View source for Orr-Sommerfeld 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>

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

<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>

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" />.

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 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

<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>

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

<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>

<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>

<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>

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

<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>

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" />).

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]]]).

====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>