Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-22T13:42:59.110Z Has data issue: false hasContentIssue false
  • English
  • Français

Yield limit analysis of particle motion in a yield-stress fluid

Published online by Cambridge University Press:  24 April 2017

Emad Chaparian  and
Ian A. Frigaard Open the ORCID record for Ian A. Frigaard [Opens in a new window]
Emad Chaparian
Affiliation:
Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada
Ian A. Frigaard*
Affiliation:
Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical and numerical study of yield-stress fluid creeping flow about a particle is presented. Yield-stress fluids can hold rigid particles statically buoyant if the yield stress is large enough. In addressing sedimentation of rigid particles in viscoplastic fluids, we should know this critical ‘yield number’ beyond which there is no motion. As we get close to this limit, the role of viscosity becomes negligible in comparison to the plastic contribution in the leading order, since we are approaching the zero-shear-rate limit. Admissible stress fields in this limit can be found by using the characteristics of the governing equations of perfect plasticity (i.e. the sliplines). This approach yields a lower bound of the critical plastic drag force or equivalently the critical yield number. Admissible velocity fields also can be postulated to calculate the upper bound. This analysis methodology is examined for three families of particle shapes (ellipse, rectangle and diamond) over a wide range of aspect ratios. Numerical experiments of either resistance or mobility problems in a viscoplastic fluid validate the predictions of slipline theory and reveal interesting aspects of the flow in the yield limit (e.g. viscoplastic boundary layers). We also investigate in detail the cases of high and low aspect ratio of the particles.

JFM classification

Non-Newtonian Flows: Non-Newtonian Flows Multiphase and Particle-laden Flows: Particle/fluid flow Non-Newtonian Flows: Plastic materials
Type
Papers
Information
Journal of Fluid Mechanics , Volume 819 , 25 May 2017 , pp. 311 - 351
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adachi, K. & Yoshioka, N. 1973 On creeping flow of a visco-plastic fluid past a circular cylinder. Chem. Engng Sci. 28 (1), 215–226.CrossRefGoogle Scholar
Balmforth, N. J., Craster, R. V., Hewitt, D. R., Hormozi, S. & Maleki, A. 2016 Viscoplastic boundary layers. J. Fluid Mech. (submitted).Google Scholar
Beris, A. N., Tsamopoulos, J. A., Armstrong, R. C. & Brown, R. A. 1985 Creeping motion of a sphere through a Bingham plastic. J. Fluid Mech. 158, 219–244.CrossRefGoogle Scholar
Boujlel, J., Maillard, M., Lindner, A., Ovarlez, G., Chateau, X. & Coussot, P. 2012 Boundary layer in pastes – displacement of a long object through a yield stress fluid. J. Rheol. 56 (5), 1083–1108.CrossRefGoogle Scholar
Chakrabarty, J. 2012 Theory of Plasticity. Butterworth-Heinemann.Google Scholar
Dubash, N. & Frigaard, I. 2004 Conditions for static bubbles in viscoplastic fluids. Phys. Fluids 16 (12), 4319–4330.CrossRefGoogle Scholar
Elgaddafi, R., Ahmed, R., George, M. & Growcock, F. 2012 Settling behavior of spherical particles in fiber-containing drilling fluids. J. Pet. Sci. Engng 84, 20–28.CrossRefGoogle Scholar
Frigaard, I., Iglesias, J. A., Mercier, G., Pöschl, C. & Scherzer, O. 2017 Critical yield numbers of rigid particles settling in Bingham fluids and Cheeger sets. SIAM J. Appl. Maths (in press).CrossRefGoogle Scholar
Frigaard, I. A. & Scherzer, O. 2000 The effects of yield stress variation in uniaxial exchange flows of two Bingham fluids in a pipe. SIAM J. Appl. Maths 60, 1950–1976.CrossRefGoogle Scholar
Hassani, R., Ionescu, I. R. & Lachand-Robert, T. 2005 Shape optimization and supremal minimization approaches in landslides modeling. Appl. Math. Optim. 52 (3), 349–364.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFEM++. J. Numer. Math. 20 (3), 251–265.CrossRefGoogle Scholar
Hild, P., Ionescu, I. R., Lachand-Robert, T. & Roşca, I. 2002 The blocking of an inhomogeneous Bingham fluid. Applications to landslides. ESAIM: Math. Modelling Numer. Anal. 36 (6), 1013–1026.CrossRefGoogle Scholar
Hill, R. 1950 The Mathematical Theory of Plasticity. Oxford University Press.Google Scholar
Huilgol, R. R. 2006 A systematic procedure to determine the minimum pressure gradient required for the flow of viscoplastic fluids in pipes of symmetric cross-section. J. Non-Newtonian Fluid Mech. 136 (2), 140–146.CrossRefGoogle Scholar
Huilgol, R. R. 2015 Fluid Mechanics of Viscoplasticity. Springer.CrossRefGoogle Scholar
Ionescu, I. R. & Lachand-Robert, T. 2005 Generalized Cheeger sets related to landslides. Calc. Var. 23 (2), 227–249.CrossRefGoogle Scholar
Jossic, L. & Magnin, A. 2001 Drag and stability of objects in a yield stress fluid. AIChE J. 47 (12), 2666–2672.CrossRefGoogle Scholar
Karimfazli, I. & Frigaard, I. A. 2013 Natural convection flows of a Bingham fluid in a long vertical channel. J. Non-Newtonian Fluid Mech. 201, 39–55.CrossRefGoogle Scholar
Karimfazli, I. & Frigaard, I. A. 2016 Flow, onset and stability: qualitative analysis of yield stress fluid flow in enclosures. J. Non-Newtonian Fluid Mech. 238, 224–232.CrossRefGoogle Scholar
Karimfazli, I., Frigaard, I. A. & Wachs, A. 2015 A novel heat transfer switch using the yield stress. J. Fluid Mech. 783, 526–566.CrossRefGoogle Scholar
Knappett, J. A., Mohammadi, S. & Griffin, C. 2010 Lateral spreading forces on bridge piers and pile caps in laterally spreading soil: effect of angle of incidence. J. Geotech. Geoenviron. Engng 136 (12), 1589–1599.CrossRefGoogle Scholar
Liu, B. T., Muller, S. J. & Denn, M. M. 2003 Interactions of two rigid spheres translating collinearly in creeping flow in a Bingham material. J. Non-Newtonian Fluid Mech. 113 (1), 49–67.CrossRefGoogle Scholar
Madani, A., Storey, S., Olson, J. A., Frigaard, I. A., Salmela, J. & Martinez, D. M. 2010 Fractionation of non-Brownian rod-like particle suspensions in a viscoplastic fluid. Chem. Engng Sci. 65 (5), 1762–1772.CrossRefGoogle Scholar
Martin, C. M. & Randolph, M. F. 2006 Upper-bound analysis of lateral pile capacity in cohesive soil. Géotechnique 56 (2), 141–145.CrossRefGoogle Scholar
Merkak, O., Jossic, L. & Magnin, A. 2006 Spheres and interactions between spheres moving at very low velocities in a yield stress fluid. J. Non-Newtonian Fluid Mech. 133 (2), 99–108.CrossRefGoogle Scholar
Mosolov, P. P. & Miasnikov, V. P. 1965 Variational methods in the theory of the fluidity of a viscous-plastic medium. Z. Angew. Math. Mech. J. Appl. Math. Mech. 29 (3), 545–577.CrossRefGoogle Scholar
Mosolov, P. P. & Miasnikov, V. P. 1966 On stagnant flow regions of a viscous-plastic medium in pipes. Z. Angew. Math. Mech. J. Appl. Math. Mech. 30 (4), 841–854.CrossRefGoogle Scholar
Murff, J. D., Wagner, D. A. & Randolph, M. F. 1989 Pipe penetration in cohesive soil. Géotechnique 39 (2), 213–229.CrossRefGoogle Scholar
Nirmalkar, N., Chhabra, R. P. & Poole, R. J. 2012 On creeping flow of a Bingham plastic fluid past a square cylinder. J. Non-Newtonian Fluid Mech. 171, 17–30.CrossRefGoogle Scholar
Okrajni, S. S. & Azar, J. J. 1986 The effects of mud rheology on annular hole cleaning in directional wells. SPE Drilling Engineering 1 (04), 297–308.CrossRefGoogle Scholar
Oldroyd, J. G. 1947 Two-dimensional plastic flow of a Bingham solid – a plastic boundary-layer theory for slow motion. Proc. Camb. Phil. Soc. 43, 383–395.CrossRefGoogle Scholar
Poulos, H. G. & Davis, E. H. 1980 Pile Foundation Analysis and Design. Wiley.Google Scholar
Prager, W. 1954 On slow visco-plastic flow. In Studies in Mathematics and Mechanics, pp. 208–216. Academic.Google Scholar
Putz, A. & Frigaard, I. A. 2010 Creeping flow around particles in a Bingham fluid. J. Non-Newtonian Fluid Mech. 165 (5), 263–280.CrossRefGoogle Scholar
Randolph, M. F. & Houlsby, G. T. 1984 The limiting pressure on a circular pile loaded laterally in cohesive soil. Géotechnique 34 (4), 613–623.CrossRefGoogle Scholar
Roquet, N. & Saramito, P. 2003 An adaptive finite element method for Bingham fluid flows around a cylinder. Comput. Meth. Appl. Mech. Engng 192 (31), 3317–3341.CrossRefGoogle Scholar
Roustaei, A., Chevalier, T., Talon, L. & Frigaard, I. A. 2016 Non-Darcy effects in fracture flows of a yield stress fluid. J. Fluid Mech. 805, 222–261.CrossRefGoogle Scholar
Tokpavi, D. L., Magnin, A. & Jay, P. 2008 Very slow flow of Bingham viscoplastic fluid around a circular cylinder. J. Non-Newtonian Fluid Mech. 154 (1), 65–76.CrossRefGoogle Scholar
Tsamopoulos, J., Dimakopoulos, Y., Chatzidai, N., Karapetsas, G. & Pavlidis, M. 2008 Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment. J. Fluid Mech. 601, 123–164.CrossRefGoogle Scholar
Wachs, A. & Frigaard, I. A. 2016 Particle settling in yield stress fluids: limiting time, distance and applications. J. Non-Newtonian Fluid Mech. 238, 189–204.CrossRefGoogle Scholar
Xiaofeng, S., Kelin, W., Tie, Y., Yang, Z., Shuai, S. & Shizhu, L. 2013 Review of hole cleaning in complex structural wells. Open Petrol. Engng J. 6, 25–32.CrossRefGoogle Scholar