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P. C.
Sousa
^{a},
F. T.
Pinho
^{b},
M. S. N.
Oliveira
^{c} and
M. A.
Alves
*^{a}
^{a}Departamento de Engenharia Química, CEFT, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. E-mail: mmalves@fe.up.pt
^{b}CEFT, Departamento de Engenharia Mecânica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
^{c}James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, G1 1XJ, UK

Received
26th May 2015
, Accepted 4th September 2015

First published on 23rd September 2015

We present an experimental investigation of viscoelastic fluid flow in a cross-slot microgeometry under low Reynolds number flow conditions. By using several viscoelastic fluids, we investigate the effects of the microchannel bounding walls and the polymer solution concentration on the flow patterns. We demonstrate that for concentrated polymer solutions, the flow undergoes a bifurcation above a critical Weissenberg number (Wi) at which the flow becomes asymmetric but remains steady. The appearance of this elastic instability depends on the channel aspect ratio, defined as the ratio between the depth and the width of the channels. At high aspect ratios, when bounding wall effects are reduced, two types of elastic instabilities were observed, one in which the flow becomes asymmetric and steady, followed by a second instability at higher Wi, in which the flow becomes time-dependent. When the aspect ratio decreases, the bounding walls have a stabilizing effect, preventing the occurrence of steady asymmetric flow and postponing the transition to unsteady flow to higher Wi. For less concentrated solutions, the first elastic instability to steady asymmetric flow is absent and only the time-dependent flow instability is observed.

The onset of purely elastic flow instabilities occurs above critical flow conditions and is due to the combination of streamline curvature and large normal stresses. McKinley and co-workers^{6,7} proposed a dimensionless criterion that can be used to estimate the critical conditions for the onset of purely elastic flow instabilities.

One geometrical configuration that has been extensively used in the investigation of nonlinear effects in complex fluids is the cross-slot arrangement.^{8–24} It consists of two perpendicular, bisecting channels with opposing inlets and opposing outlets, resulting in a flow field with a stagnation point at the centre of symmetry (cf.Fig. 1), where the flow velocity is zero and the velocity gradient is finite.^{13} A strong extensional flow is generated in the vicinity of the stagnation point and, for viscoelastic fluids, a nonlinear increase in the first normal stress difference (or similarly in the extensional viscosity) with the flow rate promotes the onset of elastic instabilities. In addition, near the stagnation point the residence time is sufficiently long for the polymer molecules to reach a steady-state elongation, allowing the estimation of the steady-state extensional viscosity, for example using birefringence measurements.^{10,11,13–18}

Arratia et al.^{8} experimentally investigated the flow of a dilute polymeric solution in a cross-slot device and observed the existence of two types of purely elastic flow instabilities that occur as the flow rate increases. In the first instability, the flow patterns become asymmetric but remain steady with the fluid entering in each of the opposing inlet channels and exiting preferentially through one outlet. At higher flow rates, a second elastic instability sets in, in which the flow becomes time-dependent. Similar experimental observations were also reported in the same year by Pathak and Hudson^{10} using a micellar solution. In the following year, Poole et al.^{9} simulated numerically the two-dimensional cross-slot flow of an upper-convected Maxwell (UCM) fluid at low and negligible Re and predicted qualitatively the flow behaviour observed experimentally. In addition, the authors demonstrated that the transition to a steady asymmetry flow is purely elastic in nature and that inertial effects reduce the strength of the asymmetry and delay the onset of the elastic instability to higher De. Following these three pioneering studies, the viscoelastic fluid flow through a cross-slot geometry has received renewed attention in order to explore the rheological conditions that can affect the onset of purely elastic flow instabilities^{11,12,17,18,20,21} and was considered an important open mathematical problem for non-Newtonian fluids.^{22}

Poole et al.^{23} investigated numerically the three-dimensional flow of an UCM fluid under creeping flow conditions by varying the depth of the geometry from conditions corresponding to the quasi-Hele Shaw flow (low aspect ratio) to quasi-two dimensional flow (high aspect ratio). Subsequently, Poole et al.^{24} predicted numerically the cross-slot flow using an Oldroyd-B model and a simplified Phan–Thien–Tanner model in order to assess the effects of the solvent viscosity and finite extensibility on the flow patterns, and they found that the steady asymmetric and unsteady flow regimes occur under different flow conditions, depending on the rheological parameters of the viscoelastic model. The effect of the type of solvent and viscosity on the viscoelastic fluid flow behaviour has been investigated experimentally in other applications, including rheometric flows^{25} or flows through microfluidic planar contractions.^{26,27} For instance, Rodd et al.^{27} used viscoelastic solutions with a constant concentration of polyethylene oxide (PEO) while varying the viscosity of the solvent, in order to vary El between 2.8 and 68. The same flow regimes (Newtonian-like flow, steady viscoelastic flow, diverging streamlines and vortex growth) were found for all solutions investigated, although at high elasticity numbers the transitions between the flow regimes occurred at higher Wi.

Motivated by the numerical studies of Poole et al.,^{23,24} here we investigate experimentally the effects of the aspect ratio of the channel, imposed by the bounding walls of the geometry, and the effects of the rheological characteristics of viscoelastic fluids on the onset of the purely elastic instabilities observed in a cross-slot device. We use several viscoelastic fluids with different rheological characteristics, in order to explore experimentally the conditions that lead to the onset of the purely elastic flow instabilities in microscale cross-slot devices.

Flow rates (Q) of equal magnitude are imposed in all four channels. For that purpose, a syringe pump with three independent modules (neMESYS, Cetoni GmbH) was used to inject with equal flow rates at the two inlets and remove the fluid from one outlet at the same flow rate. The remaining outlet is connected to a tube open to the atmosphere. For flow visualization, the fluids were seeded with 1 μm fluorescent tracer particles (Nile Red particles, Molecular Probes, Invitrogen, Ex/Em: 535/575 nm). Flow visualization was based on long exposure photography with the microchannels placed on an inverted epi-fluorescence microscope (DMI 5000M, Leica Microsystems GmbH) equipped with a 20× objective lens (numerical aperture, NA = 0.4) and a CCD camera (DFC350 FX, Leica Microsystems GmbH) or a sCMOS camera (Neo 5.5, Andor). Illumination was provided by a 100 W mercury lamp operating together with an adequate filter cube. Visualization of the flow patterns was performed at the centre plane of the microchannel, z = 0, at the mid-distance between the top and bottom bounding walls.

(1) |

(2) |

Fluid | Polymer (ppm) | Glycerol (% w/w) |
η
_{0} (Pa s) |
η
_{S} (Pa s) |
Λ (s) | n ( – ) |
λ
_{0} (s) |
β ( – ) | c* (ppm) |
λ
_{c} (s) |
---|---|---|---|---|---|---|---|---|---|---|

a Shear viscosity data fit. b Relaxation time function fit. | ||||||||||

PAA300 + 50%Glyc | 300 | 50 | 2.8 | 0.0061 | 80 | 0.44 | 13 | 0.0022 | 95 | 0.178 |

PAA300 | 300 | — | 3.50 | 0.001 | 60 | 0.30^{a} |
0.55 | 0.00029 | 45 | 0.066 |

0.70^{b} |
||||||||||

PAA200 + 70%Glyc | 200 | 70 | 3.00 | 0.023 | 140^{a} |
0.47^{a} |
2.4 | 0.0077 | 130 | 0.658 |

200^{b} |
0.73^{b} |
|||||||||

PAA190 + 80%Glyc | 190 | 80 | 1.90 | 0.063 | 32 | 0.50 | 8.0 | 0.033 | 150 | 0.390 |

PAA120 + 70%Glyc | 120 | 70 | 0.35 | 0.023 | 15 | 0.60 | 1.0 | 0.066 | 130 | 0.160 |

PAA100 + 90%Glyc | 100 | 90 | 1.70 | 0.22 | 30 | 0.55 | 5.0 | 0.13 | 170 | 0.484 |

PAA100 | 100 | — | 0.22 | 0.001 | 60 | 0.48 | 2.5 | 0.0045 | 45 | 0.012 |

PAA70 + 90%Glyc | 70 | 90 | 0.82 | 0.22 | 10 | 0.50 | 4.0 | 0.27 | 170 | 0.351 |

PEO300 | 300 | — | 0.0023 | 0.001 | 0.07^{a} |
0.87^{a} |
0.0045 | 0.43 | 350 | 0.040 |

—^{b} |
1.0^{b} |
|||||||||

PEO5000 | 5000 | — | 1.33 | 0.001 | 3 | 0.48 | 0.7 | 0.00075 | 350 | 0.228 |

The total extra-stress tensor τ is defined as the sum of the solvent and polymeric contributions, τ = τ_{p} + τ_{s}, where the solvent contribution is given by τ_{s} = η_{s}(∇u + ∇u^{T}). In the nonlinear White–Metzner model, both the shear viscosity and the first normal-stress difference depend on the shear rate, , where is the rate of deformation tensor. The shear viscosity function is given by a Carreau model,^{30}η_{p}() = (η_{0} − η_{s})/[1 + (Λ)^{2}]^{(1−n)/2} and the first normal-stress difference is N_{1} = 2η_{p}()λ()^{2}, where η_{s} is the high shear rate viscosity, which approaches the solvent viscosity of dilute polymer solutions, η_{0} is the zero-shear rate viscosity, Λ is a time constant, n is the power-law like index and λ() is also given by a Carreau model, λ = λ_{0}/[1 + (Λ)^{2}]^{(1−n)/2}. The shear viscosity and first normal-stress difference data measured in steady shear flow as well as the White–Metzner model fits are shown in Fig. 2. For most fluids, we use the same values of the parameters of the model to fit the data for both polymer shear viscosity and first normal-stress difference thus using an elastic modulus, G = η_{p}()/λ(), which is independent of the shear rate. The exceptions are the PAA300, PAA200 + 70%Glyc and PEO300 fluids, for which the parameters n and Λ used to fit the polymer viscosity function were different from those used in the relaxation time to fit the first normal-stress difference data. The rheological parameters of the White–Metzner model are listed in Table 1 and are included to allow future comparisons of the experimental data with numerical simulations. The critical overlap concentration c* is also presented in Table 1 and was calculated according to Graessley^{31} as c* = 0.77/[η], where [η] is the intrinsic viscosity, which was measured for PAA solutions using a U-tube capillary viscometer using various solutions with the same solvent and different polymer concentrations. The intrinsic viscosity was determined using the Huggins equation.^{32} The critical overlap concentration was determined experimentally for PAA solutions composed of 90%, 60% and 0% glycerol and was estimated for the remaining PAA solutions. For aqueous PEO solutions the overlap concentration was calculated from the correlation obtained by Tirtaatmadja et al.^{33} for the intrinsic viscosity ([η] = 0.072M_{w}^{0.65}, with [η] in units of cm^{3} g^{−1}). We note that for dilute polymer solutions the zero-shear rate viscosity is expected to increase linearly with the polymer concentration (η_{0}/η_{s} = 1/β ≈ 1 + c/c*) but a quadratic increase was found for the PAA solutions. The relaxation times measured in uniaxial extensional flow (λ_{c}) using a capillary break-up extensional rheometer (HAAKE CaBER™1, Thermo Scientific) are also presented in Table 1 together with the values of the solvent viscosity ratio, β, defined as β = η_{s}/η_{0}.

We use a normalized aspect ratio , defined as ,^{23} which varies in the range between (for AR → 0 corresponding to the Hele-Shaw flow limit) and (for AR → ∞ corresponding to the two-dimensional flow limit). The range of Reynolds numbers attained for each aspect ratio investigated are also given in Fig. 4 to highlight the negligible effect of inertia on the viscoelastic fluid flow. The flow patterns reported in the map of Fig. 4 were identified in three different regions: (I) symmetric flow; (II) steady asymmetric flow (first purely elastic instability); and (III) time-dependent flow (second purely elastic flow instability). For the time-dependent regime, we show three images to illustrate the oscillatory nature of the flow. It should be noted that region III can be reached from region II or directly from region I by increasing Wi, through a purely elastic instability of the second type. The different flow behaviour depends on the channel aspect ratio. For , the first elastic instability is observed, whereas for lower the steady asymmetric flow regime does not appear. The bounding walls (z = ±h/2) have a stabilizing effect, since the decrease in channel depth (or AR) increases the wall influence via enhanced shear stress thus reducing the influence of extensional effects. In addition, for the higher AR investigated, corresponding to , the critical Wi at which the two flow transitions occur seems not to depend significantly on the channel aspect ratio. In contrast, the numerical predictions of Poole et al.^{23} showed that the second elastic instability appears at gradually smaller Wi as the AR is decreased and the Hele–Shaw flow is approached , but in their case a UCM model was used in the simulations. Such a model cannot predict the shear-thinning of the viscosity and the relaxation time observed with the fluids used in this work.

Another motivation of this work was to understand why the first elastic instability sets in only for some viscoelastic fluids, whereas for other viscoelastic fluids the transition occurs directly to time-dependent flow independently of the channel AR. For that purpose, we considered all the PAA solutions described in Table 1, which have significantly different solvent viscosity ratios and different elasticity levels. The plot in Fig. 5(a) indicates the presence or absence of the first instability in a map of viscosity ratio and elasticity number for all PAA solutions tested and AR = 1.0.

The elasticity number is defined as El = Wi/Re_{0} = 2λ_{c}η_{0}/(ρw^{2}) and is independent of the flow rate. As is shown, the rheological parameter that has a major influence on the onset of the first purely elastic instability is the solvent viscosity ratio. Viscoelastic fluids with β ≲ 0.05 (more concentrated polymer solutions) experience a first transition through a steady asymmetric flow, whereas for the fluids with larger β, the flow changes directly from Newtonian-like to time-dependent flow above a critical Wi. Experiments using the two PEO solutions were also carried out. The flow visualization results are not shown here for conciseness, but the corresponding Wi − β map is qualitatively similar to Fig. 5(b), with the more concentrated solution (lower β) showing the onset of the first and second instabilities at Wi ≈ 2 and Wi ≈ 100, respectively, whereas for the dilute solution (higher β), the flow changes directly from Newtonian-like to unsteady flow at Wi ≈ 50. The types of flow transitions are identical for both PAA and PEO polymer solutions, but occur at different values of Wi for each β.

It is important to note that the critical Wi values for the onset of the first elastic instability observed in the PAA solutions are lower than 1 [cf.Fig. 5(b)], especially for the more concentrated solutions, and it may be argued that elastic effects are unimportant because under these conditions the polymer molecules remain coiled. However, the CaBER relaxation times, used in the calculation of Wi, are probably underestimated. We also calculated the shear relaxation times estimated from the rheological oscillatory shear data , where G′ and G′′ are the storage and loss moduli, and ω is the angular frequency of oscillation and we found that the extensional relaxation times are all significantly lower than these shear relaxation times, and also those that can be computed from the N_{1} data illustrated in Fig. 2. Similar results were obtained by Arnolds et al.^{38} for PEO solutions, with the authors reporting also that the ratio between the extensional and shear relaxation times decreases significantly with increasing concentration or molecular weight of the polymer. Nevertheless, instead of using a relaxation time determined from a shear flow, which leads to critical Wi values above 1, we opted to use the extensional relaxation times because the flow in the cross-slot device is strongly extensional.

The onset of steady asymmetric flow seems to be related to the compressive flow near the cross-slot centre generated by the two inlet channels.^{9} The collision of the inlet streams generates opposite normal forces, the resulting momentum decreases near the stagnation point and, due to continuity, the fluid is pushed towards the corners triggering the onset of a flow asymmetry.^{9} According to McKinley and co-workers,^{6,7} the dimensionless criterion that must be exceeded for the onset of purely elastic flow instabilities can be expressed as , where is the local radius of curvature, U the local streamwise velocity, τ_{11} the streamwise tensile stress and the local shear rate. Hence, an increase of the normal stress, combined with small streamline radii of curvature and high streamwise velocities meets the criterion for the appearance of the elastic instability. As the aspect ratio of the channel decreases, the critical region moves towards the corners of the geometry, where the streamwise velocity and shear rate increase,^{23} high tensile stress develops and the radius of curvature is small. In this case, the fluid experiences the onset of the second type of elastic instability whereas the asymmetric flow does not occur which is primarily driven by the strong extensional flow at the cross-slot centre.

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## Footnote |

† Electronic supplementary information (ESI) available: Illustrative movies of the steady symmetric (Videos S1 and S2), steady asymmetric (Video S3) and time-dependent (Video S4) flows. See DOI: 10.1039/c5sm01298h |

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