Next: Exact solutions of characteristics Up: General stability characteristics Previous: Exchange of stabilities

## A variation principle

To find the marignal states we simply put and the equations (30) and (31) becomes

 (38)

 (39)

This shows that the conditions for marginal stability are independent of the Prandtl number . Let us introduce . Then from (38) and (39) we have

 (40)

 (41)

Now we prove that for marginally stable disturbances

 (42)

where the minimum is taken over all real functions for which the integrals exist, and or vanish on boundary which might be rigid or free. For a given value of the minimum is attained when is true eigen functions at marginal state.

First we prove that the is stationary when is an eigen function. Let be variation in for a small variation in of . Here is compatiable with the BCs i.e. and and either or depending on the nature of the suraface at . From (42) we have

 (43)

where

 (44)

and

 (45)

Using integration by parts (44), (45) give

Thus from (43) we have

 (46)

Thus

 (47)

and conversely if for an arbitrary compatiable with the BCs of the problem then (47) must hold and must be a eigen function of the problem.

Next we show that the lowest eigen value of is a true mimimum. We shall assume that there is a countable infinity of marginal states for each real wave number and we denote them by corresponding to the ( ) with . We first show that is orthogonal to . To show this we multiply

by and integrate between to . After on integration by parts using BCs we have

 (48)

by symmetry in and .

Multiplying

by and on integration by parts with BCs gives

 (49)

Interchanging and in (49) and subtracting the new equation from (49) gives (using (49)

which gives

Let us normalize such that

Let be any pair of functions which satisfies the BCs of the problem (not necessarily eigen functions). We represent as

where

Also assuming uniform convergence of the series we can differentiate term by term to give

Now using orthogonality relation of and we get

Similarly

Thus

 (50)

Equality holds iff all the except are zero. Therefore are the eigen functions that gives the minimum of (42).

Next: Exact solutions of characteristics Up: General stability characteristics Previous: Exchange of stabilities
A/C for Homepage of Dr. S Ghorai 2003-01-16