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

  • G. RICHARDSON (a1) and J. R. KING (a2)

Abstract

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.

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References

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[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, 7387.
[2]Aronsson, G. (1996) On p-harmonic functions, convex duality and an asymptotic formula for injection moulding. Euro. J. Appl. Math. 8, 417437.
[3]Aronsson, G. (1968) On the partial differential equation ux 2uxx + 2uxuyuxy + uy 2uyy = 0. Arkiv für Mat. 7, 395425.
[4]Aronsson, G. (2003) Five geometric principles of injection moulding. Intern. Polymer Proc. 18, 9194.
[5]Aronsson, G. & Evans, L. C. (2002) An asymptotic model for compression molding. Indiana Univ. Math. J. 51, 136.
[6]Aronsson, G. & Janfalk, U. (1992) On Hele-Shaw flows of power law fluids. Euro. J. Appl. Math. 3, 343366.
[7]Atkinson, C. & Champion, C. R. (1984) Some boundary-value-problems for the equation ∇·(|∇φ|N∇φ) = 0. Q. J. Mech. Appl. Math. 37, 401419.
[8]Ben Amar, M. & Corveira Poire, E. (1998) Finger behaviour of shear thinning fluid in a Hele-Shaw cell. Phys. Rev. Lett. 81, 20482051.
[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, 17571767.
[10]Bergwall, A. (2002) A geometric evolution problem. Quart. Appl. Math. 50, 3773.
[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, 559571.
[12]Ceniceros, H. D., Hou, T. Y. & Si, H. (1999) Numerical study of Hele-Shaw flow with suction. Phys. Fluids 11, 24712486.
[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, 187199.
[14]Cummings, L. J. & King, J. R. (2004) Hele-Shaw flow with a point sink: Generic solution breakdown. Euro. J. Appl. Math. 15, 137.
[15]Drucker, D. & Williams, S. A. (2009) A note on Aronsson's equation. Rocky Mt. J. Math. 39, 18591869.
[16]Evans, L. C. (1991) The 1-Laplacian, the ∞-Laplacian and differential games. URL: http://math.berkeley.edu/~evans/brezis.pdf
[17]Evans, L. C. & Yu, Y. (2005) Various properties of solutions of the Infinity-Laplacian equation. Comm. Partial Differ. Equ. 30, 14011428.
[18]Galin, L. A. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk SSSR 7, 250253 (in Russian).
[19]Gurtin, M. E. (1993) Thermomechanics of Evolving Phase Boundaries in the Plane, OUP, Oxford.
[20]Howison, S. D. (1992) Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3, 209224.
[21]Howison, S. D., Lacey, A. A. & Ockendon, J. R. (1988) Hele-Shaw Free boundary problems with suction. Quart. J. Mech. Appl. Math. 41, 183193.
[22]Howison, S. D., Morgan, J. D. & Ockendon, J. R. (1997) A class of codimension two free boundary problems. SIAM Rev. 39, 221253.
[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.
[24]King, J. R. (1990) Some non-local transformations between nonlinear diffusion equations. J. Phys. A 23, 54415464.
[25]King, J. R. (1995) Development of singularities in some moving boundary value problems. Euro. J. Appl. Math. 6, 491507.
[26]Mullins, W. W. (1956) Two-dimensional motion of idealised grain boundaries. J. Appl. Phys. 27, 900904.
[27]Moser, R. (2007) The inverse mean curvature flow and p-harmonic functions. J. Eur. Math. Soc. 9, 7783.
[28]Ockendon, J. R. & Howison, S. D. (2002) Kochina and Hele-Shaw in modern mathematics, natural science and industry. J. Appl. Math. Mech. 66, 505512.
[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, 133145.
[30]Polubarinova-Kochina, P. Ya. (1945) On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47, 254257 (in Russian).
[31]Richardson, S. (1981) Some Hele Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263278.
[32]Richardson, G. & King, J. R. (2007) The Saffman-Taylor problem for an extremely shear-thinning fluid. Quart. J. Mech. Appl. Math. 60, 139160.
[33]Richardson, G. & King, J. R. (2002) Motion by curvature of a three-dimensional filament: Similarity solutions. Interfaces Free Boundaries 4, 395421.
[34]Richardson, G. & King, J. R. (2002) The evolution of space curves by curvature and torsion. J. Phys. A: Math. Gen. 35, 98579879.
[35]Sapiro, G. & Tannenbaum, A. (1992) Affine invariant scale space. Int. J. Comput. Vis. 11, 2544.
[36]Smoczyk, K. (2005) A representation formula for the inverse harmonic mean curvature flow. Elemente der Math. 60, 5765.
[37]Tanveer, S. (2000) Surprises in viscous fingering. J. Fluid Mech. 409, 273308.

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

  • G. RICHARDSON (a1) and J. R. KING (a2)

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