5 results
Shear thickening of a non-colloidal suspension with a viscoelastic matrix
- Adolfo Vázquez-Quesada, Pep Español, Roger I. Tanner, Marco Ellero
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- Journal:
- Journal of Fluid Mechanics / Volume 880 / 10 December 2019
- Published online by Cambridge University Press:
- 18 October 2019, pp. 1070-1094
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We study the rheology of a non-colloidal suspension of rigid spherical particles interacting with a viscoelastic matrix. Three-dimensional numerical simulations under shear flow are performed using the smoothed particle hydrodynamics method and compared with experimental data available in the literature using different constant-viscosity elastic Boger fluids. The rheological properties of the Boger matrices are matched in simulation under viscometric flow conditions. Suspension rheology under dilute to semi-concentrated conditions (i.e. up to solid volume fraction $\unicode[STIX]{x1D719}=0.3$) is explored. It is found that at small Deborah numbers $De$ (based on the macroscopic imposed shear rate), relative suspension viscosities $\unicode[STIX]{x1D702}_{r}$ exhibit a plateau at every concentration investigated. By increasing $De$, shear thickening is observed, which is related to the extensional thickening of the underlying viscoelastic matrix. Under dilute conditions ($\unicode[STIX]{x1D719}=0.05$), numerical results for $\unicode[STIX]{x1D702}_{r}$ agree quantitatively with experimental data in both the $De$ and $\unicode[STIX]{x1D719}$ dependences. Even under dilute conditions, simulations of full many-particle systems with no a priori specification of their spatial distribution need to be considered to recover precisely experimental values. By increasing the solid volume fraction towards $\unicode[STIX]{x1D719}=0.3$, despite the fact that the trend is well captured, the agreement remains qualitative with discrepancies arising in the absolute values of $\unicode[STIX]{x1D702}_{r}$ obtained from simulations and experiments but also with large deviations existing among different experiments. With regard to the specific mechanism of elastic thickening, the microstructural analysis shows that elastic thickening correlates well with the average viscoelastic dissipation function $\unicode[STIX]{x1D703}^{elast}$, requiring a scaling as $\langle \unicode[STIX]{x1D703}^{elast}\rangle \sim De^{\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\geqslant 2$ to take place. Locally, despite the fact that regions of large polymer stretching (and viscoelastic dissipation) can occur everywhere in the domain, flow regions uniquely responsible for the elastic thickening are well correlated to areas with significant extensional component.
Dynamic simulation of sphere motion in a vertical tube
- ZHAOSHENG YU, NHAN PHAN-THIEN, ROGER I. TANNER
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- Journal:
- Journal of Fluid Mechanics / Volume 518 / 10 November 2004
- Published online by Cambridge University Press:
- 20 October 2004, pp. 61-93
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In this paper, the sedimentation of a sphere and its radial migration in a Poiseuille flow in a vertical tube filled with a Newtonian fluid are simulated with a finite-difference-based distributed Lagrange multiplier (DLM) method. The flow features, the settling velocities, the trajectories and the angular velocities of the spheres sedimenting in a tube at different Reynolds numbers are presented. The results show that at relatively low Reynolds numbers, the sphere approaches the tube axis monotonically, whereas in a high-Reynolds-number regime where shedding of vortices takes place, the sphere takes up a spiral trajectory that is closer to the tube wall than the tube axis. The rotation motion and the lateral motion of the sphere are highly correlated through the Magnus effect, which is verified to be an important (but not the only) driving force for the lateral migration of the sphere at relatively high Reynolds numbers. The standard vortex structures in the wake of a sphere, for Reynolds number higher than 400, are composed of a loop mainly located in a plane perpendicular to the streamwise direction and two streamwise vortex pairs. When moving downstream, the legs of the hairpin vortex retract and at the same time a streamwise vortex pair with rotation opposite to that of the legs forms between the loops. For Reynolds number around 400, the wake structures shed during the impact of the sphere on the wall typically form into streamwise vortex structures or else into hairpin vortices when the sphere spirals down. The radial, angular and axial velocities of both neutrally buoyant and non-neutrally buoyant spheres in a circular Poiseuille flow are reported. The results are in remarkably good agreement with the available experimental data. It is shown that suppresion of the sphere rotation produces significant large additional lift forces pointing towards the tube axis on the spheres in the neutrally buoyant and more-dense-downflow cases, whereas it has a negligible effect on the migration of the more dense sphere in upflow.
Fully developed viscous and viscoelastic flows in curved pipes
- YURUN FAN, ROGER I. TANNER, NHAN PHAN-THIEN
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- Journal:
- Journal of Fluid Mechanics / Volume 440 / 10 August 2001
- Published online by Cambridge University Press:
- 13 August 2001, pp. 327-357
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Some h-p finite element computations have been carried out to obtain solutions for fully developed laminar flows in curved pipes with curvature ratios from 0.001 to 0.5. An Oldroyd-3-constant model is used to represent the viscoelastic fluid, which includes the upper-convected Maxwell (UCM) model and the Oldroyd-B model as special cases. With this model we can examine separately the effects of the fluid inertia, and the first and second normal-stress differences. From analysis of the global torque and force balances, three criteria are proposed for this problem to estimate the errors in the computations. Moreover, the finite element solutions are accurately confirmed by the perturbation solutions of Robertson & Muller (1996) in the cases of small Reynolds/Deborah numbers.
Our numerical solutions and an order-of-magnitude analysis of the governing equations elucidate the mechanism of the secondary flow in the absence of second normal-stress difference. For Newtonian flow, the pressure gradient near the wall region is the driving force for the secondary flow; for creeping viscoelastic flow, the combination of large axial normal stress with streamline curvature, the so-called hoop stress near the wall, promotes a secondary flow in the same direction as the inertial secondary flow, despite the adverse pressure gradient there; in the case of inertial viscoelastic flow, both the larger axial normal stress and the smaller inertia near the wall promote the secondary flow.
For both Newtonian and viscoelastic fluids the secondary volumetric fluxes per unit of work consumption and per unit of axial volumetric flux first increase then decrease as the Reynolds/Deborah number increases; this behaviour should be of interest in engineering applications.
Typical negative values of second normal-stress difference can drastically suppress the secondary flow and in the case of small curvature ratios, make the flow approximate the corresponding Poiseuille flow in a straight pipe. The viscoelasticity of Oldroyd-B fluid causes drag enhancement compared to Newtonian flow. Adding a typical negative second normal-stress difference produces large drag reductions for a small curvature ratio δ = 0.01; however, for a large curvature ratio δ = 0.2, although the secondary flows are also drastically attenuated by the second normal-stress difference, the flow resistance remains considerably higher than in Newtonian flow.
It was observed that for the UCM and Oldroyd-B models, the limiting Deborah numbers met in our steady solution calculations obey the same scaling criterion as proposed by McKinley et al. (1996) for elastic instabilities; we present an intriguing problem on the relation between the Newton iteration for steady solutions and the linear stability analyses.
Tangential flow and advective mixing of viscoplastic fluids between eccentric cylinders
- YURUN FAN, NHAN PHAN-THIEN, ROGER I. TANNER
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- Journal:
- Journal of Fluid Mechanics / Volume 431 / 25 March 2001
- Published online by Cambridge University Press:
- 22 June 2001, pp. 65-89
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This is a study on the tangential flow and advective mixing of viscoplastic fluids (Bingham plastics) between two eccentric, alternately rotating cylinders. Two geometrical configurations and various rotation modes are considered for a relatively large range of the yield stress. The hp-type finite element method with the mixed formulation is used to solve for the steady velocity and pressure fields. The bi-viscosity and the Papanastasiou models agree quantitatively with each other in predicting the velocity fields and the practically unyielded zones. However, the Papanastasiou model is more robust and economic than the bi-viscosity model in the computation using Newton iteration. In the steady flows, in addition to the motionless zones, we have discovered some plugs with rigid rotation, including rotating plugs stuck onto the outer cylinder and rotating, even counter-rotating, plugs disconnected from both cylinders. The unsteady, periodic flow is composed of a sequence of the steady flows, which is valid in the creeping flow regime. The characteristics of advective mixing in these flows have been studied by analysing the asymptotic coverages of a passive tracer, the distributions of the lineal stretching in the flow and the variations of the mean stretching of the flow with time. The tracer coverage is intuitive but qualitative and, occasionally, it depends on the initial location of the tracer. On the other hand, the distribution of stretching is quantitative and more reliable in reflecting the mixing characteristics. Interestingly, the zones of the lowest stretching in the distribution graphs are remarkably well matched with the regular zones in the tracer-coverage graphs. Furthermore, the mixing efficiency proposed by Ottino (1989) is used to characterize the advective mixing in the two geometrical configurations with various rotation modes. It is important to realize that, for plastic fluids, a major barrier to effective mixing is the unyielded fluid plugs which are controlled by the yield stress and geometrical configurations. Therefore, when designing an eccentric helical annular mixer it is important to pay attention first to the geometric issues then to the operating issues.
A numerical study of viscoelastic effects in chaotic mixing between eccentric cylinders
- YURUN FAN, ROGER I. TANNER, NHAN PHAN-THIEN
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- Journal:
- Journal of Fluid Mechanics / Volume 412 / 10 June 2000
- Published online by Cambridge University Press:
- 10 June 2000, pp. 197-225
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In this paper, we are concerned with the effect of fluid elasticity and shear-thinning viscosity on the chaotic mixing of the flow between two eccentric, alternately rotating cylinders. We employ the well-developed h-p finite element method to achieve a high accuracy and efficiency in calculating steady solutions, and a full unsteady algorithm for creeping viscoelastic flows to study the transient process in this periodic viscoelastic flow. Since the distribution of periodic points of the viscoelastic flow is not symmetric, we have developed a domain-search algorithm based on Newton iteration for locating the periodic points. With the piecewise-steady approximation, our computation for the upper-convected Maxwell fluid predicts no noticeable changes of the advected coverage of a passive tracer from Newtonian flow, with elasticity levels up to a Deborah number of 1.0. The stretching of the fluid elements, quantified by the geometrical mean of the spatial distribution, remains exponential up to a Deborah number of 6.0, with only slight changes from Newtonian flow. On the other hand, the shear-thinning viscosity, modelled by the Carreau equation, has a large impact on both the advection of a passive tracer and the mean stretching of the fluid elements. The creeping, unsteady computations show that the transient period of the velocity is much shorter than the transient period of the stress, and from a pragmatic point of view, this transient process caused by stress relaxation due to sudden switches of the cylinder rotation can be neglected for predicting the advective mixing in this time- periodic flow. The periodic points found up to second order and their eigenvalues are indeed very informative in understanding the chaotic mixing patterns and the qualitative changes of the mean stretching of the fluid elements. The comparison between our computations and those of Niederkorn & Ottino (1993) reveals the importance of reducing the discretization error in the computation of chaotic mixing. The causes of the discrepancy between our prediction of the tracer advection and Niederkorn & Ottino's (1993) experiment are discussed, in which the influence of the shear-thinning first normal stress difference is carefully examined. The discussion leads to questions on whether small elasticity of the fluid has a large effect on the chaotic mixing in this periodic flow.