Flow behavior of stratified viscoelastic laminar flow at junctions with different angles.
Interfacial phenomena between stratified multiple layers of flows with viscoelastic characters are a major concern for manufactures of film and sheet. Particularly, the deformation characteristics of the interface and flow behavior in the vicinity of stratification junctions have been topics of recent research (1-9). Both analytical and numerical methods (10-15) have been used in attempts to explain flow phenomena associated with the interfacial problems of stratified layers. In particular, recent reports (10-14) reveal the importance of interfacial instabilities, and the effects of geometric configurations, material properties and flow conditions have been studied extensively in order to give optimum co-extrusion processes.
Although some experimental and analytical results are available, they were usually carried out in a low Reynolds number region with limited test conditions. In order to compare and certify numerical results, unified and coherent experimental data are essential.
In this report, we extend our previous work (9) to include the 90[degrees] junction angle. Particularly in the present study, our attention is focused on the effect of the viscous ratio between two stratified laminar layers. In order to provide systematic data on flow behavior in the vicinity of the stratification junction, flow modes were obtained with the aid of a high-resolution video camera for typical two-dimensional flow channels with three junction angles ([theta] = 30[degrees], 60[degrees] and 90[degrees]). Test fluids used as model fluids in the present investigation include machine oil as a Newtonian fluid at the upper layer and polyacrylamide (PAA) water solutions as non-Newtonian fluids at the lower layer.
As depicted in Fig. 1, the experimental apparatus primarily consists of a piston with a cylinder [(3) and (4)], which extrudes each test fluid to the test section of the Y junction (12), in which two laminar stratified layers are formed. Test fluids are fed into the experimental apparatus through filler tanks (6) so that before the experiment, the two test fluids form a stationary stratified layer in the test section, where the level of the interfacial surface coincides with the edge of the junction (the reference line), as shown in Fig. 1. The test fluids, each of which is contained in a separate cylinder, are driven to the test section at different speeds by pushing the piston rods with electric motors (1). In order to obtain consistent data, the pressure condition of the test section is controlled and unified with the level control tank (8), where the pressures of the two test fluids are the same at the exit of the test section (each fluid is allowed to overflow at the overhead tank (9) at the flow rate se t a priori). Details of the test section are schematically shown in Fig. 2. (typically for [theta] = 90[degrees] junction). The geometric configuration is similar to that of an actual industrial film coextrusion channel, whose dimensions are scaled up with a direct ratio to the present test section. The inlet width of the upper layer (layer I) is 12 mm and the width of the lower layer (layer II) is 14 mm. The two inlet channels meet at the junction knife edge with an angle of [theta] = 30[degrees], 60[degrees] or 90[degrees] (three types of test section are considered in the present investigation). The stratified layers are then formed after the junction, toward the downstream of the channel. The width of channel after the junction is 26 mm so that before the experiment, the thickness of stationary layer I is 12 mm and the thickness of stationary layer II is 14 mm from the reference line. The x - y coordinate system is adopted as shown in Fig. 2, where x is the flow direction from the junction (knife edge) an d y is the direction orthogonal to the reference line. An adequate inlet channel length (for each fluid layer) is provided in order that prevailing flow is always fully developed at the junction, i.e., in the case of Newtonian flow (for layer I), two-dimensional Poiseuille flow persists. It is noted here that the depth of the channel is 300 mm and flow in the middle of the channel depth can be assumed as two-dimensional. The test section is made of transparent acrylic resin to allow visualization of the flow. The pressure at the junction of the inlet channels is kept equal for channels of [theta] = 30[degrees], 60[degrees] and 90[degrees] because the pressure at the exit of the channels is unified, as shown in Fig. 1.
In the experiments, flow visualization's for prevailing flow modes were made only after the flow state had reached steady state after the startup of piston movement. In the case of flow visualization, a small amount of aluminum oxide powder was added to the test fluids. A slit light was focused on the middle of the channel depth in the test section, where the flow behavior was recorded by a high-resolution video camera (10) shown in Fig. 1. By monitoring the image of the recorded data, flow modes were then determined.
Fluids and Experimental Procedure
The reference Newtonian fluid in layer I is mineral oil, as indicated in Table 1, and hereafter is called fluid A. In layer II, however, various types of fluids are tested, including Newtonian and viscoelastic fluids.
In the present study the usage of Newtonian fluid in layer I would give unified and coherent experimental data, avoiding complicated viscoelastic-viscoelastic interaction at the interface and simplying the problem. The Newtonian fluids in layer II are 80.2 wt%, 87.0 wt% and 91.2 wt% aqueous solutions of glycerin; these fluids are called fluid B, fluid C and fluid D. Thus, a combination of fluid A and fluid B (fluid C or fluid D) in layer I and layer II, respectively, gives Newtonian-Newtonian stratified laminar layers. The viscoelastic fluids in layer II are PAA-water solutions (Sanyo, Sunfroc AA-300p, [M.sub.n] [approximately equal to] 1.2 X [10.sup.7]) with concentrations of 500, 1000, 1500 and 2000 ppm, hereafter, called fluids E, F, G and H, respectively. The viscosity of the Newtonian fluid (fluid C) is adjusted so that the zero shear viscosity ([[eta].sub.0] = 0.132 Pa*s) of the viscoelastic fluid (fluid E) can be matched with the Newtonian fluid viscosity shown in Table 1. In order to characterize the viscoelastic fluids (fluid E-fluid H), the shear-thinning viscosity [eta]([gamma]) and the first normal stress difference [N.sub.1]([gamma]) are measured against shear rate [gamma] using a cone and plate rheometer. The rheological data (shear-thinning viscosity) are then fitted by the power law model as,
[eta]([gamma]) = K[[gamma].sup.(n-1)] (1)
where K and n are rheological parameters. The rheological data (shear-thinning viscosity [eta], first normal stress difference [N.sub.1], storage modulus G' and loss modulus G") are also fitted by the Kaye-Bernstein-Kearsley-Zapas (K-BKZ) model. In the present study, a separable K-BKZ type constitutive equation proposed by Larson et al. (16) is used for dealing with viscous as well as elastic characteristics of flow. In the constitutive equation, a zero value of the second normal stress difference is assumed and the solvent stress is considered as an additional Newtonian stress. The constitutive equation is written as,
[FORMULA NOT REPRODUCIBLE IN ASCII] (2)
where [C.sup.-1] is the Finger strain tensor, I the unit tensor, [I.sub.1] = trace([C.sup.-1]), [gamma] the rate-of-strain tensor, and [[eta].sub.s] the solvent viscosity. The functions that appear in the integral are as follows (16),
[FORMULA NOT REPRODUCIBLE IN ASCII] (3)
h([I.sub.1]) = [[1 + 0.2([I.sub.1] - 3)].sup.-t], (4)
where [G.sub.0] is the zero-time elastic modulus, [[lambda].sub.1], the longest relaxation time constant, [p.sub.ve] and [zeta] the model constants (16). Thus, once the parameters [p.sub.ve], [zeta], [G.sub.0], [[lambda].sub.1], and [[eta].sub.s] are determined, the fluid can be rheologically characterized. The constitutive Eq 2 includes the deformation history and has the advantage to characterizing a solvent with a single relaxation time constant (maximum time constant in the spectrum of relaxation times). It should be mentioned that the linear viscoelastic memory function M(t-t') of Eq 3 can be treated as identical to the form proposed by Segalman (cross reference in Ref.17; pp.285 and 444), setting [[lambda].sub.1] = [[lambda].sub.1], [p.sub.ve] = v and [G.sub.0] = [[eta].sub.0]/[[lambda][GAMMA](1 - v)]. It is also interesting to note that the damping function h([I.sub.1]) of Eq 4 can be replaced by h([I.sub.1]) = [([I.sub.1]/3).sup.-[zeta]] according to the suggestion by Segalman. Although many possible e xpressions for M(t-t') and h([I.sub.1]) exist (17), we shall use the expressions of Eq 3 and 4 without significantly altering the rheological characterization of the model viscoelastic fluid. The representative data (fluid H) fitted by the power low and the K-BKZ model are depicted in Fig. 3a for the shear-thinning viscosity and the first normal stress difference, and in Fig. 3b for the storage modulus and the loss modulus only by the K-BKZ model. In Table 1, the density [rho] and the zero shear viscosity are tabulated for all test fluids. In Table 2, rheological parameters of the K-BKZ model (Eqs 2-4) are listed, and in Table 3 rheological parameters of the power low model (Eq 1) are listed. Noted that the relaxation time constant [[lambda].sub.1] (the maximum relaxation time constant) and other parameters in Table 2 are determined by fitting all rheological data ([eta], [N.sub.1], G' and G") for fluids E, F, G, and H.
In presenting results obtained from experiments, the following nondimensional parameters are considered:
[Re.sub.I] = [[rho].sub.I][d.sub.I][v.sub.I]/[[eta].sub.I], [Re.sup.*.sub.II] = [[rho].sub.II][d.sup.n.sub.II][v.sup.(2-n).sub.II]/K, (5)
[De.sub.II] = [[lambda].sub.1][v.sub.II]/[d.sub.II], (6)
[alpha] = [[eta].sub.I]/K[([v.sub.II]/[d.sub.II]).sup.(n-1)], (7)
where suffixes I and II Indicate layers I and II, respectively. The Reynolds number Re is a measure of the ratio of inertial force to viscous force and the Deborah number De is a measure of the ratio of elastic force to viscous force. The use of the nondimensional parameter group ([Re.sub.I], [Re.sup.*.sub.II] and [De.sub.II]) is based on the fact that only Newtonian fluid ([De.sub.I] = 0) is used in layer I and for each layer the inertia effect is taken into account, assuming the law of similarity in hydrodynamics must hold for a prevailing flow state. In the present study, we particularly investigated the effects of the viscous ratio [alpha]; [[eta].sub.I]/[[eta].sub.II] between the two stratified layers. [alpha] is thus defined in Eq 7 for layer II from the power law fluid (Eq 1). The experimental conditions adopted in the experiments are (i) Layer I; 0.1 [less than or equal to] [Re.sub.I] [less than or equal to] 100, [De.sub.I] = 0 (ii) Layer II; 0.01 [less than or equal to] [Re.sub.II] [less than or equal to] 10, 0 [less than or equal to] [De.sub.II] [less than or equal to] 44.0. and 0.000644 [less than or equal to] [alpha] [less than or equal to] 0.0414. Three junction angles [theta] of the test section (i.e., [theta] = 30[degrees], 60[degrees] and 90[degrees]) are employed in order to verify the geometric dependence of the flow behavior.
RESULTS AND DISCUSSION
States of flow are classified as flow modes, taking into account the direction of interfacial deflection and the instability of the interfacial surface. As indicated in Table 4-1 and Table 4-2, six distinct flow modes are obtained from the results of flow visualization within the experimental conditions. In Table 4-1 and Table 4-2, representative visualization results are also displayed together with sketches. It is noted here that the deflection of the interface, either toward layer I or layer II (i.e. Mode 1 or Mode 5 respectively), is defined as when the position of interface reaches either + 1 mm or -1 mm from the datum position (Mode 3), when the flow has reached a stable and time-independent condition. The wavy interface is defined as Mode 2, Mode 4 and Mode 6, which occurs in the basic Mode 1, Mode 3 and Mode 5, respectively; Mode 2, Mode 4 and Mode 6 are judged when the position of interface is retained above [+ or -]0.5 mm amplitude after a sufficiently long time has elapsed from the start of flow st ratification. Wavy interfaces observed in the flow visualization are the nonlinear waves, which occur at the edge of the Junction and propagate toward downstream. In addition to Mode 1 to Mode 6 in Table 3, which are chiefly classified from viewpoint of states for the interfacial surface, further detailed modes are classified in Table 5. As indicated in Table 5, two additional modes are determined, based on the flow
behavior in the vicinity of junctions. Mode A, is characterized for a vortex zone that appears in the basic Mode 2 (Table 3). In Mode A, the vortex zone exists near the edge of the junction when the fluid in layer II bends toward the layer I at sharp angle. Mode B is characterized for a vortex zone at the right angle corner of layer I, and this is usually termed the hydrodynamic flow separation. Note that Mode B has another unique feature at higher Junction angle, such as [theta] = 90[degrees]: fluid in layer I invades the channel of layer II. It is mentioned here that Mode A and Mode B are strong ly dependent on the Junction angle. This will be discussed in more detail in the succeeding sections.
Flow and Interfacial Characteristics
In Fiq. 4, flow and interfacial characteristics are represented as mode map for the channel with the junction angle of [theta] = 90[degrees].
Note that the lines drawn in Fig. 4 to Fig. 8 are hand-drawn interpolation lines between mode-data. Although the boundary (lines) of mode-zones in the mode map are vague (no critical value is measured from the visual data), important information of prevailing flow and interfacial characteristics is obtained. In Fig. 4, as has been previously stated, experimental results for [theta] = 90[degrees] are particularly concerned in the present investigation in comparison with those of [theta] = 30[degrees] and [theta] = 60[degrees]. In Eq. 4a and b, Newtonian fluid (Fluid C) and viscoelastic fluid (Fluid E), which has a common zero shear viscosity with Fluid C, are used in layer H. The interfacial modes (Mode 1 to Mode 6) are plotted for [Re.sub.I] and [Re.sup.*.sub.II]. Note that in Eq. 4b the depicted mode points imply the range of Deborah number, 0.0439 [less than or equal to] [De.sub.II] [less than or equal to] 16.5. In Eq. 4a and b, quite similar mode maps are evident, indicating that effect of elasticity (which Is represented by De) is minimal on the interfacial mode in the case of [theta] = 90[degrees]. It is thought that this is because the die swell effect (which can be caused by the elasticity) in layer II is largely suppressed by descending inertial force (perpendicularly facing toward layer II) of the fluid in layer I. The deceleration momentum toward the layer II, due to right angle bend of fluid in layer I, may minimize the effect (the die swell effect) of elasticity in layer II. Also, it is speculated that the die swell effect would be small in the stratified flaw situation, while the flow with the free surface shows large surface deformation (expansion) (18). Although very small in region in the mode map, the wavy modes (Mode 2, Mode 4 and Mode 6) have little influence on elastic instability at high [Re.sub.I] and [Re.sup.*.sub.II] as shown in Fig. 4b, where Mode 4 occupies a slightly wider region in the mode map for viscoelastic fluids. Based on the observations obtained for the interfacial modes in the case of [theta] = 90[degrees], we concentrate our attention on viscous effects in the succeeding discussions, regarding that the elastic effect is small (except for a small effect of elastic instability). It should be stated that the elastic effects on the interfacial characteristics (when the interfacial surface modes and flow behavior are combined) are not to be neglected, particularly for lower junction angles, such as [theta] = 30[degrees] and 60[degrees]. This is due to the descending interfacial force, which is smaller in lower junction angles. Thus, the elastic effect would become evident, particularly for the unstable wavy modes (9). However, in the present study, we focus our attention on the viscous effect (presented as viscous ratio [alpha]) in reference to the case of [theta] = 90[degrees].
In Fig. 5, interfacial modes are displayed as mode maps for the junction of [theta] = 90[degrees], where the maps are plotted for a versus [Re.sup.*.sub.II] (Fluid B - Fluid H in layer II).
In Fig. 5a to d, each graph present's when the Reynolds number in layer I is kept constant. As seen in Fig. 5a, when [Re.sup.*.sub.I] = 0.1, three modes (Mode 1, Mode 2 and Mode 3) appear, where the interfacial deflection persists toward layer I. The wavy mode (Mode 3) exists when the Reynolds number of layer II is increased. This is due to surface instability. As the Reynolds number of layer I is increased, likewise (a) [right arrow] (b) [right arrow] (c) [right arrow] (d) in Fig. 5, interfacial modes develops. Particularly as seen in Fig. 5c, four modes can coexist, showing the complicated structure of hydrodynamic transitions. However, when [Re.sub.I] becomes higher, as shown in Fig. 5d. the interfacial modes show a similarity with Fig. 5a, where the position of the deflected interfacial surface becomes opposite to that of Fig. 5a. In consideration of [alpha], it was found that a higher viscous ratio tends to cause the surface instability, resulting in the generation of the wavy surface mode (Mode 2, Mode 4 and Mode 6).
In a similar manner, the interfacial modes are displayed in Fig. 6 and Fig. 7 for the junction angles of [theta] = 60[degrees] and [theta] = 30[degrees], respectively, considering the viscous ratio as control parameter similar to the case of [theta] = 90[degrees]. Comparing Figs. 5, 6 and 7, the angle dependences on the interfacial modes are so small that the modes are well represented by [alpha], [Re.sub.I] and [Re.sup.*.sub.II]. Note that wavy modes can be affected by the elastic effect (comparing Fig. 5c, Fig. 6c and Fig. 7c), as mentioned earlier. However, the general trends for the similarity of the interfacial modes persists without angle dependency, when represented by the combination of [alpha], [Re.sub.I] and [Re.sup.*.sub.II]: viscous and inertia effects.
It is thought that short-wavelength instability (1) may appear in the case of a melt-melt system of a coextrusion process (i.e. the short-wavelength wavy modes may dominate the unstable interfacial deformation states), whereas only the long-wavelength wavy modes seen in the flow mode classification (Table 4) persist since the interfacial tension in the present surface configuration (9) is quite high compared with a typical melt-melt coextrusion system of polymers. Thus, It is speculated that in the case of lower surface tension, such as the melt-melt system of polymers in a coextrusion process, the regions of the wavy interface modes may retreat the direction of decreasing Reynolds number (9), appearing as new short-wavelength wavy modes.
In Fig. 8, verification of Mode A and Mode B, which represent flow behavior in the vicinity of junctions, are displayed. In order to indicate the trend of occurrence of flow phenomena, mode maps are summarized in terms of [Re.sub.I] and [Re.sup.*.sub.II]. As seen in Fig. 8b, Mode B appears when the Reynolds number of layer I becomes high. Thus it is not difficult to speculate that the vortex zone is generated by the flow separation (at the right angle corner) hydrodynamically. It is interesting to note that the vortex zone, which is usually unwanted in the coextrusion process, can be overcame by increasing the Reynolds number of layer II after passing the wavy mode (Mode 4). Interfacial modes (as well as the flow Mode B) have less dependency on the fluid elasticity of layer II.
In Fig. 8a, similarly, the region of occurrence for Mode A is indicated. The general trend is that Mode A occurs at a high Reynolds number [Re.sup.*.sub.II] (with a relatively small Deborah number) at [theta] = 30[degrees]. It is further noted that an increase of the Deborah number (using higher-concentration polymer fluids) causes the Mode A region to disappear, indicating that the appearance of Mode A is largely caused by the viscous and inertia effects and is suppressed by the elastic effect. No flow behavior such as Mode A and Mode B was observed in the case of junction angle [theta] = 60[degrees]. Although no detailed study has been conducted of the reasons (the mechanism for the occurrence of modes A and B), there would be a simple explanation from the experimental verification, as explained previously in Table 5. Since mode A is characterized by a sharp (low) angle junction while mode B is characterized by a right (high) angle junction, these modes might not be induced for a mid-angle junction such as [theta] = 60[degrees].
Further study should investigate these flow characteristics of modes A and B. However, this would reveal the limitations of the present study, and we hope to report further works solving these problems, adapting numerical and analytical methods in future publications.
An experimental study was conducted in order to examine the interfacial surface modes and flow behavior in the vicinity of the junction for two stratified laminar layers. A Newtonian fluid in layer I and Non-Newtonian fluids (including Newtonian also) in layer II are used as model fluids. Particular attention was paid to the Junction angle of [theta] = 90[degrees] in the present study. Six basic interfacial modes with two flow behavior modes were observed. The elastic effect of layer II was found to be small in the interfacial modes. Detailed mode maps were obtained by using the viscous ratio and two Reynolds numbers of both layers. It was revealed that the angle dependency for the interfacial modes are minimal, except for same limited wavy and flow modes, and can be well presented by the choice of the viscous ratio and Reynolds numbers.
This work was partly supported by a grant-in-aid for Scientific Research (C) from the Ministry of Education, Japan.
(1.) B. Kohmami, J. Non-Newtonian Fluid Mech., 37, 19 (1990).
(2.) G. Sornberger, B. Vergnes. and J. F. Agassant, Polym. Eng. Sci., 26, 455 (1986).
(3.) G. Sornberger, B. Vergnes, and J. F. Agassant, Polym. Eng. Sci., 26, 682 (1986).
(4.) S. Puissant, B. Vergnes, Y. Demay. and J. F. Agassant, Polym. Eng. Sci., 32, 213 (1992).
(5.) H. Mavridis and R. N. Shroff, Polym. Eng. Sci., 34, 559 (1994).
(6.) J. F. Agassant and A. Fortin, Polym. Eng. Sci., 34, 1101 (1994).
(7.) K. Matsunaga, T. Kajiwara, and K. Funatsu, Polym. Eng. Sci., 38, 1099 (1998).
(8.) J. Perdikoulias and Tzoganakis, SPE Antec Tech. Papers, 43, 351 (1997).
(9.) H. Yamaguchi, T. Yasumoto, and H. Yamamoto, J. Rheol., 43, 1373 (1999).
(10.) J. Perdikoulias, C. Richard, J. Vlcek, and J. Viachopoulos, SPE Antec Tech. Papers, 37, 2461 (1991).
(11.) J. Vlcek, J. Perdikoulias, and J. Viachopoulos. SPE Antec Tech. Papers, 39, 3365 (1993).
(12.) A J. Rincon, A. N. Hrymak, and J. Vlachopoulos, Int. J. Numer. Meth. Fluids, 28, 1159 (1998).
(13.) H. K. Ganpule and B. Khomami, J. Non-Newtonian Fluid Mech., 80, 217 (1999).
(14.) H. K. Ganpule and B. Khomami, J. Non-Newt. Fluid Mech., 81, 27 (1999).
(15.) H. Yamaguchi, A. Mishima, T. Yasumoto, and T. Ishikawa, J. Non-Newtonian Fluid Mech., 89, 251 (2000).
(16.) R. G. Larson, S. J. Muller, and E. S. G. Shaqfeh, J. Non-Newtonian Fluid Mech., 51, 195 (1994).
(17.) R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, vol. 1, John Wiley Ai Sons, Inc. (1987)
(18.) X.-L. Luo and E. Mitsoulis, Adv. Polym. Tech., 10, 47 (1990).
[Figure 1 omitted]
[Figure 2 omitted]
Table 1 Properties of Test fluids. Test fluid Layer Density [rho] (kg/[m.sup.3]) Fluid A Mineral oil I 832.0 Fluid B Glycerin 80.2wt% II 1209 Fluid C Glycerin 87.2wt% II 1227 Fluid D Glycerin 91.2wt% II 1238 Fluid E PAA 500 ppm II 998.2 Fluid F PAA 1000 ppm II 998.7 Fluid G PAA l500 ppm II (*) 999.2 Fluid H PAA 2000 ppm Il 1006 Zero shear viscosity [[eta].sub.0] (Pa*s) Fluid A 0.002 Fluid B 0.053 Fluid C 0.132 Fluid D 0.212 Fluid E 0.132 Fluid F 0.687 Fluid G 1.87 Fluid H 3.19 Table 2 Rheological Parameters of K-BKZ Model. Viscosity of Maximum solvent relaxation time Test fluids [[eta].sub.s](Pa*s) [[lambda].sub.1](s) Fluid E PAA 500 ppm 0.001 6.50 Fluid F PAA 1000 ppm 0.001 7.25 Fluid G PAA 1500 ppm 0.001 9.25 Fluid H PAA 2000 ppm 0.001 9.40 Model Parameters [P.sub.ve] [zeta] Fluid E 0.575 0.375 Fluid F 0.425 0.500 Fluid G 0.350 0.550 Fluid H 0.340 0.550 Table 3 Model Parameters (Power-Law Model). Test fluid K(Pa*[s.sup.n]) n Range of shear rate (1/s) Fluid E PAA 500 ppm 0.078 0.517 0.5-100 Fluid F PAA 1000 ppm 0.286 0.430 0.4-100 Fluid G PAA 1500 ppm 0.657 0.334 0.3-100 Fluid H PAA 2000 ppm 1.14 0.233 0.3-100
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|Author:||Yamaguchi, Hiroshi; Yamamoto, Hideaki|
|Publication:||Polymer Engineering and Science|
|Date:||Jan 1, 2002|
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