Abstract
In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.
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References
Crochet MJ, Davies AR, Walters K (1984) Numerical simulation of non-Newtonian flow. Elsevier Science Publishers
El Kissi N, Piau JM (1989) Ecoulement de fluides polyméres enchevêtrés dans un capillaire. Modélisation du glissement macroscopique à la paroi. CR Acad Sci Paris t. 309, Série II:7–9
Georgiou GC, Crochet MJ (1994) Compressible viscous flow in slits, with slip at the wall. J Rheology 38:639–654
Hatzikiriakos SG, Dealy JM (1992a) Wall slip of molten high density polyethylenes. II. Capillary rheometer studies. J Rheol 36:703–741
Hatzikiriakos SG, Dealy JM (1992b) Role of slip and fracture in the oscillating flow of HDPE in a capillary. J Rheol 36:845–884
Hill DA, Hasegawa T, Denn MM (1990) On the apparent relation between adhesive failure and melt fracture. J Rheology 34:891–918
Hunter JK, Slemrod M (1983) Viscoelastic fluid flow exhibiting hysteritic phase changes. Phys Fluids 26:2345–2351
Kolkka RW, Malkus DS, Hansen MG, Ierley GR, Worthing RA (1988) Spurt phenomena of the Johnson-Segalman fluid and related models. J Non-Newtonian Fluid Mech 29:303–335
Larson RG (1992) Instabilities in viscoelastic flows. Rheol Acta 31:213–263
Leonov AI (1990) On the dependence of friction force on sliding velocity in the theory of adhesive friction of elastomer. Wear 141:137–145
Lin Y-H (1985) Explanation for stick-slip melt fracture in terms of molecular dynamics in polymer melts. J Rheol 29:605–637
McLeish TCB, Ball RC (1986) A molecular approach to the spurt effect in polymer melt flow. J Polym Sci B24:1735–1745
Pearson JRA, Petrie CJS (1965) On the melt-flow instability of extruded polymers. Proc. 4th Int. Rheological Congress 3:265–282
Pearson JRA, Petrie CJS (1968) On melt flow instability of extruded polymers. In: Wetton RE, Whorlow RW (eds) Polymer systems: deformation and flow. Macmillan, London 163–187
Piau JM, El Kissi N, Tremblay B (1990) Influence of upstream instabilities and wall slip on melt fracture and sharkskin phenomena during silicones extrusion through orifice dies. J Non-Newtonian Fluid Mech 34:145–180
Yerushalmi J, Katz S, Shinnar R (1970) The stability of steady shear flows of some viscoelastic fluids. Chem Eng Sci 25:1891–1902
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Georgiou, G.C. On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall. Rheol Acta 35, 39–47 (1996). https://doi.org/10.1007/BF00366551
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DOI: https://doi.org/10.1007/BF00366551