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Viscoplastic Asymptotics and Other Analytical Methods

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Lectures on Visco-Plastic Fluid Mechanics

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 583))

Abstract

The goal of these lectures is to outline some of the ideas behind the use of asymptotic analysis and other analytical methods in viscoplastic fluid mechanics

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Notes

  1. 1.

    The corresponding analysis presented in Balmforth and Craster (1999) contains erroneous powers; Putz et al. (2009) previously provide corrected formulae.

  2. 2.

    The determination of solutions of similarity form is a mathematical subject of itself. One option for the task is to introduce the rescalings, \((\hat{x},\hat{\zeta })=(kx,\ell \zeta )\), and then to establish what combinations (if any) of the parameters k and \(\ell \) lead to the same equation as (69). One finds this to be so if \(k^2=\ell ^3\). The combination \(\chi =x/\zeta ^{3/2}=\hat{x}/\hat{\zeta }^{3/2}\) is therefore invariant under the transformation and implies the existence of a self-similar solution with \(U=U(\chi )\).

  3. 3.

    The first shock condition (Rankine-Hugoniot relation; see Whitham 1974) follows from first writing the mass-conservation equation in (98) as a integral conservation law,

    $$\begin{aligned} 0 = \frac{{\mathrm d}}{{\mathrm d}t}\int _{x_1}^{x_2} \eta (x,t){\mathrm d}x + \left[ J(x,t)\right] _{x=x_1}^{x=x_2} = \int _{x_1}^{X} \eta _t(x,t) {\mathrm d}x \\ + \int _{X}^{x_2} \eta _t(x,t) {\mathrm d}x + \left[ J(x,t)-\dot{X}\eta (x,t)\right] _{x=x_1}^{x=x_2} , \end{aligned}$$

    where \(J=hu\) is the mass flux, \(x_1\) and \(x_2\) are arbitrary locations straddling the shock, and \(X^\pm \) implies the limit from either the left or right. By taking \(x_1\rightarrow X^-\) and \(x_2\rightarrow X^+\) and arguing that the remaining integrals then become negligible, we then find

    $$ \dot{X} \left[ \eta (x,t)\right] _{x=X^-}^{x=X^+} = \left[ J(x,t)\right] _{x=X^-}^{x=X^+} . $$

    The second relation follows from applying the same argument to the momentum equation in (98) (\(\Gamma =0\)):

    $$ \dot{X}{{\mathrm{Re}}}\left[ \eta u\right] _{x=X^-}^{x=X^+} = \left[ 2\eta \sigma - {1\over 2}{{\mathrm G}_{\mathrm z}}\eta ^2\right] _{x=X^-}^{x=X^+} . $$

    .

  4. 4.

    The restriction to varicose perturbations ignores the (very real) possibility that sinuous perturbations may be more unstable and dominate the break-up dynamics. Sinuous perturbations require a consideration of the bending of the midline of the fluid layer, along the lines considered in Sect. 4.5.

  5. 5.

    Note that there are typographical errors in the corresponding formulae for order-one curvature presented by Balmforth and Hewitt (2013). Specifically, the moment terms in their (95) and (116) should be \(-{\epsilon }\kappa M_s\), and that in (98) should be \(-\kappa M_s\).

References

  • Alishaev, M. G., Entov, V. M., & Segalov, A. E. (1969). Elementary solutions of plane nonlinear filtration problems. Fluid Dynamics, 4(3), 77–84.

    Article  Google Scholar 

  • Balmforth, N. J., & Craster, R. V. (1999). A consistent thin-layer theory for Bingham plastics. Journal of Non-Newtonian Fluid Mechanics, 84, 65–81.

    Article  Google Scholar 

  • Balmforth, N. J., & Hewitt, I. J. (2013). Viscoplastic sheets and threads. Journal of Non-Newtonian Fluid Mechanics, 193, 28–42.

    Article  Google Scholar 

  • Balmforth, N. J., & Kerswell, R. R. (2005). Granular collapse in two dimensions. Journal of Fluid Mechanics, 538, 399–428.

    Article  MathSciNet  Google Scholar 

  • Balmforth, N. J., Craster, R. V., Hewitt, D. R., Hormozi, S., & Maleki, A. (2017). Viscoplastic boundary layers. Journal of Fluid Mechanics, 813, 929–954.

    Article  MathSciNet  Google Scholar 

  • Balmforth, N. J., Craster, R. V., Perona, P., Rust, A. C., & Sassi, R. (2007a). Viscoplastic dam breaks and the Bostwick consistometer. Journal of Non-Newtonian Fluid Mechanics, 142, 63–78.

    Article  Google Scholar 

  • Balmforth, N. J., Craster, R. V., Rust, A. C., & Sassi, R. (2007b). Viscoplastic flow over an inclined surface. Journal of Non-Newtonian Fluid Mechanics, 142, 219–243.

    Article  Google Scholar 

  • Balmforth, N. J., Dubash, N., & Slim, A. C. (2010) Extensional dynamics of viscoplastic filaments: I and II. Journal of Non-Newtonian Fluid Mechanics, 165, 1139–1146 and 1147–1160.

    Article  Google Scholar 

  • Barnes, H. A. (1995). A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure. Journal of Non-Newtonian Fluid Mechanics, 56, 221–251.

    Article  Google Scholar 

  • Bender, C. M. & Orszag, S. A. (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill.

    Google Scholar 

  • Bernadiner, M. G. & Protopapas, A. L. (1994) Progress on the theory of flow in geologic media with threshold gradient. Journal of Environmental Science & Health Part A, 29(1), 249–275.

    Google Scholar 

  • Bittleston, S. H., Ferguson, J., & Frigaard, I. A. (2002). Mud removal and cement placement during primary cementing of an oil well: Laminar non-Newtonian displacements in an eccentric annular Hele-Shaw cell. Journal of Engineering Mathematics, 43(2–4), 229–253.

    Article  MathSciNet  Google Scholar 

  • Bleyer, J., & Coussot, P. (2014). Breakage of non-Newtonian character in flow through a porous medium: Evidence from numerical simulation. Physical Review E, 89(6), 063018.

    Article  Google 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. Journal of Rheology, 56, 1083–1108.

    Article  Google Scholar 

  • Chamberlain, J. A., Sader, J. E., Landman, K. A., & White, L. R. (2001). Incipient plane-strain failure of a rectangular block under gravity. International Journal of Mechanical Sciences, 43, 793–815.

    Article  Google Scholar 

  • Chaparian, E., & Frigaard, I. A. (2017). Yield limit analysis of particle motion in a yield-stress fluid. Journal of Fluid Mechanics, 819, 311–351.

    Article  MathSciNet  Google Scholar 

  • Chevalier, T., Chevalier, C., Clain, X., Dupla, J. C., Canou, J., Rodts, S., et al. (2013a). Darcy’s law for yield stress fluid flowing through a porous medium. Journal of Non-Newtonian Fluid Mechanics, 195, 57–66.

    Article  Google Scholar 

  • Chevalier, T., Rodts, S., Chateau, X., Boujlel, J., Maillard, M., & Coussot, P. (2013b). Boundary layer (shear-band) in frustrated viscoplastic flows. Europhysics Letters, 102, 48002.

    Article  Google Scholar 

  • Covey, G. H., & Stanmore, B. R. (1981). Use of the parallel-plate plastometer for the characterisation of viscous fluids with a yield stress. Journal of Non-Newtonian Fluid Mechanics, 8, 249–260.

    Article  Google Scholar 

  • Craster, R. V., & Matar, O. K. (2009). Dynamics and stability of thin liquid films. Reviews of Modern Physics, 81, 1131.

    Article  Google Scholar 

  • Entov, V. M. (1970). Analogy between equations of plane filtration and equations of longitudinal shear of nonlinearly elastic and plastic solids. Journal of Applied Mathematics and Mechanics, 34(1), 153–164.

    Article  MathSciNet  Google Scholar 

  • Fernández-Nieto, E. D., Noble, P., & Vila, J.-P. (2010). Shallow water equations for non-newtonian fluids. Journal of Non-Newtonian Fluid Mechanics, 165, 712–732.

    Article  Google Scholar 

  • Fusi, L., Farina, A., & Rosso, F. (2012). Flow of a Bingham-like fluid in a finite channel of varying width: a two-scale approach. Journal of Non-Newtonian Fluid Mechanics, 177, 76–88.

    Article  Google Scholar 

  • Hansen, C. J., Wu, W., Toohey, K. S., Sottos, N. R., White, S. R., & Lewis, J. A. (2009). Self-healing materials with interpenetrating microvascular networks. Advanced Materials, 21, 4143–4147.

    Article  Google Scholar 

  • Hewitt, D. R., Daneshi, M., Balmforth, N. J., & Martinez, D. M. (2016). Obstructed and channelized viscoplastic flow in a hele-shaw cell. Journal of Fluid Mechanics, 790, 173–204.

    Article  MathSciNet  Google Scholar 

  • Hewitt, I. J., & Balmforth, N. J. (2012). Viscoplastic lubrication theory with application to bearings and the washboard instability of a planing plate. Journal of Non-Newtonian Fluid Mechanics, 169, 74–90.

    Article  Google Scholar 

  • Jalaal, M. (2016). Controlled spreading of complex droplets (Doctoral dissertation, University of British Columbia).

    Google Scholar 

  • Hinch, E. J. (1991). Perturbation methods. Cambridge University Press.

    Google Scholar 

  • Jalaal, M., Balmforth, N. J., & Stoeber, B. (2015). Slip of spreading viscoplastic droplets. Langmuir, 31(44), 12071–12075.

    Article  Google Scholar 

  • Lipscomb, G. G., & Denn, M. M. (1984). Flow of Bingham fluids in complex geometries. Journal of Non-Newtonian Fluid Mechanics, 14, 337–346.

    Article  Google Scholar 

  • Liu, K. F., & Mei, C. C. (1989). Slow spreading of a sheet of Bingham fluid on an inclined plane. Journal of Fluid Mechanics, 207, 505–529.

    Article  Google Scholar 

  • Mansfield, E. H. (2005). The bending and stretching of plates. Cambridge University Press.

    Google Scholar 

  • Nye, J. F. (1952). The mechanics of glacier flow. Journal of Glaciology, 2, 82–93.

    Article  Google Scholar 

  • Nye, J. F. (1967). Plasticity solution for a glacier snout. Journal of Glaciology, 6(47), 695–715.

    Article  Google Scholar 

  • Oldroyd, J. G. (1947). Two-dimensional plastic flow of a Bingham solid. Mathematical Proceedings of the Cambridge Philosophical Society, 43, 383–395.

    Article  MathSciNet  Google Scholar 

  • Pelipenko, S., & Frigaard, I. A. (2004). Two-dimensional computational simulation of eccentric annular cementing displacements. IMA journal of applied mathematics, 69(6), 557–583.

    Article  MathSciNet  Google Scholar 

  • Piau, J.-M. (2002). Viscoplastic boundary layer. Journal of Non-Newtonian Fluid Mechanics, 102, 193–218.

    Article  Google Scholar 

  • Pinkus, O., & Sternlicht, B. (1961). Theory of hydrodynamic lubrication. McGraw-Hill.

    Google Scholar 

  • Prager, W., & Hodge, P.G. (1968). Theory of perfectly plastic solids. Dover.

    Google Scholar 

  • Putz, A., Frigaard, I. A., & Martinez, D. M. (2009). The lubrication paradox & use of regularisation methods for lubrication flows. Journal of Non-Newtonian Fluid Mechanics, 163, 62–77.

    Article  Google 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.

    Article  Google Scholar 

  • Roustaei, A., Gosselin, A., & Frigaard, I. A. (2015). Residual drilling mud during conditioning of uneven boreholes in primary cementing. part 1: Rheology and geometry effects in non-inertial flows. Journal of Non-Newtonian Fluid Mechanics, 220, 87–98.

    Article  MathSciNet  Google Scholar 

  • Schoof, C., & Hewitt, I. A. (2013). Ice-sheet dynamics. Annual Review of Fluid Mechanics, 45, 217–239.

    Article  MathSciNet  Google Scholar 

  • Smyrnaios, D. N., & Tsamopoulos, J. A. (2001). Squeeze flow of Bingham plastics. Journal of Non-Newtonian Fluid Mechanics, 100, 165–190.

    Article  Google Scholar 

  • Smyrnaios, D. N., & Tsamopoulos, J. A. (2006). Transient squeeze flow of viscoplastic materials. Journal of Non-Newtonian Fluid Mechanics, 133, 35–56.

    Article  Google Scholar 

  • Sneddon, I. N. (1957). Elements of partial differential equations. McGraw-Hill.

    Google Scholar 

  • Talon, L., & Bauer, D. (2013). On the determination of a generalized Darcy equation for yield-stress fluid in porous media using a Lattice-Boltzmann TRT scheme. The European Physical Journal E, 36, 139.

    Article  Google Scholar 

  • Tokpavi, D. L., Magnin, A., & Jay, P. (2008). Very slow flow of Bingham viscoplastic fluid around a circular cylinder. Journal of Non-Newtonian Fluid Mechanics, 154, 65–76.

    Article  Google Scholar 

  • Walton, I. C., & Bittleston, S. H. (1991). The axial flow of a Bingham plastic in a narrow eccentric annulus. Journal of Fluid Mechanics, 222, 39–60.

    Article  Google Scholar 

  • Whitham, G. B. (1974). Linear and nonlinear waves. Wiley.

    Google Scholar 

  • Wilson, S. D. R. (1999). A note on thin-layer theory for Bingham plastics. Journal of Non-Newtonian Fluid Mechanics, 1, 29–33.

    Article  Google Scholar 

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Acknowledgements

I thank two Hewitts for their contributions to the work summarized in these notes: Ian Hewitt computed and prepared Figs. 5, 20 and 21. Duncan Hewitt computed and prepared Figs. 8, 11, 12, 13 and 14. Section 3 is a prelude to a more thorough discussion by Balmforth et al. (2017). I thank Richard Craster for the construction of the slipline field in Fig. 12. Lujia Liu contributed to the developments of Sect. 4.4.

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  1. University of British Columbia, Vancouver, B.C., Canada

    Neil J. Balmforth

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  1. Neil J. Balmforth
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Correspondence to Neil J. Balmforth .

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  1. Laboratory of the Future, University of Bordeaux, Pessac, France

    Guillaume Ovarlez

  2. Department of Mechanical Engineering, Ohio University, Athens, Ohio, USA

    Sarah Hormozi

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Balmforth, N.J. (2019). Viscoplastic Asymptotics and Other Analytical Methods. In: Ovarlez, G., Hormozi, S. (eds) Lectures on Visco-Plastic Fluid Mechanics. CISM International Centre for Mechanical Sciences, vol 583. Springer, Cham. https://doi.org/10.1007/978-3-319-89438-6_2

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