# 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