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Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data
- Rahul N. Chacko, Romain Mari, Suzanne M. Fielding, Michael E. Cates
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- Journal:
- Journal of Fluid Mechanics / Volume 847 / 25 July 2018
- Published online by Cambridge University Press:
- 29 May 2018, pp. 700-734
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Dense suspensions of hard particles are important as industrial or environmental materials (e.g. fresh concrete, food, paint or mud). To date, most constitutive models developed to describe them are, explicitly or effectively, ‘fabric evolution models’ based on: (i) a stress rule connecting the macroscopic stress to a second-rank microstructural fabric tensor $\unicode[STIX]{x1D64C}$; and (ii) a closed time-evolution equation for $\unicode[STIX]{x1D64C}$. In dense suspensions, most of the stress comes from short-ranged pairwise steric or lubrication interactions at near-contacts (suitably defined), so a natural choice for $\unicode[STIX]{x1D64C}$ is the deviatoric second moment of the distribution $P(\boldsymbol{p})$ of the near-contact orientations $\boldsymbol{p}$. Here we test directly whether a closed time-evolution equation for such a $\unicode[STIX]{x1D64C}$ can exist, for the case of inertialess non-Brownian hard spheres in a Newtonian solvent. We perform extensive numerical simulations accessing high levels of detail for the evolution of $P(\boldsymbol{p})$ under shear reversal, providing a stringent test for fabric evolution models. We consider a generic class of these models as defined by Hand (J. Fluid Mech., vol. 13, 1962, pp. 33–46) that assumes little as to the micromechanical behaviour of the suspension and is only constrained by frame indifference. Motivated by the smallness of microstructural anisotropies in the dense regime, we start with linear models in this class and successively consider those increasingly nonlinear in $\unicode[STIX]{x1D64C}$. Based on these results, we suggest that no closed fabric evolution model properly describes the dynamics of the fabric tensor under reversal. We attribute this to the fact that, while a second-rank tensor captures reasonably well the microstructure in steady flows, it gives a poor description during significant parts of the microstructural evolution following shear reversal. Specifically, the truncation of $P(\boldsymbol{p})$ at second spherical harmonic (or second-rank tensor) level describes ellipsoidal distributions of near-contact orientations, whereas on reversal we observe distributions that are markedly four-lobed; moreover, ${\dot{P}}(\boldsymbol{p})$ has oblique axes, not collinear with those of $\unicode[STIX]{x1D64C}$ in the shear plane. This structure probably precludes any adequate closure at second-rank level. Instead, our numerical data suggest that closures involving the coupled evolution of both a fabric tensor and a fourth-rank tensor might be reasonably accurate.
Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions
- Michael E. Cates, Elsen Tjhung
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- Journal:
- Journal of Fluid Mechanics / Volume 836 / 10 February 2018
- Published online by Cambridge University Press:
- 18 December 2017, P1
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Binary fluid mixtures are examples of complex fluids whose microstructure and flow are strongly coupled. For pairs of simple fluids, the microstructure consists of droplets or bicontinuous demixed domains and the physics is controlled by the interfaces between these domains. At continuum level, the structure is defined by a composition field whose gradients – which are steep near interfaces – drive its diffusive current. These gradients also cause thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects the composition field, while thermal noise creates additional random fluxes that allow the system to explore its configuration space and move towards the Boltzmann distribution. This article introduces continuum models of binary fluids, first covering some well-studied areas such as the thermodynamics and kinetics of phase separation, and emulsion stability. We then address cases where one of the fluid components has anisotropic structure at mesoscopic scales creating nematic (or polar) liquid-crystalline order; this can be described through an additional tensor (or vector) order parameter field. We conclude by outlining a thriving area of current research, namely active emulsions, in which one of the binary components consists of living or synthetic material that is continuously converting chemical energy into mechanical work. Such activity can be modelled with judicious additional terms in the equations of motion for simple or liquid-crystalline binary fluids. Throughout, the emphasis of the article is on presenting the theoretical tools needed to address a wide range of physical phenomena. Examples include the kinetics of fluid–fluid demixing from an initially uniform state; the result of imposing a steady macroscopic shear flow on this demixing process; and the diffusive coarsening, Brownian motion and coalescence of emulsion droplets. We discuss strategies to create long-lived emulsions by adding trapped species, solid particles, or surfactants; to address the latter, we outline the theory of bending energy for interfacial films. In emulsions where one of the components is liquid-crystalline, ‘anchoring’ terms can create preferential orientation tangential or normal to the fluid–fluid interface. These allow droplets of an isotropic fluid in a liquid crystal (or vice versa) to support a variety of topological defects, which we describe, altering their interactions and stability. Addition of active terms to the equations of motion for binary simple fluids creates a model of ‘motility-induced’ phase separation, where demixing stems from self-propulsion of particles rather than their interaction forces, altering the relation between interfacial structure and fluid stress. Coupling activity to binary liquid crystal dynamics creates models of active liquid-crystalline emulsion droplets. Such droplets show various modes of locomotion, some of which strikingly resemble the swimming or crawling motions of biological cells.
Contributors
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- By Graham Allan, Donna M. Allen, Irwin Altman, Arthur Aron, Donald H. Baucom, Steven R. H. Beach, Ellen Berscheid, Rosemary Blieszner, Jeffrey Boase, Tyfany M. J. Boettcher, Barbara B. Brown, Abraham P. Buunk, Lorne Campbell, Daniel J. Canary, Rodney Cate, John P. Caughlin, Mahnaz Charania, Jennie Y. Chen, F. Scott Christopher, Jennifer A. Clarke, Marilyn Coleman, W. Andrew Collins, Michael K. Coolsen, Nathan R. Cottle, Carolyn E. Cutrona, Marianne Dainton, Valerian J. Derlega, Lisa M. Diamond, Pieternel Dijkstra, Steve Duck, Pearl A. Dykstra, Norman B. Epstein, Beverley Fehr, Frank D. Fincham, Helen E. Fisher, Julie Fitness, Garth J. O. Fletcher, Myron D. Friesen, Lawrence Ganong, Kelli A. Gardner, Jenny de Jong Gierveld, Robin Goodwin, Christine R. Gray, Kathryn Greene, David W. Harris, Willard W. Hartup, John H. Harvey, Kathi L. Heffner, Ted L. Huston, William J. Ickes, Emily A. Impett, Michael P. Johnson, Deborah J. Jones, Deborah A. Kashy, Janice K. Kiecolt‐Glaser, Jeffrey L. Kirchner, Brighid M. Kleinman, Galena H. Kline, Mark L. Knapp, Ascan Koerner, Jean‐Philippe Laurenceau, Kim Leon, Timothy J. Loving, Stephanie D. Madsen, Howard J. Markman, Alicia Mathews, Mario Mikulincer, Patricia Noller, Nickola C. Overall, Letitia Anne Peplau, Daniel Perlman, Sally Planalp, Urmila Pillay, Nicole D. Pleasant, Caryl E. Rusbult, Barbara R. Sarason, Irwin G. Sarason, Phillip R. Shaver, Alan L. Sillars, Jeffry A. Simpson, Susan Sprecher, Susan Stanton, Greg Strong, Catherine A. Surra, Anita L. Vangelisti, C. Arthur VanLear, Theo van Tilburg, Barry Wellman, Amy Wenzel, Carol M. Werner, Adam R. West, Sarah W. Whitton, Heike A. Winterheld
- Edited by Anita L. Vangelisti, University of Texas, Austin, Daniel Perlman, University of British Columbia, Vancouver
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- Book:
- The Cambridge Handbook of Personal Relationships
- Published online:
- 05 June 2012
- Print publication:
- 05 June 2006, pp xvii-xxii
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Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study
- VIVIEN M. KENDON, MICHAEL E. CATES, IGNACIO PAGONABARRAGA, J.-C. DESPLAT, PETER BLADON
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- Journal:
- Journal of Fluid Mechanics / Volume 440 / 10 August 2001
- Published online by Cambridge University Press:
- 13 August 2001, pp. 147-203
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The late-stage demixing following spinodal decomposition of a three-dimensional symmetric binary fluid mixture is studied numerically, using a thermodynamically consistent lattice Boltzmann method. We combine results from simulations with different numerical parameters to obtain an unprecedented range of length and time scales when expressed in reduced physical units. (These are the length and time units derived from fluid density, viscosity, and interfacial tension.) Using eight large (2563) runs, the resulting composite graph of reduced domain size l against reduced time t covers 1 [lsim ] l [lsim ] 105, 10 [lsim ] t [lsim ] 108. Our data are consistent with the dynamical scaling hypothesis that l(t) is a universal scaling curve. We give the first detailed statistical analysis of fluid motion, rather than just domain evolution, in simulations of this kind, and introduce scaling plots for several quantities derived from the fluid velocity and velocity gradient fields. Using the conventional definition of Reynolds number for this problem, Reϕ = ldl/dt, we attain values approaching 350. At Reϕ [gsim ] 100 (which requires t [gsim ] 106) we find clear evidence of Furukawa's inertial scaling (l ∼ t2/3), although the crossover from the viscous regime (l ∼ t) is both broad and late (102 [lsim ] t [lsim ] 106). Though it cannot be ruled out, we find no indication that Reϕ is self-limiting (l ∼ t1/2) at late times, as recently proposed by Grant & Elder. Detailed study of the velocity fields confirms that, for our most inertial runs, the RMS ratio of nonlinear to viscous terms in the Navier–Stokes equation, R2, is of order 10, with the fluid mixture showing incipient turbulent characteristics. However, we cannot go far enough into the inertial regime to obtain a clear length separation of domain size, Taylor microscale, and Kolmogorov scale, as would be needed to test a recent ‘extended’ scaling theory of Kendon (in which R2 is self-limiting but Reϕ not). Obtaining our results has required careful steering of several numerical control parameters so as to maintain adequate algorithmic stability, efficiency and isotropy, while eliminating unwanted residual diffusion. (We argue that the latter affects some studies in the literature which report l ∼ t2/3 for t [lsim ] 104.) We analyse the various sources of error and find them just within acceptable levels (a few percent each) in most of our datasets. To bring these under significantly better control, or to go much further into the inertial regime, would require much larger computational resources and/or a breakthrough in algorithm design.