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A linear ordinary differential equation
 
A linear ordinary differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi  ^ {(} 4) - 2 \alpha  ^ {2} \phi  ^ {\prime\prime} + \alpha  ^ {4} \phi  = \
 +
i \alpha R[( w- c)( \phi  ^ {\prime\prime} - \alpha  ^ {2} \phi ) - w  ^ {\prime\prime} \phi ],
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702502.png" /> is the [[Reynolds number|Reynolds number]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702503.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702504.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702505.png" />-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702506.png" /> is constant, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702507.png" /> is a spectral parameter. For the Orr–Sommerfeld equation, the boundary value problem
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702508.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\phi (- 1)  = \phi  ^  \prime  (- 1)  = \phi ( 1)  = \phi  ^  \prime  ( 1) =  0
 +
$$
  
 
is examined. The Orr–Sommerfeld equation arose from the research by W. Orr
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o0702509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025010.png" />, with rigid boundaries; for the stream function, the disturbance takes the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025011.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025012.png" /> for all eigen values, and unstable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025013.png" /> for some of them. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025015.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025016.png" /> is a small parameter. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025017.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025018.png" /> is a turning point (see [[Small parameter, method of the|Small parameter, method of the]]). The appropriate parameter is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025019.png" />. In the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025020.png" /> the equation becomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025021.png" />, with a solution of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025022.png" /></td> </tr></table>
+
$$
 +
\phi ( \eta )  = \int\limits _ {- \infty } ^  \eta  \int\limits _ {- \infty } ^ { {\eta ^ {\prime\prime} } } ( \eta  ^  \prime  )  ^ {1/2} H _ {1/3}  ^ {(} 1) [ 2 ( i \eta  ^  \prime  )  ^ {2/3} /3 ]  d \eta  ^  \prime  d \eta ^ {\prime\prime } ,
 +
$$
  
which is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025023.png" />. In general, at a finite distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025024.png" /> one obtains a fundamental system of solutions of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025025.png" /></td> </tr></table>
+
$$
 +
\phi _ {1,2} ( y)  = \
 +
\phi _ {1,2}  ^ {0} ( y) + O(( \alpha R)  ^ {-} 1 ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025026.png" /></td> </tr></table>
+
$$
 +
\phi _ {3,4} ( y)  =   \mathop{\rm exp} \left [ \pm  \int\limits ^ { y }  \sqrt {
 +
\frac{i( w- c) }{\alpha R }
 +
}  dy \right ] \times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025027.png" /></td> </tr></table>
+
$$
 +
\times
 +
\left [ ( w- c)  ^ {-} 5/4 + O(( \alpha R)  ^ {-} 1/2 ) \right ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025028.png" /> is a fundamental system of solutions of the non-viscous (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025029.png" />) equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025030.png" /></td> </tr></table>
+
$$
 +
( w- c) ( \phi  ^ {\prime\prime} - \alpha  ^ {2} \phi ) - w  ^ {\prime\prime} \phi  = 0.
 +
$$
  
Research into the problem (1), (2) entails, among others, the following difficulties: 1) the non-viscous equation in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025031.png" /> has a holomorphic solution and a solution with a logarithmic singularity; 2) for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025032.png" /> (i.e. in the most important instance) the turning points merge with the end points of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025033.png" /> (for example, for a quadratic profile of velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025034.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070250/o07025035.png" />, a strict proof of instability has been obtained (see [[#References|[3]]], [[#References|[4]]]).
+
When $  \alpha R \gg 1 $,  
 +
a strict proof of instability has been obtained (see [[#References|[3]]], [[#References|[4]]]).
  
 
====References====
 
====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>
 
<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====
 
====Comments====

Revision as of 08:04, 6 June 2020


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)
How to Cite This Entry:
Orr-Sommerfeld equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orr-Sommerfeld_equation&oldid=48071
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article