M.A. Alves

Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions

The paper presents very accurate numerical results for the vortex size, the vortex intensity and the Couette correction, in planar contraction flows of Oldroyd-B and PTT fluids ( e = 0.25) with both the linear and the exponential stress function, and with a solvent viscosity ratio equal to 1/9. The accuracy of these results is quantified, being generally below 1% (0.3% for most results), and the finest mesh employed had over 1 million degrees of freedom. Such degree of mesh fineness is shown to be required for accurate results with the Oldroyd-B fluid, especially at high Deborah numbers, but the shear-thinning PTT fluid in general does not require the finest meshes. In terms of level of elasticity, steady results for the PTT fluid could be obtained for values of the Deborah number in excess of 100 (linear PTT) and 10,000 (exponential PTT).

The flow of viscoelastic fluids past a cylinder: finite-volume high-resolution methods

Accurate solutions are obtained with the numerical method of Oliveira et al. [J. Non-Newtonian Fluid Mech. 79 (1998) 1] for the inertialess plane flow around a confined cylinder. This numerical procedure is based on the finite-volume method in non-orthogonal block-structured meshes with a collocated arrangement of the dependent variables, and makes use of a special interpolation practice to avoid stress‐velocity decoupling. Two high-resolution schemes (MINMOD and SMART) are implemented to represent the convective terms in the constitutive equations for the upper convected Maxwell and Oldroyd-B fluids, and the resulting predictions of the drag coefficient on the cylinder are shown to be as accurate as existing finite-element method predictions based on the supposedly very accurate h-p refinement technique. Numerical uncertainties are quantified with help of Richardson’s extrapolation technique and the orders of convergence of the differencing schemes are established and shown to be second-order accurate. Calculations performed with a wake-refined mesh predicted the variation of the maximum longitudinal normal stress in the wake as De 3 and De 5 depending on Deborah number.

Effect of a high-resolution differencing scheme on finite-volume predictions of viscoelastic flows

Improved accuracy and enhanced convergence rate are achieved when a finite-volume method (FVM) is used in conjunction with a high-resolution scheme (MINMOD) to represent the stress derivatives in the constitutive equation, because it avoids oscillations of the solution field near sharp stress gradients. Calculations for the benchmark flow of an upper-convected Maxwell fluid through a 4:1 plane contraction were carried out at a constant Reynolds number of 0.01 and varying Deborah numbers in four consistently refined meshes, the finest of which had a normalised cell size of 0.005 in the vicinity of the re-entrant corner. The MINMOD scheme was able to provide converged solutions up to Deborah numbers well beyond those attained by other second-order accurate schemes. The asymptotic behaviour of velocity and stresses near the re-entrant corner was accurately predicted as compared with Hinch’s theory [1]. The simulations improved previous results for the same flow conditions obtained with less accurate schemes, and the present results can be used as benchmark values up to a Deborah value of 3 with quantified numerical uncertainties.

Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid

Analytical solutions are derived for the steady state channel and pipe flows of viscoelastic fluids obeying the complete single-mode Phan-Thien–Tanner (PTT) constitutive equation with a linear stress coefficient in the absence of a solvent viscosity contribution. The results include the profiles of all relevant stresses, the axial velocity and the viscosity across the gap. The three material functions of the single-mode PTT model in steady Couette flow are also given and it is shown that the conditions of the maximum point in the shear stress versus shear rate curve are related to the conditions for existence of steady state solutions in the channel and pipe flows. The range of model parameters for which a classical steady solution exists is established.

Extensional flow of blood analog solutions in microfluidic devices

In this study, we show the importance of extensional rheology, in addition to the shear rheology, in the choice of blood analog solutions intended to be used in vitro for mimicking the microcirculatory system. For this purpose, we compare the flow of a Newtonian fluid and two well-established viscoelastic blood analog polymer solutions through microfluidic channels containing both hyperbolic and abrupt contractions∕expansions. The hyperbolic shape was selected in order to impose a nearly constant strain rate at the centerline of the microchannels and achieve a quasihomogeneous and strong extensional flow often found in features of the human microcirculatory system such as stenoses. The two blood analog fluids used are aqueous solutions of a polyacrylamide (125 ppm w∕w) and of a xanthan gum (500 ppm w∕w), which were characterized rheologically in steady-shear flow using a rotational rheometer and in extension using a capillary breakup extensional rheometer (CaBER). Both blood analogs exhibit a shear-thinning behavior similar to that of whole human blood, but their relaxation times, obtained from CaBER experiments, are substantially different (by one order of magnitude). Visualizations of the flow patterns using streak photography, measurements of the velocity field using microparticle image velocimetry, and pressure-drop measurements were carried out experimentally for a wide range of flow rates. The experimental results were also compared with the numerical simulations of the flow of a Newtonian fluid and a generalized Newtonian fluid with shear-thinning behavior. Our results show that the flow patterns of the two blood analog solutions are considerably different, despite their similar shear rheology. Furthermore, we demonstrate that the elastic properties of the fluid have a major impact on the flow characteristics, with the polyacrylamide solution exhibiting a much stronger elastic character. As such, these properties must be taken into account in the choice or development of analog fluids that are adequate to replicate blood behavior at the microscale.

Steady viscoelastic fluid flow between parallel plates under electro-osmotic forces: Phan-Thien-Tanner model.

The electro-osmotic flow of a viscoelastic fluid between parallel plates is investigated analytically. The rheology of the fluid is described by the Phan-Thien-Tanner model. This model uses the Gordon-Schowalter convected derivative, which leads to a non-zero second normal stress difference in pure shear flow. A nonlinear Poisson-Boltzmann equation governing the electrical double-layer field and a body force generated by the applied electrical potential field are included in the analysis. Results are presented for the velocity and stress component profiles in the microchannel for different parametric values that characterize this flow. Equations for the critical shear rates and maximum electrical potential that can be applied to maintain a steady fully developed flow are derived and discussed.

Extensional rheology and elastic instabilities of a wormlike micellar solution in a microfluidic cross-slot device

Wormlike micellar surfactant solutions are encountered in a wide variety of important applications, including enhanced oil recovery and ink-jet printing, in which the fluids are subjected to high extensional strain rates. In this contribution we present an experimental investigation of the flow of a model wormlike micellar solution (cetyl pyridinium chloride and sodium salicylate in deionised water) in a well-defined stagnation point extensional flow field generated within a microfluidic cross-slot device. We use micro-particle image velocimetry (μ-PIV) and full-field birefringence microscopy coupled with macroscopic measurements of the bulk pressure drop to make a quantitative characterization of the fluids rheological response over a wide range of deformation rates. The flow field in the micromachined cross-slot is first characterized for viscous flow of a Newtonian fluid, and μ-PIV measurements show the flow field remains symmetric and stable up to moderately high Reynolds number, Re ≈ 20, and nominal strain rate, nom ≈ 635 s−1. By contrast, in the viscoelastic micellar solution the flow field remains symmetric only for low values of the strain rate such that nom ≤ λM−1, where λM = 2.5 s is the Maxwell relaxation time of the fluid. In this stable flow regime the fluid displays a localized and elongated birefringent strand extending along the outflow streamline from the stagnation point, and estimates of the apparent extensional viscosity can be obtained using the stress-optical rule and from the total pressure drop measured across the cross-slot channel. For moderate deformation rates (nom ≥ λM−1) the flow remains steady, but becomes increasingly asymmetric with increasing flow rate, eventually achieving a steady state of complete anti-symmetry characterized by a dividing streamline and birefringent strand connecting diagonally opposite corners of the cross-slot. Eventually, as the nominal imposed deformation rate is increased further, the asymmetric divided flow becomes time dependent. These purely elastic instabilities are reminiscent of those observed in cross-slot flows of polymer solutions, but seem to be strongly influenced by the effects of shear localization of the micellar fluid within the microchannels and around the re-entrant corners of the cross-slot.

Extensional flow of hyaluronic acid solutions in an optimized microfluidic cross-slot device.

We utilize a recently developed microfluidic device, the Optimized Shape Cross-slot Extensional Rheometer (OSCER), to study the elongational flow behavior and rheological properties of hyaluronic acid (HA) solutions representative of the synovial fluid (SF) found in the knee joint. The OSCER geometry is a stagnation point device that imposes a planar extensional flow with a homogenous extension rate over a significant length of the inlet and outlet channel axes. Due to the compressive nature of the flow generated along the inlet channels, and the planar elongational flow along the outlet channels, the flow field in the OSCER device can also be considered as representative of the flow field that arises between compressing articular cartilage layers of the knee joints during running or jumping movements. Full-field birefringence microscopy measurements demonstrate a high degree of localized macromolecular orientation along streamlines passing close to the stagnation point of the OSCER device, while micro-particle image velocimetry is used to quantify the flow kinematics. The stress-optical rule is used to assess the local extensional viscosity in the elongating fluid elements as a function of the measured deformation rate. The large limiting values of the dimensionless Trouton ratio, Tr ∼ O(50), demonstrate that these fluids are highly extensional-thickening, providing a clear mechanism for the load-dampening properties of SF. The results also indicate the potential for utilizing the OSCER in screening of physiological SF samples, which will lead to improved understanding of, and therapies for, disease progression in arthritis sufferers.

Dynamics of high-Deborah-number entry flows: a numerical study

A. M. AFONSO, P. J. OLIVEIRA, F. T. P INHO AND M. A. ALVES† Departamento de Engenharia Quı́mica, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal Departamento de Engenharia Electromecânica, Unidade de Materiais Texteis e Papeleiros, Universidade da Beira Interior, 6201-001 Covilhã, Portugal Departamento de Engenharia Mecânica, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal

Adaptive multiresolution approach for solution of hyperbolic PDEs

This paper establishes an innovative and efficient multiresolution adaptive approach combined with high-resolution methods, for the numerical solution of a single or a system of partial differential equations. The proposed methodology is unconditionally bounded (even for hyperbolic equations) and dynamically adapts the grid so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed, thus possessing the two desired properties of a numerical approach: stability and accuracy. Numerical results for five test problems are presented which clearly show the robustness and cost effectiveness of the proposed method. 2002 Elsevier Science B.V. All rights reserved.