3.1 Bifurcations and chaotic transition to melt fracture

For three different values of $De$ , figure 3 discloses representative extrudate profiles including streamlines and pressure contours at fixed instants (left) and the time variation of $Q_{asym}$ , the degree of flow rate asymmetry (right). In order to probe the degree of elastically driven flow asymmetry, $Q_{asym}\equiv (Q_{up}-Q_{low})/(Q_{up}+Q_{low})$ is defined, in which $Q_{up}$ and $Q_{low}$ are flow rates through the upper and lower halves of the channel exit, respectively. Only the mesh that is perfectly symmetric about the $x$ -axis is used (figure 2 a), hence the instability manifested in the form of symmetry breaking of solutions is not at all associated with any artefact of the spatial discretization.

Figure 3.

Streamlines and extrudate surfaces on the contour of pressure at time instants indicated (left) and the degree of flow rate asymmetry $Q_{asym}$ for $\unicode[STIX]{x1D6FC}<t<\unicode[STIX]{x1D6FC}+3$ (right): (a) $De=15.88$ , (b) $De=16.7$ , (c) $De=22$ .

Since every solution in figure 3 conforms to the elastically unstable regime, the flows never attain steady state and exhibit conspicuous fluctuations that certify the unstable flow dynamics after the subcritical bifurcation from the steady state with increasing $De$ . Whereas the extrudate profiles for $De=15.88$ and 16.7 (figure 3 a,b left) depict a more or less regular structure of the melt fracture, presumably indicative of sharkskin, at higher $De=22$ (figure 3 c left) gross melt fracture appears (the simulated animation can be found in the supplementary movies available at https://doi.org/10.1017/jfm.2018.659). Creeping elastic flow in the straight channel and therefore with straight streamlines is known to be linearly stable, however, the singular points located at the die lip induce a high pressure gradient (and also high stress due to the molecular stretching caused by the abrupt change of boundary conditions from the no-slip wall to free interface) as well as finite streamline curvature, which is explicitly demonstrated near the exit wall on the left of figure 3. Even recurring tiny vortices (alternating with a short lifetime) are existent and can be observed near singularities. The high stress near these tips at the channel exit is thought to regulate the onset of instability that corresponds to the subcritical transition. On the right of figure 3 the symmetry breaking of the flow field due to elastic instability is more quantitatively expressed in terms of $Q_{asym}$ in the time interval $\unicode[STIX]{x1D6FC}<t<\unicode[STIX]{x1D6FC}+3$ (for instance in the case of symmetric flow $Q_{asym}=0$ ). According to the temporal behaviour of this parameter, the solution corresponds to three such distinct flow regimes as a periodic state or limit cycle at $De=15.88$ (figure 3 a), a quasiperiodic state at 16.7 (figure 3 b) and an aperiodic state at 22 (figure 3 c).

The periodic limit cycle of figure 3(a) indeed emanates from the second steady state via the subcritical bifurcation. In the lower $De$ , another weak periodic fluctuation exists followed by this second steady flow with increasing $De$ , and it is discussed in more detail with figures 4 and 5. In real viscoelastic flow and in regard to figure 3(c), such aperiodic chaotic flow seems to mainly originate from the high extensional stress incurred in the contraction flow or by strong wall slip (or via subcritical dynamic transition inside the channel described in Meulenbroek et al. Reference Meulenbroek, Storm, Bertola, Wagner, Bonn and van Saarloos2003), and thus a developed elastic instability or wall slip may regulate the melt fracture outside the die. Nevertheless, in this study, possible gross melt fracture modelled simply by chaotic dynamic transition (due to singularity at the die lip) is naïvely demonstrated, however low its practical probability might be. For rigorous analysis of elastic flow instability including melt fracture, every possible origin of instability, including the current chaotic regime, has to be considered altogether in formulating and modelling the problem.

Figure 4.

Calculated flow curve ( $De$ versus $\unicode[STIX]{x0394}p/\unicode[STIX]{x0394}x$ in red with symbols) and the corresponding curve for fully developed steady flow of the straight channel (in blue dash-dot). The left inset is a magnified view of the curves inside the green circle. The right inset is the ratio between the corresponding values for the calculated and fully developed flow curves. $\unicode[STIX]{x0394}p$ is the time-averaged pressure difference in the channel exit region ( $\unicode[STIX]{x0394}p=p(x=-6)-p(x=0)$ and thus $\unicode[STIX]{x0394}x=6$ ).

Figure 5.

Modified flow curve of $De$ versus $(\unicode[STIX]{x0394}p/\unicode[STIX]{x0394}x)-(c_{1}\times De+c_{2})$ calculated from the hysteretic flow analysis by slowly decreasing $De$ from the solution at $De=15.55$ (red curve) and slowly increasing $De$ from the solution at $De=15.07$ (blue dashed curve). Green and blue solid curves represent solutions for steady and fully developed unstable flows, respectively.

Figure 4 showing the dependence of the pressure gradient on $De$ , may elucidate the nature of dynamic flow transition explicitly, where the red curve with symbols represents the pressure gradient $-\text{d}p/\text{d}x$ (time averaged in the case of instability) calculated for the channel end region of length 6 (i.e. in the region $-6<x<0$ ) and the equivalent curve in the fully developed steady flow of a straight die is also shown by the blue dash-dot curve for comparison. The solution expresses overall downward concavity, especially in the region $De<7$ , which is chiefly incurred by the shear thinning characteristic inherent in the Leonov constitutive equation. On the other hand, careful examination of the red flow curve confirms bifurcations noted as functional or differential discontinuities and also several crossings with the blue curve that indeed originate from the dynamic transition of flow regimes. The magnified detail of the graph inside the green circle is redrawn in the left figure inset, and even if small in magnitude, the discontinuous jump in the flow curve deduced as the subcritical bifurcation is clearly indicated at $De=15.5$ , which denotes the transition from a time-invariant steady solution to the periodic limit cycle (the sharkskin instability of the extrudate). More precisely, a very small fluctuation indeed starts near $De=10.4$ , where its existence is evident, but due to extremely slow development into instability and weakness of the fluctuation signals, its correct identification is very difficult and thus the detailed analysis is not included in the current study. When we further increase $De$ in this start-up flow simulation, another peculiar transition from a periodic limit cycle to a stable steady flow emerges at $De=12.1$ . Then, as soon as $De$ exceeds the critical value $De=15.5$ , the second periodic instability appears and the amplitude of fluctuations for $Q_{asym}$ becomes approximately 20 times larger than that of the previous fluctuation with $De<12.1$ (here all the points are obtained from the steady solution or the time average of the fully developed instability initiated as a start-up flow). The apparent loss of continuity of data at the transition of $De=15.5$ results from the subcritical dynamic transition concluded from the hysteretic flow simulation exhibited in figure 5. Therefore, this subcritical Hopf bifurcation transforms the second steady flow again into the periodic limit cycle presumed herein as the sharkskin melt fracture.

Concerning the result interpreted as the sharkskin instability, a rather critical remark should be made. This instability may be associated with a type of melt fracture distinct from sharkskin fracture, that is, in the real 3-D case it may represent a spiral pattern of melt fracture or another 3-D periodic kind of unstable flow. In the study of 2-D flow we cannot identify its exact type and evidently this expresses the limitation of the 2-D flow simulation. Notwithstanding the unavoidable uncertainty, this melt fracture with periodic fluctuations of relatively small amplitude is termed the sharkskin instability, which seems the most appropriate choice with the current restricted amount of information.

The right inset of figure 4 replots the flow curve of the pressure gradient divided by the value corresponding to fully developed steady flow (the blue curve), which more sensitively detects the conversion of the dynamics. In addition to the subcritical bifurcation, there discernibly exists another transition at $De=17.3$ , which leads to chaotic flow converted from the quasiperiodic regime, and it is expressed as an abrupt slope change which is not clear in the left inset of the normal flow curve but is in the right inset of the modified plot. Past this transition, there appears a crossing of the blue and red curves near $De=20$ (in the main figure), which means that, owing to this instability, the power required for unstable flow begins to exceed that for the corresponding developed steady flow of a straight die. The current phenomenon is similar to that present in the turbulence of a Newtonian fluid although a strict analogy unquestionably necessitates flow modelling of 3-D elastic turbulence which is not the subject of this paper.

According to the current result combined with past research results reported elsewhere, it is concluded that there possibly exist two competing mechanisms for sharkskin fracture; the subcritical transition herein suggested and the wall slip near the die lip. In principle, a distinction between them can be made in experiments even though this may be hard in practice. If the sharkskin is initiated by the wall slip, then the slope of the flow curve (the common flow curve of pressure drop versus apparent shear rate, or the current flow curve of pressure gradient versus $De$ ) would decrease at the transition due to reduction of the wall shear rate (and thus decrease in wall shear stress) in comparison with the nominal one. On the other hand, in the case of sharkskin fracture caused by the subcritical bifurcation described in this study, the apparent slope would slightly increase due to a small discontinuous jump, as shown in figure 4, since inducing temporally periodic behaviour is presumed to necessitate an extra pressure drop. However, this kind of subtle change in the slope of the flow curve may be easily hidden in comparison with the overall behaviour of total pressure variation along the entire flow channel measured experimentally, and thus its detection may be hard in practice.

In order to certify more convincing evidence for subcritical transition, a curve of flow hysteresis is shown in figure 5 by slowly decreasing $De$ (the decreasing direction denoted with a red arrow) from the developed periodic flow at $De=15.55$ as the initial value, where the solution consists of the unstable state corresponding to sharkskin fracture just above the subcritical transition. In more detail, the inlet boundary located at $x=-20$ was assigned steady Poiseuille flow profile for the straight pipe corresponding to a desired $De$ slightly lower than the previous setting. Then the computation was started from the previous fully developed solution until a new fully developed instability or fully relaxed steady state was attained. This process continued up to a $De$ as low as 10.4 and the solution is drawn as the red curve with empty diamond symbols while the original solution of start-up flow is given with the blue curve and filled symbols. For clear illustration, the values of the ordinate are modified by subtracting the linear regression line (regression of data for the lower branch steady flow from $12.1<De<15.5$ ) from the pressure gradient, that is, the vertical axis assigns values of $|\text{d}p/\text{d}x|-(c_{1}\times De+c_{2})$ where $c_{1}=0.172$ and $c_{2}=3.0385$ . The red curve distinctly exhibits flow hysteresis, that is, the current system contains multiple solutions in the range of $12.1<De<15.5$ , where the lower branch (lower green curve) expresses a steady flow solution but the upper branch (red curve) corresponds to a periodic sharkskin melt fracture.

The lower branch solution of steady flow (green symbols with a green curve) is obtained when the start-up flow simulation from the rest state explained in § 2 is performed or the test of hysteresis initiated from the steady branch itself (solution corresponding to the lower branch for $12.1<De<15.5$ ) is carried out. However, the upper branch of unstable periodic solution (red diamond symbols with a red curve in $12.1<De<15.5$ ) is attainable only in the case of a hysteretic flow simulation starting from the unstable periodic solution (the upper branch solution itself or the developed unstable flow solution for $De>15.5$ ). On the other hand, although the simulation starts from the lower branch of the steady state, the solution enters again into weak periodic instability when the $De$ is set lower than 12.1 by the decrease of flow rate. This weak fluctuation persists at a $De$ as low as 10.4, and then the flow becomes steady and stable as $De$ further decreases below 10.4. In the unstable domain with periodicity $(10.4<De<16)$ , the amplitude of the fluctuations continuously decreases with $De$ decreasing in the direction of red arrow in the figure. Unlike the typical subcritical transition, below $De=15.5$ there has not been observed any abrupt change (the discontinuous jump at the saddle-node point from the upper to the lower branch) on the hysteresis curve on decreasing $De$ , and this is suspected to result from unresolved complex dynamics initiated at two bifurcations at $De=10.4$ and 12.1. The existence of the lower branch solution of steady state in the middle of periodic instability is quite unique and peculiar, and thus its further investigation, including the identification of the bifurcation at $De=10.4$ , suggests a good subject of future study.

For more detailed analysis, an additional curve of flow hysteresis in the opposite sense has also been obtained by slowly increasing $De$ from the solution on the lower branch at $De=15.07$ (the increasing direction denoted with blue arrow in figure 5), where this initial condition is the solution corresponding to fully relaxed steady flow below the subcritical transition. The solution is drawn as the blue dashed curve with large circular symbols. In this numerical test on the other hand, an abrupt discontinuous jump of pressure gradient is markedly recognizable near $De=15.5$ , where an increase of $De$ as small as 0.45 % (from 15.48 to 15.55 in terms of $De$ ) brings about the rather large step variation illustrated in the figure. This is equivalent to the discontinuous jump already observed in the curve of the start-up flow simulation from rest shown in figure 4 and confirms the subcritical bifurcation involved in the nonlinear flow dynamics leading to periodic melt fracture. When one takes into consideration the result of supercritical bifurcation demonstrated experimentally (Arratia et al. Reference Arratia, Thomas, Diorio and Gollub2006) and computationally (Poole et al. Reference Poole, Alves and Oliveira2007) in the elongation dominated cross-slot flow, two bifurcations at $De=10.4$ and 12.1 in addition to the subcritical one at 15.5 as well as the chaotic transition at 17.3 herein associated with melt fracture signify the complexity and variety of nonlinear dynamic phenomena possibly present in elastic flow instability.

Figure 6.

FFT spectrum (power normalized by the amplitude of maximum peak). Figure insets are the magnifications of the spectrum that illustrate the main and subharmonic peaks: (a) $De=15.88$ , (b) $De=16.7$ , (c) $De=22$ .

The time-dependent function $Q_{asym}$ of figure 3 is transformed into a frequency spectrum in figure 6 by fast Fourier transform (FFT), where the amplitude is normalized by the magnitude at the maximum peak. In the perfectly periodic case of $De=15.88$ , distinct modes at $f_{0}$ and at odd multiples of $3f_{0}$ and $5f_{0}$ are observed and essentially no other frequency modes exist, which can be clearly seen in the inset of figure 6(a). As the flow rate increases to $De=16.7$ , the main mode slightly shifts to higher frequency and small peaks of subharmonic modes appear at integral fractions of the main and odd multiple frequencies, which are denoted in the inset of figure 6(b) as peaks marked with small blue circles. These small amplitude modes seem to show signs of period doubling bifurcation (doubling at $f_{0}/2$ and $3f_{0}/2$ ; quadrupling at $f_{0}/4$ and $3f_{0}/4$ ). Other smaller peaks probably indicate signs of quasiperiodic transition with the appearance of incommensurate subharmonic motions in the flow dynamics. Figure 6(c), on the contrary, evinces a densely distributed frequency spectrum, especially the slow mode appears with notable amplitude, and thus this offers an affirmative proof of the chaotic dynamics involved in the flow. In turn, figure 6(b) provides evidence of chaotic (figure 6 c) transition through period doubling as well as a quasiperiodic route from a limit cycle (figure 6 a).

At this point several comments should be made. The current subcritical bifurcation from the steady (shown by the extrudate profile portrayed in figure 1) to periodic flow state of figure 3(a) is specifically distinguished from the subcritical route to melt fracture observed in other studies (Bertola et al. Reference Bertola, Meulenbroek, Wagner, Storm, Morozov, van Saarloos and Bonn2003; Meulenbroek et al. Reference Meulenbroek, Storm, Bertola, Wagner, Bonn and van Saarloos2003). On frequent occasions, the real extrudate instability starts in a periodic form of sharkskin fracture depicted in figures 3(a) or 6(a) presumably by way of the subcritical transition, and then, as the flow rate or $De$ further increases, various types of large amplitude melt fracture emerge, one of which belongs to the gross melt fracture (cf. Leonov & Prokunin Reference Leonov and Prokunin1994). The gross melt fracture is thought to result mainly from the other type of subcritical instability generated by the intense flow inside the die (Meulenbroek et al. Reference Meulenbroek, Storm, Bertola, Wagner, Bonn and van Saarloos2003) or the spurt flow (Adewale & Leonov Reference Adewale and Leonov1997). However, this study suggests additional instance of its occurrence via the chaotic transition depicted in figures 3(c) and 6(c) as long as the finite disturbance remains sufficiently small to prohibit the subcritical bifurcation along the die and thus, under no-slip conditions, the dynamics takes the route of period-doubling/quasiperiodic transition to chaos. Without wall slip taken into account (by the assumption made here), the die lip singularity becomes the sole origin of melt fracture in this work. At the end tip of the channel wall, the finite curvature of the streamlines results from the sudden expansion of the flow stream, and furthermore, the abrupt change of boundaries from no-slip on the wall to the free surface of the extrudate causes concentrated extension of liquid elements and accordingly high tensile stress. As a result, this further intensifies the singular character of the die exit wall. This unavoidable streamline curvature even induces tiny vortices attached to the die lip, which readily lose stability and start to fluctuate under high stress, and this is thought to give rise to nonlinear instability via the chaotic transition as well as subcritical bifurcation at some critical $De$ . Such vortices rarely or weakly appear in the steady state, however in the unstable regime they are manifestly formed and oscillate with periodic, quasiperiodic or aperiodic duration (snapshots on the left of figure 3 portray such tiny vortices with high pressure gradients near the exit wall and their chronological dynamics can be observed in the animation of supplementary movies).

Actually, the main result of this section does not consist of the complete verification of the complicated mechanism of sharkskin and melt fracture instability. To the contrary, we assert that description or analysis of sharkskin fracture based on the fundamental assumption only of wall slip may be misleading and has to be incorporated with the current nonlinear dynamic mechanism, since both mechanisms, wall slip and nonlinear dynamic transition (subcritical or chaotic bifurcation), will compete for the initiation and further evolution of sharkskin instability. In other words, this work proves a possibility that even without any wall slip almost every kind of melt fracture can be realized simply in terms of the high stress along the curved streamlines near the die lip. Although the mechanism is not certain yet, the wall slip may become the most crucial element determining the onset of sharkskin. Nevertheless, when rigorously and realistically modelling melt fracture, one has to consider the problem based on a slip boundary, including all the nonlinear dynamics herein performed.

3.2 Effect of geometrical exit modification on melt fracture

In this section, we examine the effect of geometrical modification of the die exit domain on melt fracture, especially on the sharkskin instability. Since the occurrence of melt fracture instability usually sets the upper bound of manufacturing speed, in the real industrial process, the control of melt fracture or delay of the onset of sharkskin is valuable. In the current geometry of a straight channel under Poiseuille flow, the die lip, that is, the end point of channel wall, becomes the sole origin of sharkskin or melt fracture under the boundary condition specified as no slip. These lip points in the mathematical approximation of the elastic flow designate a strong singularity where the flow boundary abruptly changes from no slip to free surface and, accordingly, intense extensional deformation exists. Introducing a short expansion channel element at the tip of the die we make an attempt to attenuate the degree of strength of the die lip singularity. For this purpose, a basic computational domain with no expansion element is constructed with triangular mesh elements that are not symmetric about the centreline (line of $y=0$ ), as shown in figure 2(b), in order to examine whether the introduction of the exit modification can recover the flow symmetry even in an asymmetrically discretized flow domain. A short expansion flow domain is introduced with a length of $H/2$ (i.e. in the region of $-0.5<x<0$ ) and variable slope, and one example of the modified channel domain is exhibited in figure 2(c), where the slope of channel exit is 0.34 ( $-0.34$ for the lower wall). As before, every wall of the channel, including the diverging wall elements, is assigned as a no-slip boundary, and therefore the following results for the flow field, instability and possible stabilization induced by this die modification are valid and meaningful only under the current computational regime of strictly no slip.

As an example, the flow at $De=16$ is computationally modelled with the asymmetric mesh of figure 2(b) and the resultant extrudate profile with streamlines and pressure contour is drawn in figure 7(a) at the instant $t=5$ . The elastic flow instability consists in a domain of periodic sharkskin occurring above the subcritical transition, however, in addition, the extrudate displays overall skewness (that is, the computed extrudate profile is bent upwards) possibly due to an artefact of the asymmetric discrete spatial resolution. Figure 7(b–d) illustrates at the same temporal instant the results for the modified exit geometry, which demonstrate the dependence of the interface evolution on the slope of diverging wall and the general tendency of decreasing sharkskin for the introduction of this expansion channel element. Such a diminishing amplitude of fluctuation notwithstanding, the case of figure 7(b) (slope 0.2) still exhibits a rough interface due to the remaining unstable flow dynamics and figure 7(d) (slope 0.5) shows the skewness remaining in the extrudate liquid. On the other hand, figure 7(c), of slope 0.34, depicts a smooth interface as well as an extrudate symmetry that is almost recovered.

A similar analysis has been performed with the equivalent spatial approximation for $De=22$ , in which the unstable flow corresponds to the chaotic regime. Figure 8 represents the solution at $t=2$ for the exit wall slope (a) 0 (i.e. no exit modification), (b) 0.2, (c) 0.4 and (d) 0.6. As a result, figure 8(a) obviously presents the surface profile corresponding to gross melt fracture. Even under this heavy elastic effect, the introduction of a short expansion channel at the exit dramatically alleviates the instability, which can be observed in figure 8(b–d). Especially the channel part of slope 0.6 (figure d) is shown to be the most effective at recovering an almost symmetric and smooth profile.

Figure 7.

Streamlines and extrudate surfaces on the contour of pressure at $t=5$ and $De=16$ : (a) original flow domain without die exit modification, and flow domain with exit modification of slope of exit wall: (b) 0.2, (c) 0.34, (d) 0.5.

Figure 8.

Streamlines and extrudate surfaces on the contour of pressure at $t=2$ and $De=22$ : (a) original flow domain without die exit modification, and flow domain with exit modification of slope of exit wall: (b) 0.2, (c) 0.4, (d) 0.6.

Although it is at this stage too early to estimate the practical efficiency and to draw conclusions, the current study verifies that the geometrical modification of the channel exit wall has a drastic influence on the flow dynamics and instability leading to melt fracture. Now that notable wall slip may occur in the chaotic instability, the effect of the diverging wall element is doubtable for the reduction of gross melt fracture. However, in the weakly unstable flow exhibiting sharkskin, this channel modification is judged to be feasible. Indeed, the removal or reduction of the sharkskin melt fracture has been experimentally observed by the introduction of a diverging taper at the die exit similar to that of the current modelling (Pol et al. Reference Pol, Joshi, Tapadia, Lele and Mashelkar2007). Whereas this numerical outcome, supported by experimental evidence of Pol et al. (Reference Pol, Joshi, Tapadia, Lele and Mashelkar2007), suggests the potential feasibility of die exit modification, the expansion channel could possibly raise the degree of extrudate swelling and thus this apparent improvement may simply amount to stable flow through a wider channel where sharkskin disappears due to lowered shear strength. In addition, the following needs to be mentioned: the experimental verification by Pol et al. (Reference Pol, Joshi, Tapadia, Lele and Mashelkar2007) may not be regarded as a strict proof for the validity of the instability mechanism provided by the current analysis since the sharkskin caused by wall slip may also be reduced in the modification into the diverging die.