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The stability of two-dimensional linear flows. (English) Zbl 0585.76045
(From authors’ summary.) A theoretical investigation is made of the linear stability of a viscous incompressible fluid undergoing a steady, unbounded two-dimensional flow in which the velocity field is a linear function of position. Such flows can be characterized completely by a single parameter $$\lambda$$ which ranges from $$\lambda =0$$ for simple shear flow to $$\lambda =1$$ for pure extensional flow. The linearized velocity disturbance equations are analyzed for an arbitrary spatially periodic initial disturbance to give the asymptotic behavior of the disturbance at large time for $$0\leq \lambda \leq 1$$. In addition, a complete analytical solution of the vorticity disturbance equation is obtained for the case $$\lambda =1$$. It is found that unbounded flows with $$0<\lambda \leq 1$$ are unconditionally unstable and an instability criterion is obtained.
Reviewer: M.Boudourides

##### MSC:
 7.6e+06 Parallel shear flows in hydrodynamic stability
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##### References:
 [1] Benjamin, Proc. R. Soc. London Ser. A 359 pp 1– (1978) [2] Benjamin, Proc. R. Soc. London Ser. A 359 pp 27– (1978) [3] Benjamin, Proc. R. Soc. London Ser. A 377 pp 221– (1981) [4] Benjamin, J. Fluid Mech. 121 pp 219– (1982) [5] Taylor, Proc. R. Soc. London Ser. A 146 pp 501– (1934) [6] Rumscheidt, J. Colloid Sci. 16 pp 210– (1961) [7] Grace, Chem. Eng. Commun. 14 pp 225– (1982) [8] Flumerfelt, Ind. Eng. Chem. Fundam. 11 pp 312– (1972) [9] Pope, Colloid Poly. Sci. 255 pp 633– (1977) [10] Fuller, Rheol. Acta 19 pp 580– (1980) [11] Hopf, Ann. Phys. (4) 44 pp 1– (1914) [12] Gallagher, J. Fluid Mech. 13 pp 91– (1962) [13] Reid, Stud. Appl. Math. 61 pp 83– (1979) · Zbl 0431.76038 [14] Pearson, J. Fluid Mech. 5 pp 274– (1959) [15] Marrucci, AIChE J. 13 pp 931– (1967) [16] Tanner, Rheol. Acta 14 pp 959– (1975) [17] Tanner, AIChE J. 22 pp 910– (1976) [18] Olbricht, J. Non-Newt. Fluid Mech. 10 pp 291– (1982) [19] Levinson, Duke Math J. 15 pp 111– (1948)
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