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Full text of "Scientific Papers - Vi"

1915]             NOTE  ON THE  FORMULA. FOR  THE  GRADIENT  WIND                  277
and
dv       dv        dv                      ,,
_  +u- -- f- fl _ -|- 2 6)16  CO-?/*.
at        dx       dy                 J
Since the relative motion is supposed to be steady, du/dt, dv/dt disappear, and the dynamical equations are
1 dp      2l0            du       du                             n.
 f- = &ro3 + 2wv u-^ -- v -T~ . ..................... (1)
p dx                         dx       dy
I dp                         dv       dv                           /0.
-- r- = ft)2?/  2d)U  U^j -- V -r- ...................... (2)
p dy       J                 dx       dy                           N
The velocities u, v may be expressed by means of the relative stream-function -\/r :
u = dtyjdy,              v 
Equations (1), (2) then become
1 4        *
_i*J(**)V*m+vv.^, ...... W
2 cfcc (V<&/      \dyJ }         r    dx
-+             + V^.; ...... (4)
p dy                  dy     2 dy   \dxj      \dy) f         Y    rfy
and on integration, if we leave out the part of p independent of the relative motion,
in which by a known theorem V2-v|r is a function of ^ only. If <a be omitted, (5) coincides with the equation given long ago by Stokes f. It expresses p in terms of x/r ; but it does not directly allow of the expression of -\|r in terms of p, as is required if the data relate to a barometric chart.
We may revert to the more usual form, if in (1) or (3) we bake the axis of x perpendicular to the direction of (relative) motion at any point. Then u = 0, and
1 dp               dty dty                                   /ftN
- -/- = 2wv + -3- ^ ......................... (6)
p dx              dx  dy*
But since dty/dy = 0, the curvature at this place of the stream-line (^ = const.)
s
^        . ~ r " dyz  ' dx'
and thus                                   f = 2cov +  , .............................. (7)
p dx            ~ r
* Lamb's Hydrodynamics,  206i
f Carnb. Phil. Trans. Vol. vn. 1812 ;  Math, and, Phys. Papers, Vol. i. p. 9.t yet been proved theoretically that the stability can fail for finite disturbances on the supposition of perfectly smooth walls ; but such failure seems probable. We know from the work of Keynolds, Lorentz, and Orr that no failure of stability can occur unless @D*jv > 177, where D is the distance between, the walls, so that j3D represents their relative motion.