The Gas-assisted Expelled Fluid Flow in the Front of a Long Bubble in a Channel

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1 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo The Gas-assisted Epelled Fluid Flow in the Front of a Long Bubble in a Channel C.H. Hsu 1, P.C. Chen 3,.Y. ung, and G.C. uo 1 1 Department of Mechanical Engineering Chung-Yuan Christian Universit. Department of Mechanical Engineering Nana Institute of Technolog. 414 Sec. 3 Chung Shang East R. Chung Li 30 TAIWAN. 3 Nuclear Fuels and Materials Division, Institute of Nuclear Energ Research k.kung@msa.hinet.net Abstract: - Investigated in this stud is the stead-state flow field of a long bubble penetrating into a region filled with a viscous fluid confined b two closel located parallel plates. Instead of using the complicated procedure for iterativel computing the free surface and flow patterns, we simpl use a theoretical profile of the bubble so that the influence of bubble shape on the flow field can be eamined directl. Due to the simplification, flow fields with higher Renolds number are easier to be included and different flow phenomenon is found. The numerical techniques emploed are finite difference method (FDM) with successive over-relaation (SOR). The simulation results show coincidentl with others the two tpical flow patterns (complete bpass flow and recirculation flow). The graduall moving of the stagnation point in the front of the bubble tip between two tpical flow patterns is clearl presented and eplainable. Both of the position of the stagnation point parallel and perpendicular to the flow, and, depends on Renolds number, Re, and, the ratio of asmptotic bubble width to the distance between two parallel plates. As or Re increases, increases too, but decreases. A quasi-linear relationship between and is found in a recirculation flow region. Be ware that the stagnation point is ver sensitive with for Re>100. As Re increases, the maimum value of stream function becomes bigger too. ma increases, and the recirculation zone near the bubble tip e-words: - long bubble, epelled fluid flow, stagnation point, two-phase flow, inertia forces. 1 Introduction Tsing long bubble to assist manufacture or medical processes is common in various industries, such as gas-assisted injection molding, bio-mechanic process and medical treatment process, etc. Particularl, the information about the epelled fluid flow in front of the bubble and the contour of the bubble front are of great interests in the simulating processes. Using the fundamental dnamic equations of the bubble b some theoretical or eperimental deduced empirical equations can practicall simplif simulation procedures. The flow patterns, and the migration of stagnation point and the influence of the inertia forces on the flow field are presented in this stud. The impacts of the bubble size on the flow field draw a great interest of man researchers [,9-10]. Mavridis et al. [11] studied the movement of the polmer melts front in both two-dimensional channels and tubes. The computed the inner flow fields of both the Newtonian and the non-newtonian fluids and compared the difference of these melt front shapes. Fairbrother and Stubbs [1] first studied the penetration of a long bubble in a tube. The empirical formula between the fractional coverage ISSN: Issue 1, Volume 4, Januar 009

2 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo are obtained. Hsu et.al. [] numericall studied the Long Gas Bubble Front. Talor [3] presented three fluid flow patterns, two kinds of re-circulating flows with a low capillar number, and a b-pass flow with high capillar number. Bretherton [4] developed a theoretical formula relating the fractional coverage and the pressure drop across the bubble. Goldsmith and Mason [5] visualized the flow pattern, the coating thickness and the shape of the bubble front. Co [6-7] etended Talor s eperiment and proposed a numerical solution of the momentum equation. Co also developed an empirical equation of the bubble profile, verified the eistence of the two flow patterns in the Poiseuille flow. In the Hele- Shaw cell model, Pitts [8] reported a theoretical bubble shape equation for smaller than However, the stud of the effects of, using atheoretical bubble s profile is absent. In present work, the flow patterns previousl considered b Hsu et al [9-10] in front of a semi-infinite gas bubble were computed. The streamlines demonstrate how the flow field changes when is varied and evidentl eist recirculating flow and the migration of stagnation point. Governing Equations This paper is closel followed Saffman and Geoffre Talor [] and Hsu et al [9-10] assumptions, studies a bubble steadil with moving eed U epels the viscous fluid confined in two closel parallel plates as illustrated in Fig. 1, and the channel is open. The continuit equation is epressed as u u + = 0 (1) The momentum equations are u u 1 p u u u + u = + υ + ρ u u 1 p u u u + u = + υ + ρ ) () The stream function (, of the flow is defined as u =, u = and vorticit ω is defined as u u ω = curl V = = (3) (4) Substituting equation (3) and (4) into (), the vorticit equation can be obtained as ω ω ω ω υ = + (5) B introducing the following dimensionless variables =, =, H H u u u =, u = U(1 ) U(1 ) ω u u ω = =, = U (1 ) UH (1 ) H UH (1 ) Re = υ where υ is the kinematic viscosit and Re is the Renolds number reectivel, equations (4) and (5) can be written as + + ω = 0 (6) ω ω ω ω + Re 0 = (7) An inverse bubble velocit U is added in the flow to fi the bubble in the computation domain so that the boundar conditions are epressed as follows. No slip condition on the channel wall( AB ) Δ 1 + i, js 1 ( 1 ) = 1; ωi, j = s ( Δ) The Hagen-Poiseuille flow at the far upstream ( BC ) = ; ω = ( ) ( ) ( ) The smmetr conditions at the centerline ( CD ) (8) (9) = 0; ω = 0 (10) At the interface between the two fluids ( DE ). In general, for a given bubble profile, the net force acting on the interface should be zero due to the balance between the surface tension on the interface and stresses on both sides of the interface. B using the theoretical bubble profile equation proposed b Pitts [8], the present stud imposes two conditions suggested b Co[6]: no diffusion across the interface (Eq. 11), and zero tangential stress on ISSN: Issue 1, Volume 4, Januar 009

3 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo the interface between gas and the fluid (Eq. 1). Therefore, two conditions at the interface can be deduced as u sinθ + u cosθ = 0 (11) u u u u sinθcosθ + + ( cos θ sin θ) 0 = (1) B combining equations (11) and (1), the equation of vorticit on the interface can be obtained as ω d = tan θ θ (13) d Thus, the conditions on the interface are d = 0; ω = tan θ θ (14) d The flow in the region is rectilinear at the far downstream ( EA ) + =, 1; ω = 0 (15) 1.1 Numerical Procedure The finite-difference method (FDM) with successive over-relaation (SOR) is emploed in the present stud. The finite difference formulas are obtained from equation (6) as ω ω i 1, j i, j+ ω + i 1, j + ω ω i, j 1 i, j+ ω + i, j 1 ( Δ) Δ ( d+δ) Δ d d( d+δ) (16) ω d i 1, j ωi 1, j ωi 1, j ( d ) ωi, j ω + + Δ Δ i, j 1 Re = 0 Δ dδ ( d+δ) where i, j denote indices at -direction and - direction, Δ and Δ are even grid sizes at - direction and -direction, and d is the uneven grid size along the interface ( 0 < d < Δ ), reectivel. The uneven grid d is introduced in order to fit the curvilinear interface as shown in Figure. Two different grids, one is curvilinear conformal to the interface and the other is rectilinear parallel to the solid boundaries, are adopted in the numerical computational scheme. The finite difference equation with SOR is solved b using Gauss-Seidal iteration method. Rearrange equation (16), the Gauss-Seidal iteration form with SOR is obtained as ( a +b +c +d ) ( 1 ) α + + ωi, j = ωi+1,j ωi,j+1 ωi-1,j ωi,j-1 + α ωi, j(17) e where 1 1 d ( Δ) Δ Δ ( d+δ) Δ ( d+δ) 1 1 Δ ( Δ) Δ d ( d +Δ) dδ ( d +Δ) d Δ Re ( Δ) dδ dδ ( d +Δ) a = Re, b = + Re c = + Re, d = Re e= + + = = i, j i 1, j Δ i, j i, j 1 d α and denote the over-relaation factor and iterative time, reectivel. Discrete and rearrange the stream-vorticit equation (7), the Gauss-Seidal iteration form with SOR is obtained as α = a +b +c +d + ω + 1 α (18) ( + + ) ( ) i, j 1 i 1, j 1 i, j 1 1 i 1, j 1 i, j 1 i, j i, j e1 where 1 a1 = c1 = Δ b1 = Δ d +Δ ( ) d1 = d d ( + Δ) e1 = + Δ dδ Starting with an initial guess of values of vorticit and stream function (all set at zero values in the computation domain) and introducing boundar conditions, the value of ω is computed from equation (17) at each grid point, and then substituting the ω value into equation (18) to obtain new value. The iteration process is repeated to calculate the values of ω and until the relative residual reaches the convergent criterion. The criterion is defined as k+1 k i,j i,j (19) k i,j The vorticit on the channel wall is derived from the wall values of stream function, i, j in backward s form b using the Talor epansion. The epansion becomes 1 3 i, j 1, ( s = i j Δ + Δ + O Δ )(0) s i, js i, js j s where the inde denotes the -position of the channel wall. Substituting the known value = on the i, j s 1 boundar and rearranging equation (0), the boundar value of the vorticit can be obtained as ISSN: Issue 1, Volume 4, Januar 009

4 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo ω 1 + i, js 1 i, j = s ( Δ ) Δ ( 1 ) (1) 3 The numerical results Man previous studies verified two tpicall flow patterns [5,7]. Eeciall, Hsu [9,10] dilaed the third pattern which suggested b Talor [3]. Present stud discusses the impacts of the values of and Renolds number Re on the position of stagnation point and the maimum value on the flow field. The influences of the upstream boundar condition on the fluid flows are discussed below. It is known from Co [7] that the upstream distribution of can be epressed as = 1 1 ( ) ( ) () B setting in equation (), the relative equation of and can be obtained as = (3) 3 From equations () and (3), one can find that the location of varies as a function of and. If one set that the position of locates on, it results in. The increases from to 1 as increases from zero to unit, i.e., the recirculating flow pattern will be observed if the value of is higher than. Thus the critical situation will signif the two tpical flow patterns proposed b Talor [3], Co [7]. The numerical results presented here verif the flow fields mentioned in the previous studies. Phenomena in the creeping flow ( Re = 0 ) and the effects of the inertia force ( Re > 0 ) to the flows are discussed. The effects of and Re to the position of the stagnation point and the value of the maimum are also presented. 3.1 The inertial effects We assume the profile of the bubble still obe the result proposed b Pitts [8]. Figure 3 illustrates three kind of flow patterns with five different values. The complete bpass flow is shown in Fig. 3(a) and 3(b). The flow pattern suggested b Talor [3] is shown in Fig. 3(c), and the recirculation flow is presented in Fig. 3(d) and 3(e). All streamlines in the figures are drawn from = 1 to with an increment of 0.1. ma In Figure 3(a) and 3(b), the streamlines are quite uniform and parallel in the rear region within It shows that the flow velocit in the region is nearl equal. Also, the contour of the bubble is nearl parallel to the channel wall in far downstream, as proposed b Co [6] and Pitts [8]. Comparing Fig. 3(a) and 3(b), the results show that the flow region in the front of the bubble tip, which is affected b the bubble, decreases as value increases. The third flow pattern, which is the flow with the stagnation point located in the front of the bubble on the centerline, is shown in Fig. 3(c). It is ver interesting to notice that a recirculation region is formed when value is just a little higher than /3. The recirculation region las between the centerline and the streamline, = 0, including the stagnation point. In addition, the third flow pattern seems to be a transition state between the complete bpass flow pattern and recirculation flow pattern, because this flow phenomenon onl can be observed when value is just a little higher than /3. More details will be discussed in Figure 1. Figures 3(d) and 3(e) are the tpical recirculation flow pattern verified b man previous studies such as Co [7]. The ma value in the recirculation region in both Fig. 3(d) and 3(e) is greater than zero. There is a stagnation streamline ( = 0 ) starts from the front of the bubble and outreaches into the fluid with a distance awa from the centerline. Between the centerline and the stagnation streamline, the fluid flows reversel awa from the bubble. The downstream bpass path of the main flow becomes narrower and the size of the recirculation zone grows larger while the value increases. It is worth noting that the stagnation point moves downstream along the bubble profile and the streamlines show notable changes in both main flow region and recirculation region while the value increases. Figure 3 present the distribution of streamline with different under zero Renolds number value in complete bpass flow (</3) region, the transient region and recirculation flow ( /3) region, reectivel. the stagnation point locates on the centerline between the upstream and the bubble tip when is just above the critical value /3. Figure 4 shows the correonding velocit distributions of Figure 3. It is found that the fluid in the region with 1 0 flows downstream in the ISSN: Issue 1, Volume 4, Januar 009

5 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo same direction toward the bpass path at the rear part of the bubble. The reverse flow is in the region with > 0. The value of dominates the present of the complete bpass flow or recirculation flows. As shown in Fig. 4(a) and 4(b), the velocit in the front of bubble tip decreases while increases up to /3. The recirculation flow is present with >/3. The maimum recirculation velocit and the recirculation zone increase while increases from /3 to 1, and reach their maimum values when =1. The stagnation point moves along the centerline, as shown in Fig. 4(c), and it moves downstream along the bubble profile, as observed in Fig. 4(d) and 4(e), while increases. In other words, the stagnation point moves downstream along the centerline and the bubble profile while the cross-sectional area of the bubble increases. Also, while the crosssectional area of the bubble increases, it is difficult for the viscous fluid at the center region to move downstream. Finall, the fluid reaches a stagnant state and then starts to move upstream. As mentioned, the maimum recirculation velocit increases while increases. However, the effect of to the velocit distribution in the range with 1 0 is minor. The velocit distributions near the stagnation point in Fig. 4(c), 4(d) and 4(e) are enlarged to show in Fig. 5(a), 5(b) and 5(c) for further insight. Fig. 5(a), 5(b) and 5(c) show that the positions of stagnation point with =0.67, 0.70 and 0.77 are located at ( , 0), ( , ) and ( , ), reectivel. The velocit near the stagnation point, as shown in Fig. 5(a), decreases to zero when coordinate varies from larger to = 0. The figure also shows a tin recirculation flow region, which is not clearl presented in Fig. 3(c) and 4(c). The reason for not showing the tin recirculation zone is that, for Fig 3(c), the value of ma is less than 0.1, which makes a difficult to plot streamlines and, for Fig. 4(c), onl a zero velocit is shown. The velocit between the bubble tip and stagnation point near the centerline also presents in Fig. 5(a). The flow velocities move nearl parallel to the centerline onl in a thin range not over 0.05, while the fluid move abruptl upward along the bubble. Because the position of the stagnation point is located on the bubble profile with >/3, there is no fluid velocit parallel to the centerline in Fig. 5(b) and 5(c). In Fig. 5(b), the fluid in the front of the bubble tip shows a stagnant state, which is not clearl observed in Fig. 3(d) and 4(d). Fig. 5(c) shows that the fluid in the range, 0.4, moves abruptl toward the centerline in the region near the bubble. In conclusion, the fluid in the front of the stagnation point shows the transition phenomena when increases. The stagnation point moves downstream along the centerline to the bubble tip, and then moves further downstream along the bubble profile. In the recirculation region, the fluid moves graduall toward the centerline, turns around and then flows upstream. In order to reduce the loading on calculation, man previous studies such as Co [6], added an inverse bubble velocit, U, in the flow to set the bubble fied in the calculation domain. Co[7] used same concept to measure the velocit profile in order to verif the coincidence of eperiments with the theoretical and numerical simulations. The present stud also adopts the same concept to simplif the computations. In addition, the bubble is assumed to etend downstream to infinit ( ). One can set a zero velocit at the far downstream section, i.e., the viscous fluid either adheres to the channel wall or moves upstream awa from the bubble. As a result, if a constant bubble velocit U is added back, which means to subtract the velocit presents in Fig. 4(a-e) with the bubble velocit, the fluid onl moves upstream. The concept was also used in Mavridis et al. [11] to investigate the fountain flow phenomenon in the airassisted polmer injection process. Figures 6(a-e) show the results of adding U to the velocit distributions presented in figures 4(a-e), reectivel. These figures show that the velocit distribution in the front of the bubble tip, > 0, is less affected b the bubble, the velocit distribution are the same in various cross section along -coordinate under same value. The velocit variations are evident in the region of < 0. This also indicates that the epelling effect of the bubble to the viscous fluid flow fields is onl a local effect. Furthermore, the velocit distributions in the region from the bubble tip ( = 0 ) to downstream section ( = 3) can be divided into two patterns. One is in the region surrounded b the downstream section, bubble profile and channel wall, in which shows a nearl stagnant flow, and the other is near the bubble tip where shows a bubble driven flow. The stagnant range decreases while increases. Figures 7 and 8 present the distribution of ISSN: Issue 1, Volume 4, Januar 009

6 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo streamline with different Renolds numbers under constant value in complete bpass flow (</3) region and recirculation flow ( /3) region, reectivel. In Figure 7, the increasing of Renolds number doesn t have notable effect on the flow field in complete bpass flow region. In Figures 8, as Renolds number increases, the maimum value increases, and the recirculation zone near the bubble tip become bigger, too. The range of the recirculation flow region increase as the Renolds number further increases, this shows Renolds number has notable effect on the recirculation flow and the maimum value. The velocit distributions of Figure 8 in both moving frame of reference and fied frame of reference are shown in Figure 9 and Figure 10, reectivel. The velocit distributions in Figure 9 are similar to those of creeping flow shown in Fig. 4(e). In Figure 10, fast fluid particles in the central region of the fluid flow will induce the fluid near the channel wall to move toward the central core and upstream region. This tendenc increases while the Renolds number increases. In conclusion, the variation of the Renolds number does not have notable effect in the complete bpass flow. But in the recirculation flow, the fluid near the bubble tip is epelled to move toward the core region while the Renolds number increases. 3. The relations between and the location of stagnation point, maimum value Based upon Pitts [8] which reported that the eperimental results and theoretical profiles agree well under the condition 0.77, thus the present results all calculated until the reaches the maimum value 0.8 to avoid deviation from the eperimental results. Figure 11 depicts maimum streamline value as a function of for various Re values. The value of ma in Figure 11 is larger than zero and it means the recirculation flow occurred. If ma is less than zero, the bpass flow shown. It is found that ma converges to the same values ( 0.685) for all Re values, then it deviates obviousl as increasing value. The higher value and the higher Re values results in the higher ma value. In the case of increasing, the flow rate at far downstream gets decreasing, so the losing mass flow rate in the far downstream should be supplied from the far upstream. Furthermore, ma is monotonous increasing while the Renolds number increasing at the same value and the ma gets closer to each other at higher Renolds number ( 00 Re 300 ) and deviates for lower Re. Figure 1(a) shows the location of stagnation point on -direction versus the, and Figure 1(a) presents the stagnation point correonding to bubble tip locates on the centerline ( > 0 ). In Figure 1(a) the stagnation point moves from = 3 toward the bubble tip area along the centerline for all Re as the increment deviates from = /3. In other words, the increment of stimulate the movement of stagnation point from = 3 to = 0 is not over 0.0 ( /3< < ) in creeping flow region, and in relative high Renolds number region ( Re >100 ) the stagnation point is ver sensitive to an increment of, a small increment of will cause stagnation point to move toward the bubble tip area. The situation of stagnation point moves in the range 0 3 is a transient state between complete bpass flow and recirculation flow. It is found that the at = The relation between and with various Re in the tpical recirculation flow region ( /3 ) is shown in Figure 10(b). The value of decreases when or Re increases. In addition, the value of decreases almost linearl while increases with fied. The stagnation point moves further downstream awa from the bubble tip (i.e., is more negative) while the Renolds number increases with fied. A quasi-linear relation between and is found in the recirculation flow region. The relation between the -coordinate (i.e., ) of the stagnation point and is shown in Figure 13. The solid line represents the -position of the streamline = 0 on the far upstream section with various. All the other lines represent the relation between and with various Renolds number. The value of increases while Re or increases, but the gradient of curves decreases while increases. Comparing to the -coordinate ( ) of = 0 on the far upstream section, the difference between them decreases while decreases. Also, the difference between and ISSN: Issue 1, Volume 4, Januar 009

7 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo of = 0 decreases while Re increases with fied. 4 Conclusion The stud not onl verifies the flow pattern proposed b previous studies but also detailed investigates the third flow pattern between the complete bpass flow and recirculation flow. It shows that the third flow pattern onl observed in a small range ( /3< < ) and low Renolds number. In a moving frame of reference, it is found that the velocit near the front of bubble tip decreases while the increases until the reaches the /3, and as the is larger than /3 the velocit increases (toward the upstream). The stagnation point divides the fluid flow in two directions: flow toward the upstream or downstream. When the Renolds number increases, the velocit near the bubble tip increases and the affection in the region is similar to the case of high in creeping flow. In a fied frame of reference, it is found that the velocit is onl affected in a small region in front of bubble tip. The velocit in the region from bubble tip ( = 0 ) to the downstream ( = 3) can be divided into two regions, one is the velocit shows nearl stagnant in the downstream and the other is the velocit shows the obviousl driven b the bubble near the bubble tip. The stagnant range decreases as the increases, and it also verifies the real phsical phenomenon of zero velocit at the far downstream. As the Renolds number increases, the fluids near the bubble tip epelled to move toward the centerline and the velocities increase in recirculation flow region. But the increasing of Renolds number does not have notable effect on the flow field in complete bpass flow region. Increasing of Renolds number, the affection of inertial forces, the range of the recirculation flow region increases for the same value and the range of in the transient flow becomes smaller, but no notable affections were found in the complete bpass flow. When the or Re increases, the ma and get increasing and the gets decreasing. Nomenclature: H The half distance between the two parallel plates U The constant velocit of the bubble n t The normal unit vector on the bubble interface The tangential unit vector on the bubble interface The aial direction in coordinate sstem The radial direction in coordinate sstem Greek letters α The over-relaation factor θ The angle between the normal of the interface and the aial direction The ratio of asmptotic bubble width to half distance of the two parallel plates Stream function ω Vorticit Dimensionless parameters Re Renolds number Dimensionless form Subscript i j s The number of grid in aial direction The number of grid in radial direction Solid wall Stagnation point Acknowledgment The Authors are grateful for the financial support provided b the National Science Council, NSC-97-1-E References: [1] F.P. Fairbrother, and A.E. Stubbs, Studies in electroendosmosis. Part VI. The bubble tube method of measurement, J. Chem. Sci. Vol. 1, 1935, pp [] C.H. Hsu, P.C. Chen,.Y. ung and G.C. uo, Pressure Field in Front of Long Gas Bubble in a Fluid Filled Channel, WSEAS ISSN: Issue 1, Volume 4, Januar 009

8 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo TRANSACTIONS ON MATHEMATICS, Vol.5, pp (006). [3] G.I. Talor, Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech. Vol. 10, 1961, pp [4] F.P. Bretherton, The motion of long bubbles in tubes, J. Fluid Mech., Vol.10, 1961, pp [5] H.L. Goldsmith, and S.G. Mason, The movement of single large bubbles in closed vertical tubes, J. Fluid Mech., Vol.14, 196, pp [6] B.G. Co, On driving a viscous fluid out of a tube, J. Fluid Mech.,Vol.14, 196, pp [7] B.G. Co, An eperimental investigation of the streamlines in viscous fluids epelled from a tube, J. Fluid Mech., Vol.0, 1964, pp [8] E. Pitts, Penetration of gluid into a Hele-Shaw cell: The Saffman-Talor eperiment, J. Fluid Mech., Vol.97, 1980, pp [9] C.H. Hsu, P.C. Chen,.Y. ung and C. Lai, mpact of the Ratio of Asmptotic Bubble Width to Radius of Circular Tube and Renolds Number in a Gas Bubble Driven Flow, Chem. Eng. Sci., Vol. 60, p [10] C.H. Hsu,.Y. ung, P. C. Chen and S. D.Huang, The Gas-assisted Fluid Flow Epelled in Front of a Long Bubble in a Circular Tube, Trans. of the CSME, Vol. 30 No. 1, 006, pp [11] H. Mavridis, A.N. Hrmak, and J. Vlachopoulos, Finite element simulation of fountain flow in injection molding, Polm. Eng. Sci. 6, 1986, Bubble Profile Δ d Δ Figure. Diagram of the uneven grid near the interface Channel wall A B = H n E θ Bubble profile t C = 0 F = 3H D = 3H Figure 3. Distributions of streamline with different and Re = 0 Figure 1. Diagram of the gas bubble steadil epelled viscous fluid and the coordinate sstem ISSN: Issue 1, Volume 4, Januar 009

9 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo Fig. 4 Distributions of velocit in the moving frame of reference with different and Re = 0 Fig. 6 Distributions of velocit in the fied frame of reference with different and Re = 0 Fig. 5 Distributions of velocit near the stagnation point in the moving frame of reference with different and Re = 0 Figure 7. Distribution of dimensionless streamlines for various Renolds numbers in complete bpass flow region ISSN: Issue 1, Volume 4, Januar 009

10 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo Figure 9. Distribution of velocit of the recirculation flow in the moving frame of reference with different Re Figure 8. Distribution of dimensionless streamlines for various Renolds numbers in recirculation flow region Figure 10. Distribution of velocit of the recirculation flow in the fied frame of reference with different Re Re=0 Re=50 Re=100 Re=00 Re=50 Re= Figure 11. versus the maimum ma value with ISSN: Issue 1, Volume 4, Januar 009

11 C. H. Hsu, P. C. Chen,. Y. ung, G. C. uo various Re -coordinate (a) Re=0 Re=10 Re=5 Re=50 Re=100 Re=300 -coordinate upstream Re=0 Re=10 Re=5 Re=50 Re=100 Re= Figure 13. versus the location of stagnation point on -direction with various Re coordinate Re=0 Re=10 Re=5 Re=50 Re=100 Re= (b) Figure 1. versus the location of stagnation point on -direction with various Re ISSN: Issue 1, Volume 4, Januar 009