; 2;::: ;N and. Thi ytem i alo olved by Newton method. 4. Numerical reult The numerical cheme of Section 3 were ued to compute olution for variou value of F,! and þ. We found that the coefficient a n decreaerapidly a n increae. For example, a ¾ :6ð, a ¾ 5:7ð 3, a 3 ¾ 2:2ð 3 and a 5 ¾ 2:5ð 4 for þ D 7³=2,! D and F D :4 when N D 7. Mot of the computation were performed with 7 N 2. We alo repeated the calculation with C 6D :2 and checked that the reult were independent of the value of < C < :5. I. Solution without urface tenion. When urface tenion i neglected, there i a olution for each value of F and þ. A explained in Section 3, the angle take only the value, ³=3 and³ þ. The numerical reult how that there i a critical Froude number F c for each <þ<2³=3uchthat D ³ þ when F < F c ; D ³=3when F D F c ; D when F > F c ; (4.) and that for 2³=3 þ<³there i only one configuration with D for each F >.

34 J. Lee and J.-M. Vanden-Broeck [9] q =U.9.8.7.6.5.4.3.2. = = =3 = =2 =7=2 =3=4 F c??????????????????????????????????????????????????????????????????????????????????..2.3.4.5.6.7.8.9 F FIGURE 5. Value of the peed q =U veru F for variou value of þ. :9 :8 :7 :6 F :5 :4 :3 :2 : 3 6 9 2 5 8 FIGURE 6. Value of F c veru þ. We denote by q S the velocity at the eparation point S. The olution with D

[] Bubble riing in an inclined tube 34 are alo characterized by q S 6D, and thoe with D ³=3and D ³ þ by q S D. For 2³=3 þ<³, q S tend to zero a F approache zero. Figure 5 how the value of q S veru F for variou þ. The reult of Figure 5, retricted to <þ<³=2, are imilar to thoe in Figure of Couët and Strumolo [3]. Note that the inclination þ ued by Couët and Strumolo (which we denote by þ c ) i the complement of our angle þ. Thu þ c D ³=2 þ. Figure 6 how the value of F c veru þ. Forþ D ³=2, F c ³ :56, in accordance with Vanden-Broeck [2] reult. A þ approache 2³=3 from below, larger and larger value of N are needed to obtain accurate value of F c. Thi numerical difficulty i related to the fact that there are no olution with D ³=3forþ>2³=3. However an extrapolation of the reult of Figure 6 confirm that F c approache zero a þ tend to 2³=3 from below. In Figure 7a, 7b and 7c, we preent typical profile for everal value of þ. A F!, the profile of the bubble approache the wall. A F!, the profile of the bubble approache a horizontal line. When <þ<2³=3, the bubble urface at the contact point S i tangent to the wall for each value of F c < F < and leave horizontally from the wall for each < F < F c (ee Figure 7a and 7b). On the other hand, for 2³=3 þ<³, the free urface leave tangentially from the point S for all F > ( ee Figure 7c). :5,:5,,:5,2,2:5,2,:5,,:5 :5 :5 FIGURE 7A. Computed olution for þ D ³=4andT D. The free urface profile from left to right correpond to F D :2; :45, F D F c ³ :6262 and F D :8 (q ³ :49).

342 J. Lee and J.-M. Vanden-Broeck [] :5,:5,,:5,2,2:5,2:5,2,:5,,:5 :5 FIGURE 7B. Computed olution for þ D 7³=2 and T D. The free urface profile from left to right correpond to F D :8, F D F c ³ :393 and F D :8. :5,:5,,:5,2,2:5,2:5,2,:5,,:5 :5 FIGURE 7C. Computed olution for þ D 3³=4andT D. The free urface profile from left to right correpond to F D :2 (q ³ :2), F D :5 (q ³ :53) and F D :9(q ³ :76).

[2] Bubble riing in an inclined tube 343 II. Solution with urface tenion. When urface tenion i included in the free urface condition, the numerical computation how that there i a flow for each value of F, þ and!. AmentionedinSection3, theangle come a part of the olution and it value are no longer retricted to, ³=3 and³ þ. In Figure 8, we preent value of veru F for! D and everal value of <þ<³. Thee reult confirm the calculation of Couët and Strumolo [3] for <þ<³=2and extend them to the range <þ<³(ee their Figure 6). For each value of þ we found that tend to zero a F tend to infinity. Figure 8 how that for a fixed value of!, the amplitude of the ocillation around D ³ þ on each curve die out a þ increae. Similar curve were obtained for other value of! ufficiently large. We oberved that the amplitude of the ocillation decreae a! increae when þ i fixed. FIGURE 8. Value of veru F for! D. The curve from the top to the bottom correpond to þ D ³=3;³=2; 9³=36; 7³=2; 23³=36; 25³=36 and 3³=4. The dotted line from the top to the bottom correpond to the value D ³ þ for þ D ³=2; 23³=36 and 25³=36, repectively. Typical profile for þ D 3³=4 with! D are hown in Figure 9. We note that equation (2.4) implie that the free urface approache a horizontal line for any! and þ a F tend to zero.

344 J. Lee and J.-M. Vanden-Broeck [3] :5,:5,,:5,2,2,:5,,:5 :5 FIGURE 9. Computed olution for þ D 3³=4and! D. The curve from left to right correpond to F D :2; :4 and:6. 5. Dicuion of the reult, election technique and comparion with experiment A we mentioned in the introduction, the flow configuration tudied in thi paper decribe two different phyical problem. The firt i a jet falling down along a wall (ee Figure b). In thi cae, the numerical reult with T D how that there i a jet for each value of F and <þ<³. The econd phyical problem i a bubble riing in an inclined tube (ee Figure a). The numerical reult with T D how that there i a mathematical olution for each value of <þ<³and F. Recall that þ i defined a the angle between the left wall and the horizontal (ee Figure a). For any value of <þ <³=2, the angle þ D þ and þ D ³ þ correpond to the ame inclined tube viewed from the front or the back. Therefore there are two olution for a given inclined tube. However, olution with ³=2 <þ<³have not been oberved experimentally. Alo, the experimental data (Maneri [9]) how that for each <þ<³=2, there i only one value of F for which a bubble exit. Thi doe not agree with the numerical reult which predict a olution for each value of F and <þ<³=2. The dicrepancy can removed by generalizing the procedure derived by Vanden-Broeck [3] for the configuration of Figure 2 (ee the introduction for a ummary of the method). Thu we introduce

[4] Bubble riing in an inclined tube 345 urface tenion and take the limit a the urface tenion approache zero. The detail are decribed below. Couët and Strumolo [3] chooe for each value of þ and!, the particular olution correponding to the larget value of F for which D ³=2. Our reult how that there are no olution with D ³=2 whent D unle þ D ³=2 (ee (4.)). Thi finding doe not invalidate the election criterion of Couët and Strumolo becaue they ue it with T mall but different from zero. In fact their elected olution are in very good agreement with experiment a hown for example in the Figure 5, 6 and 7 of their paper. Here we how that an equally good agreement with experiment can be obtained by a different election criterion in which we take the limit a T! intead of keeping T 6D aincouët and Strumolo [3]. More preciely we elect for each <þ<³=2 the value F Ł of the Froude number defined by F Ł D lim!! F þ.!/; (5.) where F þ.!/ i the larget Froude number at which D, for given value of <þ<³=2and!. The reult of Section 4I how that can only take one of the value ³ þ, ³=3 and (that i, one of the only three poible value in the abence of urface tenion). :9 :8 :7 F :6 :5 N N N N :4 :3 :2 : 3 6 9 2 5 8 FIGURE. Value of F Ž veru þ. The dot are the experimental value of Maneri (97); N, methanol; ý, water.

346 J. Lee and J.-M. Vanden-Broeck [5] Figure 8 how that F þ.!/ Dfor D and! ufficiently large. Thu our criterion (4.2) with D lead to the unintereting value F Ł D. The only olution with D ³=3 in the abence of urface tenion correpond to F D F c (ee (4.)). Therefore our criterion with D ³=3 lead to F Ł D F c. In Figure, we compare thee elected value with the experimental value of Maneri [9]. The agreement i not very atifactory and the elected value are too high. For each value of! 6D and <þ<³=2, Figure 8 how that there i a dicrete et of value for which D ³ þ. ThevalueF Ł elected by our criterion for D ³ þ are hown in Figure. In order to find thee value for each <þ<³=2, we computed firt F þ.!/ by uing a variant of the cheme of Section 3 in which the problem i reduced to a ytem of N algebraic equation for N unknown F and a n, n D ; 2;::: ;N tobeolvedfor D ³ þ and given value of þ and!. We then determined the value of F Ł for each þ by evaluating F þ.!/ for larger and larger value of!. :9 :8 :7 F :6 :5 N N N N :4 :3 :2 : 3 6 9 2 5 8 FIGURE. Value of F Ł veru þ. The dot are the experimental value of Maneri (97); N, methanol; ý, water. We alo how in Figure the experimental value of Maneri [9]. The agreement between experiment and the theoretical curve i uniformly good. Thi finding indicate that our criterion hould be ued with D ³ þ. The agreement i better for methanol than water. Thi i to be expected ince the value of the urface tenion T for methanol i 3 time maller than for water (recall that our election criterion

[6] Bubble riing in an inclined tube 347 applie in the limit a T approache zero)..5 -.5 - -.5-2 -2.5 -.5 - -.5.5.5 FIGURE 2. Computed olution for þ D ³=3 andf D F Ł ³ :527. The dot are the experimental value of Maneri (97). A free urface profile for þ D ³=3andF D F Ł ³ :527 i hown in Figure 2. We alo compared in Figure 2 our elected profile for þ D ³=6 with the ame experimental value a Couët and Strumolo in their Figure 5. The agreement i alo very good and upport our choice D ³ þ in (4.2). It i worthwhile mentioning that our election criterion can be generalized by replacing F þ.!/ in (4.2), by the n th larget Froude number at which D ³ þ for given value of <þ<³=2and!. Heren denote an integer. Since both the amplitude and the wavelength of the ocillation around ³ þ in Figure 8 tend to zero a T!, the criterion yield the ame value of F Ł independently of the value of n. Finally we conider further the cae þ D ³=2. A decribed in the introduction there are two different poible bubble. The firt one i a bubble bounded on the left by a free urface and on the right by a wall of the tube (ee Figure a with þ D ³=2). The correponding elected value of F Ł i :325. The econd one i the free bubble of Figure 2. The experiment of Collin [2] and Maneri [9] how that the econd bubble i the one which i oberved. The elected value of the Froude number baed on the width of the tube for the free bubble of Figure 2 wa found by Vanden-Broeck [3] to be F r D :23 (ee introduction again for detail). Thi value of :23 doe not agree with the value :325 predicted in Figure for þ D ³=2. The reaon for thi

348 J. Lee and J.-M. Vanden-Broeck [7] dicrepancy i that the free bubble olution of Figure 2 i part of another family of olution which i not decribed by our model, namely a free bubble in an inclined tube (ee Figure 3 for a ketch). It i only in the cae þ D ³=2 that the free bubble can be obtained from the configuration of Figure a by a imple reflection. The experiment of Maneri [9] ugget that the branch of free bubble exit only for value of þ cloe to ³=2. A þ decreae from ³=2, there i a tranition to the olution decribed in thi paper. FIGURE 3. Sketch of a free bubble riing in an inclined tube. 6. Concluion We have preented numerical olution for a flow bounded by two wall and a free urface. The configuration model a bubble riing in a tube or a jet falling down from a nozzle along a wall. When urface tenion i neglected there are olution correponding to three different value of the angle. Thee three type of olution are conitent with the local analyi of Vanden-Broeck and Tuck [7]. When urface tenion i included in the free urface condition, the angle can take arbitrary value. It wa found that for given value of!, F and þ, there i a unique olution. Thi i in agreement with our everyday experience with jet falling from a nozzle. However, if we interpret the flow a a model for a bubble riing in a tube, thi finding i in contradiction with experiment which predict that there i only one bubble for given value of! and þ. Couët and Strumolo [3] preented a criterion to elect the phyically relevant olution which require T to be different from zero. We have preented an alternative criterion which work in the limit a T!. We have hown that our predicted value of the Froude number and our elected profile are in a good an agreement with experimental data a thoe of Couët and Strumolo.

[8] Bubble riing in an inclined tube 349 Acknowledgement Thi work wa ponored by the National Science Foundation. The firt author thank the GARC and Department of Mathematic, Seoul National Univerity, Korea for their upport and the econd author thank the Tel Aviv Univerity, the Technion, Haifa and the Hebrew Univerity of Jerualem for their hopitality during the time when thi paper wa put in the final form. Special thank are alo due to Mirna Džamonja for many contructive comment and her help with the Figure. Reference [] G. Birkhoff and D. Carter, Riing plane bubble, J. Math. and Mech. 6 (957) 769 779. [2] R. Collin, A imple model of a plane ga bubble in a finite liquid, J. Fluid Mech. 22 (965) 763 77. [3] B. Couët and G. S. Strumolo, The effect of urface tenion and tube inclination on a twodimenional riing bublle, J. Fluid Mech. 84 (987) 4. [4] F. Dia and J.-M. Vanden-Broeck, Flow emerging from a nozzle and falling under gravity, J. Fluid Mech. 23 (99) 465 477. [5] P. R. Garabedian, On teady-tate bubble generated by taylor intability, Proc. R. Soc. London Ser. A 24 (957) 423 43. [6] J. B. Keller, Teapot effect, J. Appl. Phy. 28 (957) 859 864. [7] J. B. Keller and J. Gee, Flow of thin tream with free boundarie, J. Fluid Mech. 59 (973) 47 432. [8] J. Lee and J.-M. Vanden-Broeck, Two-dimenional jet falling from funnel and nozzle, Phy. Fluid A 5 (993) 2454 246. [9] C. C. Maneri, The motion of plane bubble in inclined duct, Ph. D. Thei, Polytechnic Intitute of Brooklyn, New York. [] M. Reiner, The teapot effect...a problem, Phy. Today 9 (956) 6 2. [] E.O. Tuck, Efflux from a lit in a vertical wal, J. Fluid Mech. 987 (987) 253 264. [2] J.-M. Vanden-Broeck, Bubble riing in a tube and jet falling from a nozzle, Phy. Fluid 27 (984) 9 93. [3] J.-M. Vanden-Broeck, Riing bubble in a tube with urface tenion, Phy. Fluid 27 (984) 264 267. [4] J.-M. Vanden-Broeck, Pointed bubble riing in a two-dimenional tube, Phy. Fluid 29 (986) 343 344. [5] J.-M. Vanden-Broeck and J. B. Keller, Pouring flow, Phy. Fluid 29 (986) 3958 396. [6] J.-M. Vanden-Broeck and J. B. Keller, Pouring flow with eparation, Phy. Fluid A (989) 56 58. [7] J.-M. Vanden-Broeck and E. O. Tuck, Flow near the interection of a free urface with a vertical wall, SIAM J. Appl. Math. 54 (954) 3.