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Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type. (English) Zbl 0701.76011

Summary: In this paper we discuss shearing motions and Poiseuille flows of Oldroyd (Johnson-Segalman) fluids with retardation time. We show that the motion exists for arbitrary time and arbitrary initial data. We investigate the (Lyapunov) stability of the basic steady flow to one-dimensional finite amplitude perturbations.

MSC:

76A10 Viscoelastic fluids
35Q35 PDEs in connection with fluid mechanics
76E99 Hydrodynamic stability
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References:

[1] G. ASTARITA, G. MARRUCCI, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London, 1974. · Zbl 0316.73001
[2] C. GUILLOPÉ, J. C. SAUT, Résultats d’existence pour des fluides viscoélastiques à loi de comportement de type différentiel. C.R. Acad. Sci. Paris, 305, série I (1987), 489-492, and article to appear in Nonlinear An., T.M.A. Zbl0624.76008 MR916317 · Zbl 0624.76008
[3] P. HENRICI, Applied and Computational Complex Analysis, vol. I, John Wiley, New York, 1974. Zbl0313.30001 MR372162 · Zbl 0313.30001
[4] G. IOOSS, Bifurcation et stabilité, Publications Mathématiques d’Orsay, 1973. MR487634
[5] D. D. JOSEPH, Stability of Fluid Motions, vol. I and II, Springer, Berlin-Heidelberg-New York, 1976. Zbl0345.76023 · Zbl 0345.76023
[6] T. KATO, Perturbation Theory for Linear Operators, Springer, Berlin-Heidel-berg-New York, 1966. Zbl0148.12601 MR203473 · Zbl 0148.12601
[7] R. W. KOLKKA, G. R. IERLEY, M. G. HANSEN, R. A. WORTHING, On the stability of viscoelastic parallel shear flows, Technical Report, F.R.O.G., Michigan Technological University, 1987.
[8] R. W. KOLKKA, D. S. MALKUS, M. G. HANSEN, G. R. IERLEY, R. A. WORTHING, Spurt phenomena of the Johnson-Segalman fluid and related models, J. Non-Newt. Fl. Mech., 29 (1988), 303-335.
[9] J. G. OLDROYD, On the formulation of rheological equations of state, Proc. Roy. Soc. London, A 200 (1950), 523-541. Zbl1157.76305 MR35192 · Zbl 1157.76305
[10] [10] G. PRODI, Theoremi di tipo locale per il sistema di Navier-Stokes e la stabilita delle soluzione stazionarie, Rend. Sem. Univ. Padova, 32 (1962), 374-397. Zbl0108.28602 MR189354 · Zbl 0108.28602
[11] M. RENARDY, W. J. HRUSA, J. A. NOHEL, Mathematical Problems in Viscoelasticity, Longman, New York, 1987. Zbl0719.73013 MR919738 · Zbl 0719.73013
[12] D. H. SATTINGER, Topics in Stability and Bifurcation Theory, Lectures Notes in Mathematics, 309, Springer, Berlin-Heidelberg-New York, 1973. Zbl0248.35003 MR463624 · Zbl 0248.35003
[13] W. R. SCHOWALTER, Behavior of complex fluids at solid boundaries, J. Non-Newt. Fl. Mech., 29 (1988), 85.
[14] J. YERUSHALMI, S. KATZ, R. SHINNAR, The stability of steady shear flows of some viscoelastic fluids, Chem. Eng. Sc., 25 (1970), 1891-1902.
[15] J. K. HUNTER, M. SLEMROD, Viscoelastic fluid flow exhibiting hysteritic phase changes, Phys. Fluids 26 (1983), 2345-2351. Zbl0529.76009 · Zbl 0529.76009
[16] D. S. MALKUS, J. A. NOHEL, B. J. PLOHR, Time-dependent shear flow of a non-Newtonian fluid, in Contemporary Mathematics, vol. 100, ed. W. B. Lindquist, A.M.S. (1989), 91-110. Zbl0683.76003 MR1033511 · Zbl 0683.76003
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