Ill posedness of Bingham-type models for the downhill flow of a thin film on an inclined plane

Fusi, L.; Farina, A.; Rosso, F.

doi:10.1090/qam/1391

Authors: L. Fusi, A. Farina and F. Rosso
Journal: Quart. Appl. Math. 73 (2015), 615-627
MSC (2010): Primary 76A05, 74D10, 76D08
DOI: https://doi.org/10.1090/qam/1391
Published electronically: September 15, 2015
MathSciNet review: 3432275
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the flow of a thin layer of a Bingham-type material over an inclined plane with “small” tilt angle. A Bingham-type continuum is a material which behaves as a viscous fluid above a certain threshold (tied to the shear stress) and as a solid below such a threshold. We consider creeping flow and that the ratio between the thickness and the length of the layer is small, so that the lubrication approach is suitable. The unknowns of the model are the layer thickness, the position of the yield surface and the position of the advancing front. We first show that, though diverging in a neighborhood of the wetting front, the shear stress is integrable so that total dissipation is bounded. We then prove that the mathematical problem is inherently ill posed independently on the constitutive model selected for the solid domain. We therefore conclude that either the Bingham-type models are inappropriate to describe the thin film motion on an inclined surface or the lubrication technique fails in approximating such flows.

References

R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport phenomena, Wiley, 1960.

N.J., Balmforth and R.V. Craster, A consistent thin-layer theory for Bingham plastics, J. Non-Newtonian Fluid Mech. 57 (1999), 65-81.

P.G. De Gennes, Wetting: statics and dynamics, Rev. Mod. Phys. 57 (1985), 827-863.

S. Cochard and C. Ancey, Experimental investigation of the spreading of viscoplastic fluids on inclined planes, J. Non-Newtonian Fluid Mech. 158 (2009), 73-84.

Lorenzo Fusi and Angiolo Farina, An extension of the Bingham model to the case of an elastic core, Adv. Math. Sci. Appl. 13 (2003), no. 1, 113–163. MR 2002398

Lorenzo Fusi and Angiolo Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004), no. 5, 826–847. MR 2087767, DOI https://doi.org/10.1007/s00033-004-3056-5

Angiolo Farina and Lorenzo Fusi, On a parabolic free boundary problem arising from a Bingham-like flow model with a visco-elastic core, J. Math. Anal. Appl. 325 (2007), no. 2, 1182–1199. MR 2270078, DOI https://doi.org/10.1016/j.jmaa.2006.02.052

Lorenzo Fusi and Angiolo Farina, Modelling of Bingham-like fluids with deformable core, Comput. Math. Appl. 53 (2007), no. 3-4, 583–594. MR 2323711, DOI https://doi.org/10.1016/j.camwa.2006.02.033

A. Farina, A. Fasano, L. Fusi, and K. R. Rajagopal, Modeling materials with a stretching threshold, Math. Models Methods Appl. Sci. 17 (2007), no. 11, 1799–1847. MR 2372339, DOI https://doi.org/10.1142/S0218202507002480

A. Farina, A. Fasano, L. Fusi, and K. R. Rajagopal, On the dynamics of an elastic-rigid material, Adv. Math. Sci. Appl. 20 (2010), no. 1, 193–217. MR 2760725

Lorenzo Fusi and Angiolo Farina, A mathematical model for an upper convected Maxwell fluid with an elastic core: study of a limiting case, Internat. J. Engrg. Sci. 48 (2010), no. 11, 1263–1278. MR 2760984, DOI https://doi.org/10.1016/j.ijengsci.2010.06.001

L. Fusi and A. Farina, Pressure-driven flow of a rate type fluid with stress threshold in an infinite channel, Inter. J. Nonlin. Mech. 46 (2011), 991-1000.

A. Farina, A. Fasano, L. Fusi, and K. R. Rajagopal, The one-dimensional flow of a fluid with limited strain-rate, Quart. Appl. Math. 69 (2011), no. 3, 549–568. MR 2850745, DOI https://doi.org/10.1090/S0033-569X-2011-01249-7

L. Fusi, A. Farina, and F. Rosso, Flow of a Bingham-like fluid in a finite channel of varying width: a two-scale approach, J. Non-Newtonian Fluid Mech. 177-178 (2012), 76-88.

L. Fusi, A. Farina, and F. Rosso, The lubrication paradox for the flow of a Bingham fluid on an inclined surface, Inter. J. Nonlin. Mech, 58 (2014), 139-150.

G. Lipscomb and M. Denn, Flow of Bingham fluids in complex geometries, J. Non-Newtonian Fluid Mech. 14 (1984), 337–346.

H. E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121 (1982), 43–58.

C. W. Macosko, Rheology: Principles, Measurements and Applications, Wiley, 1994.

K. R. Rajagopal and A. R. Srinivasa, On the thermodynamics of fluids defined by implicit constitutive relations, Z. Angew. Math. Phys. 59 (2008), no. 4, 715–729. MR 2417387, DOI https://doi.org/10.1007/s00033-007-7039-1

S.D.R. Wilson, Squeezing flow of a Bingham material, J. Non-Newtonian Fluid Mech. 47 (1993), 211-219, 715–729.