Orr-Sommerfeld equation

A linear ordinary differential equation

$$ \tag{1 } \phi ^ {(} 4) - 2 \alpha ^ {2} \phi ^ {\prime\prime} + \alpha ^ {4} \phi = \ i \alpha R[( w- c)( \phi ^ {\prime\prime} - \alpha ^ {2} \phi ) - w ^ {\prime\prime} \phi ], $$

where $ R $ is the Reynolds number, $ w( y) $ 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 $ [- 1, 1] $ in the complex $ y $- plane, $ \alpha > 0 $ is constant, and $ c $ is a spectral parameter. For the Orr–Sommerfeld equation, the boundary value problem

$$ \tag{2 } \phi (- 1) = \phi ^ \prime (- 1) = \phi ( 1) = \phi ^ \prime ( 1) = 0 $$

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 $ - \infty < x < \infty $, $ - 1 < y < 1 $, with rigid boundaries; for the stream function, the disturbance takes the form $ \phi ( y) e ^ {i \alpha ( x- ct) } $.

The eigen values of the problem (1), (2), generally speaking, are complex; the flow is stable if $ \mathop{\rm Im} c < 0 $ for all eigen values, and unstable if $ \mathop{\rm Im} c > 0 $ for some of them. The curve $ \mathop{\rm Im} c ( \alpha , R) = 0 $ 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 $ \alpha R \gg 1 $.

The asymptotic theory of the Orr–Sommerfeld equation is based on the assumption that $ ( \alpha R) ^ {-} 1 \rightarrow 0 $ is a small parameter. A point $ y _ {c} $ at which $ w( y _ {c} ) = c $ is a turning point (see Small parameter, method of the). The appropriate parameter is $ \epsilon = ( \alpha R w _ {c} ^ \prime ) ^ {- 1/3 } $. In the local coordinates $ \eta = ( y- y _ {c} )/ \epsilon $ the equation becomes $ i \phi ^ {iv} + \eta \phi ^ {\prime\prime } = 0 $, with a solution of the form

$$ \phi ( \eta ) = \int\limits _ {- \infty } ^ \eta \int\limits _ {- \infty } ^ { {\eta ^ {\prime\prime} } } ( \eta ^ \prime ) ^ {1/2} H _ {1/3} ^ {(} 1) [ 2 ( i \eta ^ \prime ) ^ {2/3} /3 ] d \eta ^ \prime d \eta ^ {\prime\prime } , $$

which is valid for $ \eta > 0 $. In general, at a finite distance from $ y= y _ {c} $ one obtains a fundamental system of solutions of the form

$$ \phi _ {1,2} ( y) = \ \phi _ {1,2} ^ {0} ( y) + O(( \alpha R) ^ {-} 1 ), $$

$$ \phi _ {3,4} ( y) = \mathop{\rm exp} \left [ \pm \int\limits ^ { y } \sqrt { \frac{i( w- c) }{\alpha R } } dy \right ] \times $$

$$ \times \left [ ( w- c) ^ {-} 5/4 + O(( \alpha R) ^ {-} 1/2 ) \right ] , $$

where $ \phi _ {1} ^ {0} ( y), \phi _ {2} ^ {0} ( y) $ is a fundamental system of solutions of the non-viscous (i.e. $ \alpha R = 0 $) equation

$$ ( w- c) ( \phi ^ {\prime\prime} - \alpha ^ {2} \phi ) - w ^ {\prime\prime} \phi = 0. $$

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

When $ \alpha R \gg 1 $, 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)