Orr-Sommerfeld equation

A linear ordinary differential equation

(1)

where is the Reynolds number, is a given function (the profile of the velocity of the undisturbed flow) which is usually taken to be holomorphic in a neighbourhood of the segment in the complex -plane, is constant, and is a spectral parameter. For the Orr–Sommerfeld equation, the boundary value problem

(2)

is examined. The Orr–Sommerfeld equation arose from the research by W. Orr

and A. Sommerfeld [2] concerning the stability in a linear approximation of a plane Poiseuille flow — a flow of a viscous incompressible liquid in a tube , , with rigid boundaries; for the stream function, the disturbance takes the form .

The eigen values of the problem (1), (2), generally speaking, are complex; the flow is stable if for all eigen values, and unstable if for some of them. The curve is called a neutral curve. The Poiseuille flow is stable for small Reynolds numbers. W. Heisenberg [6] was the first to propose that a Poiseuille flow is unstable for large Reynolds numbers, and calculated four points of the neutral curve. For a quadratic profile of velocity, it has been established that the flow is unstable for .

The asymptotic theory of the Orr–Sommerfeld equation is based on the assumption that is a small parameter. A point at which is a turning point (see Small parameter, method of the). The appropriate parameter is . In the local coordinates the equation becomes , with a solution of the form

which is valid for . In general, at a finite distance from one obtains a fundamental system of solutions of the form

where is a fundamental system of solutions of the non-viscous (i.e. ) equation

Research into the problem (1), (2) entails, among others, the following difficulties: 1) the non-viscous equation in a neighbourhood of has a holomorphic solution and a solution with a logarithmic singularity; 2) for small (i.e. in the most important instance) the turning points merge with the end points of the segment (for example, for a quadratic profile of velocity ).

When , a strict proof of instability has been obtained (see [3], [4]).

References

[1a]	W.McF. Orr, "The stability or instability of the steady motions of a liquid I" Proc. R. Irish Acad. A , 27 (1907) pp. 9–68
[1b]	W.McF. Orr, "The stability or instability of the steady motions of a perfect liquid and of a viscous liquid II" Proc. R. Irish Acad. A , 27 (1907) pp. 69–138
[2]	A. Sommerfeld, , Proc. 4-th Internat. Congress of Mathematicians Rome, 1908 (1909) pp. 116–124
[3]	C.C. Lin, "Theory of hydrodynamic stability" , Cambridge Univ. Press (1955)
[4]	G. Birkhoff (ed.) et al. (ed.) , Hydrodynamic instability , Proc. Symp. Appl. Math. , 13 , Amer. Math. Soc. (1962)
[5]	J.M. Gersting, D.F. Janowski, "Numerical methods for Orr–Sommerfeld problems" Internat. J. Numer. Methods Eng. , 4 (1972) pp. 195–206
[6]	W. Heisenberg, Ann. of Phys. , 74 : 15 (1924) pp. 577–627

Comments

See also Poiseuille flow.

References

[a1]	W.O. Criminale, "Stability of parallel flows" , Acad. Press (1967)
[a2]	H. Schlichting, "Fluid dynamics I" S. Flügge (ed.) , Handbuch der Physik , VIII/1 , Springer (1959) pp. 351–450
[a3]	A. Georgescu, "Hydrodynamic stability theory" , M. Nijhoff (1985)