Viscoelastic Flow into a Slit at

Transcription

1 Viscoelastic Flow into a Slit at Low Reynolds Numbers Part 2 Velocity Profiles By Kiyoji Nakamura, Yasuhiro Yamamoto, Fujio Nakamura and Akira Horikawa, Members, TMSJ Department of Mechanical Engineering, Osaka University, Suita, Osaka Besed on the Journal of the Textile Machinery Society of Japan, Transactions, Vol. 29, No. 11, T (1976) Abstract When melted polyethylene liquid having long relaxation time flows from a reservoir into a slit, the circulating secondary flow is observed near the slit even at low deformation velocity. In the present paper, the velocity of main flow and that of circulating secondary flow were traced with aluminium powder and the influence of the slit width and the slit length on flow behaviour was investigated. The results are as follows: Significant difference is found between the velocity of main flow and that of circulating secondary flow, and the circulating velocity is very small. The velocity at the boundary area between main and circulating secondary flows varies continuously, and the velocity of circulating secondary flow is the heighest at the boundary area. The region of circulating secondary flow varies a little according to the change of the slit width or of the slit length. Flow behaviour is, however, more influenced by deformation velocity than by the two factors above mentioned. 1. Introduction In the previous paper~l1, flow patterns were investigated by changing deformation velocity, temperature, melt-index or wall angle, when low-density polyethylene melt flowed from a reservoir into a slit at low Reynolds numbers. The circulating secondary flow near the slit was also observed with small melt-index polyetylene. From these flow patterns, however, it was impossible to estimate the velocity. So the velocity measurement may be useful for understanding viscoelastic flows. L. H. Drexler et al. 21 measured in 1973 the velocity of melted polyethylene flow near a slit, but he did not study the circulating secondary flow caused by viscoelasticity and only measured the radial flow into a slit. In the present paper, we measure the velocity of small melt-index polyethylene flows having large relaxation time at low Reynolds numbers; they have circulating secondary flows. The measurement will make clear the velocity difference between the main flow and the circulating secondary flow. The velocity measurement is carried out by tracing small particles mixed in polyethylene melt. In the previous paper, the influence of wall angle on the flow pattern was described. In the present paper, the influence of the slit width or length on circulating secondary flows is investigated. 2. Experimental Apparatus and Procedure 2.1 Extruder The extruder used was the same that appeared in the Vol. 24 No.1(1978) previous paper', but the heat-resistant glass was modified : A sheet of black paper was put on the face of the back window not to reflect the light. The front side of the die was painted white to take clear pictures. The velocity was measured by the movement of aluminium powder in reflected light. The black paper was renewed every time the experiment was over because the apparatus was often heated up to 260 C. The polymer was premolded into pieces, 82 mm x 10 mm x 500 mm, so that they would slide easily into the reservoir. The same polymer but containing a very small amount of aluminium powders was extruded through a 1 mm diameter hole, and the filaments thus obtained were passed through holes drilled transversely on the center line of the polymer sample at intervals of 5 mm. These filaments were also threaded along the flow direction at the tip of the polymer sample where circulating secondary flows were observed. The content of aluminium powder was as small as possible within the limit of velocity measurement so as not to interfere the polyethylene flow. 2.2 Apparatus of a light source A slide projector was used for a light source. A circular plate having a hole was put between the projector and the die and was rotated with a motor at a constant rate so that the die was illuminated intermittently. These details were shown in Fig. 1. A camera was placed in front of the die to which the light was cast obliquely not to hinder to take pictures. 2.3 Experimental method 1

2 The extruder was operated in the same way as in the previous paper~i1, and so its explanation is omitted here. Velocity measurement was carried out by tracing the movement of aluminium powder; we measured its position on the photographic pictures taken at regular intervals. In Fig. 2 is shown one example of the pictures. If the positions of a particle at time t and t+4t were xi (t) and xi(t+4t), the velocity vi is calculated by Vg - xi(t+4t)-xi(t) 4t. The image of a scale set at the same plane as the aluminium powder was used as the length standard. 3. Fully-Developed Region In Fig. 3, 2H and Ware widths of the flow channel, 0 inlet angle, l length of the slit and 2h width of the slit. Suffix 2 indicates the direction of the width of the flow channel (direction of H) and suffix 1 indicates the flow direction. In order to see the accuracy of this method with aluminium powder, we measured the powder velocity in the fullydeveloped region far from the slit, and compared it with the velocity vi calculated theoretically, vi is expressed by the power law model as vi - 2n+1 1- x2 (n+1)in V n+1 H ' where n is flow index, V mean velocity and x2 distance from the center of the flow channel. The flow index here is 0.46 (at melt-index 0.5, 190 C) as was obtained in the previous paper~i~. As shown in Fig. 4, the method is accurate enough for our purpose. The fully-developed region can be obtained if l xi ~/H is larger than 2.5 as will be mentioned in the following section. 4. Velocity Profiles at the Constant Ratio of Slit to Channel Width and with the Constant Slit Length In the previous paperm, flow patterns were investigated at the constant ratio of slit to channel width (h/h) and with the constant slit length 1. We also observed circulating secondary flows in the polyethylene liquid having large relaxation time (melt-index 0.5), but none in liquid having short Fig. 1 Apparatus of a light source (~ voltage stabilizer, 02 Revolution regulator, Electric motor, Reduction gear, 05 Circular plate, Projector) Fig. 3 Schematic diagram of the flow channel 2H=10.2 mm) (W=83mm, Fig. 2 Locus of aluminium powders Fig. 4 Velocity profile between parallel 190 C) plates (melt-index 0.5, 2 Journal of The Textile Machinery Society of Japan

3 relaxation time (melt-indexes 4, 20 and 35). Considering these results, we used polyethylenes having melt-indexes 0.5 and 4 to investigate the velocity profile. Fig. 5 shows velocity profiles having no circulating secondary flows at 190 C, for such polyethylenes as deformation velocity V/H=0.026 s-i and melt-index 4. I xi I/H is the dimensionless distance towards the upper stream from the slit entrance. vi/ V at l xi (/H=1.71 agrees with that at l xi =2.56. This figure shows that the velocity increases at the center, but decreases near the wall when the fluid approaches the slit. Fig. 6 shows dimensionless velocity v2/ V in the direction of x2 axis. Values of v2/ V at l xi I/H=1.71 and 2.56 were not plotted in Fig. 6; these were nearly equal to 0. In the following we show velocity profiles with circulating secondary flows. Fig. 7 shows velocity profiles in xi direction at 190 C for fluids having melt-index 0.5 and deformation velocity v/h=0.032 s-i. In this case too, vi/ V at l xi l/h=1.71 agrees with that at l xi l/h=2.56. The velocity is larger at the center and smaller near the slit than the velocity with no circulating secondary flows. Velocity differences between main and circulating flows are shown in Figs. 8 and 9. Fig. 8 shows the flow at 190 C for fluid having melt-index 0.5 and deformation velocity V/H=0.025 s-i. Fig. 9 shows that when deformation velocity V/H=0.051 s-i. The arrow shows the magnitude of the velocity at each starting point. In both figures, there are large differences between main and circulating flows; the latter is very slow and not constant, reaching the maximum on the boudary of the main flow. The velocity changes continuously on the boundary of main and circulating flows. Fig. 10 shows a velocity diagram at inlet angle 8=110. Compared with Fig. 9, the circulating flow region becomes smaller with incresasing the inlet angle, and the main flow is hardly influenced by its change. Fig. 5 Velocity profiles (melt-index 4,190 C, deformation velocity s-1, ratio of slit to channel width 0.1, dimensionless slit length 1.0) Fig. 6 Velocity profiles (melt-index 4,190 C, deformation velocity s-1, ratio of slit to channel width 0.1, dimensionless slit length 1.0) Fig. 7 Velocity profiles (melt-index 0.5, 190 C, deformation velocity s-1, ratio of slit to channel width 0.1, Vol. 24 No.1(1978) 3

4 Fig, 8 Velocity diagram (melt-index 05, 190 C, deformation velocity s-i, ratio of slit to channel width 0.1, Fig. 10 Velocity diagram (inlet angle 8=110, melt-index 0.5, 190 C, deformation velocity s-1, ratio of slit to channel width 0.1, Fig. 9 Velocity diagram (melt-index 0.5, 190 C, deformanton velocity s-1, ratio of slit to channel width 0.1, Fig. 11 Velocity diagram (ratio of slit to channel width 0.53, melt-index 0.5, 190 C, deformation velocity s'1, 5. Velocity Profiles when Changing Ratio of Slit to Channel Width the flow pattern; ~5, the flow angle into a slit, increases a the ratio of slit to channel width h/h influences a little on In Figs. 11, 12 and 13 are shown flow patterns dimen- little with decreasing h/h. The influence of deformation ~innlpecl i when hip-fl 5~ f Zd and.411 b4 (1 V.1V 1h rpe 1VUpVV411 PrtivPly P111V11 V1,, 14/11 {J1V11IVUUl' V.JJ, 1 1 velocity VIH_ on 0 is much larger than that of h/i_t. where dimensionless slit length l/h=1.0 and deformation velocity V/H=0.013 s-1. Melted polyethylene used here has melt-index 0.5 at 190 C. Now we introduce a parameter ~S to show the influence of the ratio of slit to channel width h/h on flow patterns of main and circulating flows; the parameter ~5 in Fig. 14 is a variable to show the flow shape. In Fig. 15 is shown the relation between h/h and ~5 using deformation velocity V/H as a parameter, indicating that 6. Velocity Profiles when Changing the Slit Length If the slit length changes, the velocity profile and the stress distribution near the slit change, and the flow pattern is influenced a little. We investigated velocity profiles by changing dimensionless slit lengths, i.e. t/h=0.20, 1.00 and 3. 19, when the ratio of slit to channel width h/h was 0.16 and deformation velocity V/H was s-1. Melted polyethylene used here had melt-index 0.5 at 190 C 4 Journal of The Textile Machinery Society of Japan

5 Fig. 12 Velocity diagram (ratio of slit to channel width 0.34, melt-index 0.5, 190 C, deformation velocity s-1, Fig. 14 Schematic diagram of circulating secondary flow Fig. 15 Relation between ratio of slit to circulating secondary flow channel wi dth and Fig. 13 Velocity diagram (ratio of slit to channel width 0.16, melt-index 0.5, 190 C, deformation velocity s-1, dimensionless sligt length 1.0) In Fig. 16 is shown the relation between dimensionless slit length 1/H and ai. i becomes a little ll~',~ll Jll~ b/ll 4Lll ~ 3 T~iahi(~ shows that ~ ~ v c..mes large with increasing l/h, but deformation velocity V/H has more influence on c than i/h. The relation between V/H and ~ shown in Fig. 17 indicates that ~5 becomes large enough with increasing V/H. 7. Conclusions Main and circulating velocities near a slit were investigated experimentally by tracer particles, and the following results obtained. Fig. 16 Relation between slit length and circulating secondary flow (melt-index 0.5, 190 C, ratio of slit to channel width 0.16) Vol. 24 No.1(1978) 5

6 (1) Circulating velocity is much smaller than main velocity. (2) Circulating velocity is the maximum on the boundary of main flow. (3) If the ratio of slit to channel width (h/h) increases at a constant deformation velocity and with a constant slit length, the inlet angle of main flow (q5) tends to become a little small. (4) If dimensionless slit length (i/h) increases at a constant deformation velocity and with a constant ratio of slit to channel width, the inlet angle of main flow (~b) tends to become a little large. (5) If deformation velocity (V/H) increases with a consstant ratio of slit to channel width and a constant slit length, the inlet angle of main flow (c) increases with it. Fig. 17 Relation between deformation velocity and circulating secondary flow (melt-index 0.5, 190 C) References [1] K. Nakamura, S. Umegaki and A. Horikawa; J. Text. Mach. Soc. Jap. 23,101 (1977). [2] C. H. Drexler and C. D. Han; J. Appl. Poly. Sc., 17, 2355 (1973). 6 Journal of The Textile Machinery Society of Japan