View source for Orr-Sommerfeld equation

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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|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 [[#References|[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 [[#References|[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|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 [[#References|[3]]], [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Sommerfeld,  , ''Proc. 4-th Internat. Congress of Mathematicians Rome, 1908''  (1909)  pp. 116–124</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C.C. Lin,   "Theory of hydrodynamic stability" , Cambridge Univ. Press  (1955)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Birkhoff (ed.)  et al. (ed.) , ''Hydrodynamic instability'' , ''Proc. Symp. Appl. Math.'' , '''13''' , Amer. Math. Soc.  (1962)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.M. Gersting,   D.F. Janowski,   "Numerical methods for Orr–Sommerfeld problems"  ''Internat. J. Numer. Methods Eng.'' , '''4'''  (1972)  pp. 195–206</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Heisenberg,   ''Ann. of Phys.'' , '''74''' :  15  (1924)  pp. 577–627</TD></TR></table>

====Comments====
See also [[Poiseuille flow|Poiseuille flow]].

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