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A variation principle

To find the marignal states we simply put $ \sigma=0$ and the equations (30) and (31) becomes

$\displaystyle (D^2-a^2)\Theta = -W$ (38)

$\displaystyle (D^2-a^2)^2W=a^2 R_a \Theta$ (39)

This shows that the conditions for marginal stability are independent of the Prandtl number $ P_r$. Let us introduce $ F=a^2 R_a \Theta$. Then from (38) and (39) we have

$\displaystyle (D^2-a^2)^2W = F$ (40)

$\displaystyle (D^2-a^2)F = -a^2 R_a W$ (41)

Now we prove that for marginally stable disturbances

\begin{displaymath}\begin{array}{ll} R_a &= \min\frac{\displaystyle\int_0^1\left...
...\min\frac{\displaystyle K_1}{\displaystyle a^2 K_2} \end{array}\end{displaymath} (42)

where the minimum is taken over all real functions $ W$ for which the integrals exist, $ F=(D^2-a^2)^2 W$ and $ F,W,DW$ or $ D^2W$ vanish on boundary which might be rigid or free. For a given value of $ a$ the minimum is attained when $ W$ is true eigen functions at marginal state.

First we prove that the $ R_a$ is stationary when $ W$ is an eigen function. Let $ \delta R_a$ be variation in $ R_a$ for a small variation in $ \delta W$ of $ W$. Here $ \delta W$ is compatiable with the BCs i.e. $ \delta W=\delta F=0$ and $ z=0,1$ and either $ D\delta w=0$ or $ D^2\delta W=0$ depending on the nature of the suraface at $ z=0,1$. From (42) we have

$\displaystyle \delta R_a = \frac{1}{a^2 K_2}(\delta K_1 - R_a a^2 \delta K_2)$ (43)


$\displaystyle \delta K_1 = 2\int_0^1(DF D\delta F + a^2 F \delta F) dz$ (44)


$\displaystyle \delta K_2 = 2\int_0^1[(D^2-a^2)W][(D^2-a^2)\delta W] dz$ (45)

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

$\displaystyle \delta K_1 = -2\int_0^1\delta F(D^2 - a^2)F dz

$\displaystyle \delta K_2 = 2\int_0^1 W(D^2-a^2)^2\delta W dz = 2\int_0^1 W\delta F dz

Thus from (43) we have

$\displaystyle \delta R_a = \frac{1}{a^2 K_2}\delta F[(D^2-a^2)F+R_a a^2 W]dz$ (46)

Thus $ \delta R_a=0$

$\displaystyle (D^2-a^2)F = - R_a a^2 W$ (47)

and conversely if $ \delta R_a=0$ for an arbitrary $ \delta F[=(D^2-a^2)^2\delta W]$ compatiable with the BCs of the problem then (47) must hold and $ F$ must be a eigen function of the problem.

Next we show that the lowest eigen value of $ R_a$ is a true mimimum. We shall assume that there is a countable infinity of marginal states for each real wave number $ a$ and we denote them by $ F_j,W_j$ corresponding to the $ R_{aj}$ ( $ j=1,2,\cdots$) with $ 0<R_{a1}<R_{a2}<\cdots$. We first show that $ F_j$ is orthogonal to $ W_k$. To show this we multiply

$\displaystyle (D^2-a^2)^2W_j=F_j

by $ W_k$ and integrate between $ z=0$ to $ 1$. After on integration by parts using BCs we have

$\displaystyle \int_0^1 F_j W_k dz=\int_0^1(D^2-a^2)W_j(D^2-a^2)W_k dz = \int_0^1F_k W_j dz$ (48)

by symmetry in $ j$ and $ k$.


$\displaystyle (D^2-a^2)F_j = - R_{aj} a^2 W_j

by $ F_k$ and on integration by parts with BCs gives

$\displaystyle \int_0^1(DF_j DF_k + a^2 F_j F_k) dz$ (49)

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

$\displaystyle a^2 (R_{aj}-R_{ak}) \int_0^1 F_j W_k dz = 0

which gives

$\displaystyle \int_0^1 F_j W_k dz = 0 \qquad j\ne k

Let us normalize $ W_k$ such that

$\displaystyle \int_0^1 F_j W_k dz = \delta_{jk}

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

$\displaystyle W = \sum_{j=1}^{\infty}A_j W_j$   where$\displaystyle \quad A_j=\int_0^1 F_j W dz

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

$\displaystyle F=(D^2-a^2)^2 W=\sum_{j=1}^{\infty}A_j(D^2-a^2)^2 W_j= \sum_{j=1}^{\infty}A_j F_j

Now using orthogonality relation of $ F_j$ and $ W_k$ we get

K_1 &=& \displaystyle\int_0^1[(DF)^2+a^2 ...
...=& \displaystyle\sum_{j=1}^{\infty}a^2 R_{aj} A_j^2


K_2 &=& \displaystyle\int_0^1[(D^2-a^2)W]...
...F dz\\
&=& \displaystyle\sum_{j=1}^{\infty} A_j^2


$\displaystyle \frac{K_1}{a^2 K_2}-R_{a1} = \frac{\sum_{j=1}^{\infty}(R_{aj}-R_{a1})A_j^2}{K_2}\ge 0$ (50)

Equality holds iff all the $ A_j$ except $ A_1$ are zero. Therefore $ W_1,F_1$ are the eigen functions that gives the minimum $ R_{a1}$ of (42).

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