Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T07:19:41.980Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hele-Shaw injection problem for an extremely shear-thinning fluid

Published online by Cambridge University Press:  23 July 2015

G. RICHARDSON  and
J. R. KING
G. RICHARDSON
Affiliation:
Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK email: [email protected]
J. R. KING
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider Hele-Shaw flows driven by injection of a highly shear-thinning power-law fluid (of exponent n) in the absence of surface tension. We formulate the problem in terms of the streamfunction ψ, which satisfies the p-Laplacian equation ∇·(|∇ψ|p−2∇ψ) = 0 (with p = (n+1)/n) and use the method of matched asymptotic expansions in the large n (extreme-shear-thinning) limit to find an approximate solution. The results show that significant flow occurs only in (I) segments of a (single) circle centred on the injection point, whose perimeters comprise the portion of free boundary closest to the injection point and (II) an exponentially small region around the injection point and (III) a transition region to the rest of the fluid: while the flow in the latter is exponentially slow it can be characterised in detail.

Keywords

power law fluidsmatched asymptotic expansionsfree boundary problemsp-Laplace equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Special Anniversay Issue 5: Celebrating 75 years , October 2015 , pp. 563 - 594
Copyright
Copyright © Cambridge University Press 2015 

References

[1]Alexandrou, A. N. & Entov, V. (1997) On the steady-state advancement of fingers and bubbles in a Hele-Shaw cell filled by a non-Newtonian fluid. Euro. J. Appl. Math. 8, 73–87.CrossRefGoogle Scholar
[2]Aronsson, G. (1996) On p-harmonic functions, convex duality and an asymptotic formula for injection moulding. Euro. J. Appl. Math. 8, 417–437.CrossRefGoogle Scholar
[3]Aronsson, G. (1968) On the partial differential equation ux 2uxx + 2uxuyuxy + uy 2uyy = 0. Arkiv für Mat. 7, 395–425.CrossRefGoogle Scholar
[4]Aronsson, G. (2003) Five geometric principles of injection moulding. Intern. Polymer Proc. 18, 91–94.CrossRefGoogle Scholar
[5]Aronsson, G. & Evans, L. C. (2002) An asymptotic model for compression molding. Indiana Univ. Math. J. 51, 1–36.CrossRefGoogle Scholar
[6]Aronsson, G. & Janfalk, U. (1992) On Hele-Shaw flows of power law fluids. Euro. J. Appl. Math. 3, 343–366.CrossRefGoogle Scholar
[7]Atkinson, C. & Champion, C. R. (1984) Some boundary-value-problems for the equation ∇·(|∇φ|N∇φ) = 0. Q. J. Mech. Appl. Math. 37, 401–419.CrossRefGoogle Scholar
[8]Ben Amar, M. & Corveira Poire, E. (1998) Finger behaviour of shear thinning fluid in a Hele-Shaw cell. Phys. Rev. Lett. 81, 2048–2051.Google Scholar
[9]Ben Amar, M. & Corveira Poire, E. (1999) Pushing a non-Newtonian fluid in a Hele-Shaw cell: From fingers to needles. Phys. Fluids 11, 1757–1767.CrossRefGoogle Scholar
[10]Bergwall, A. (2002) A geometric evolution problem. Quart. Appl. Math. 50, 37–73.CrossRefGoogle Scholar
[11]Brewster, M. A., Chapman, S. J., Fitt, A. D. & Please, C. P. (1995) Asymptotics of slow flow of very small exponent shear thinning fluids in a wedge. Euro. J. Appl. Math. 6, 559–571.CrossRefGoogle Scholar
[12]Ceniceros, H. D., Hou, T. Y. & Si, H. (1999) Numerical study of Hele-Shaw flow with suction. Phys. Fluids 11, 2471–2486.CrossRefGoogle Scholar
[13]Chapman, S. J., Fitt, A. D. & Please, C. P. (1997) Extrusion of power law shear thinning fluids with small exponent. Int. J. Non-Linear Mech. 32, 187–199.CrossRefGoogle Scholar
[14]Cummings, L. J. & King, J. R. (2004) Hele-Shaw flow with a point sink: Generic solution breakdown. Euro. J. Appl. Math. 15, 1–37.CrossRefGoogle Scholar
[15]Drucker, D. & Williams, S. A. (2009) A note on Aronsson's equation. Rocky Mt. J. Math. 39, 1859–1869.CrossRefGoogle Scholar
[16]Evans, L. C. (1991) The 1-Laplacian, the ∞-Laplacian and differential games. URL: http://math.berkeley.edu/~evans/brezis.pdfGoogle Scholar
[17]Evans, L. C. & Yu, Y. (2005) Various properties of solutions of the Infinity-Laplacian equation. Comm. Partial Differ. Equ. 30, 1401–1428.CrossRefGoogle Scholar
[18]Galin, L. A. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 7, 250–253 (in Russian).Google Scholar
[19]Gurtin, M. E. (1993) Thermomechanics of Evolving Phase Boundaries in the Plane, OUP, Oxford.CrossRefGoogle Scholar
[20]Howison, S. D. (1992) Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3, 209–224.CrossRefGoogle Scholar
[21]Howison, S. D., Lacey, A. A. & Ockendon, J. R. (1988) Hele-Shaw Free boundary problems with suction. Quart. J. Mech. Appl. Math. 41, 183–193.CrossRefGoogle Scholar
[22]Howison, S. D., Morgan, J. D. & Ockendon, J. R. (1997) A class of codimension two free boundary problems. SIAM Rev. 39, 221–253.CrossRefGoogle Scholar
[23]Kelly, E. D. & Hinch, E. J. (1997) Numerical simulations of sink flow in the Hele-Shaw cell with small surface tension. Euro. J. Appl. Math. 8, 553–550.CrossRefGoogle Scholar
[24]King, J. R. (1990) Some non-local transformations between nonlinear diffusion equations. J. Phys. A 23, 5441–5464.CrossRefGoogle Scholar
[25]King, J. R. (1995) Development of singularities in some moving boundary value problems. Euro. J. Appl. Math. 6, 491–507.CrossRefGoogle Scholar
[26]Mullins, W. W. (1956) Two-dimensional motion of idealised grain boundaries. J. Appl. Phys. 27, 900–904.CrossRefGoogle Scholar
[27]Moser, R. (2007) The inverse mean curvature flow and p-harmonic functions. J. Eur. Math. Soc. 9, 77–83.CrossRefGoogle Scholar
[28]Ockendon, J. R. & Howison, S. D. (2002) Kochina and Hele-Shaw in modern mathematics, natural science and industry. J. Appl. Math. Mech. 66, 505–512.CrossRefGoogle Scholar
[29]Piscotti, F., Boldizar, A., Righdal, M. & Aronsson, G. (2002) Evaluation of a model describing the advancing flow front in injection moulding. Intern. Polymer Proc. 17, 133–145.CrossRefGoogle Scholar
[30]Polubarinova-Kochina, P. Ya. (1945) On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47, 254–257 (in Russian).Google Scholar
[31]Richardson, S. (1981) Some Hele Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278.CrossRefGoogle Scholar
[32]Richardson, G. & King, J. R. (2007) The Saffman-Taylor problem for an extremely shear-thinning fluid. Quart. J. Mech. Appl. Math. 60, 139–160.CrossRefGoogle Scholar
[33]Richardson, G. & King, J. R. (2002) Motion by curvature of a three-dimensional filament: Similarity solutions. Interfaces Free Boundaries 4, 395–421.CrossRefGoogle Scholar
[34]Richardson, G. & King, J. R. (2002) The evolution of space curves by curvature and torsion. J. Phys. A: Math. Gen. 35, 9857–9879.CrossRefGoogle Scholar
[35]Sapiro, G. & Tannenbaum, A. (1992) Affine invariant scale space. Int. J. Comput. Vis. 11, 25–44.CrossRefGoogle Scholar
[36]Smoczyk, K. (2005) A representation formula for the inverse harmonic mean curvature flow. Elemente der Math. 60, 57–65.CrossRefGoogle Scholar
[37]Tanveer, S. (2000) Surprises in viscous fingering. J. Fluid Mech. 409, 273–308.CrossRefGoogle Scholar