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Full text of "Scientific Papers - Vi"

HORIZONTAL  LAYER OP  FLUID                                     439
For a two-dimensional disturbance we may make m=0 and £ = 2?r/A, here A is the wave-length along x. The A, of maximum instability is thus >proximately
A, = 2£     .................................(28)
Again, if Z = m = 27r/A, as for square cells,
X = 2 Y 2 . £",.................................(^9)
•eater than before in the ratio \/2 :1.
We have considered especially the cases where K is relatively small and latively large. Intermediate cases would need to be dealt with by a imerical solution of (23).
When w is known in the form
being now a known function of I, m, s, u and v are at once derived by eansof (11) and (12). Thus
il     diu               im    dw                             /r>1.
^ —_______________nt  —._______________                                                  / QT   \
ie connexion between w and d is given by (15) or (16). When ft is sgative and n positive, 0 and w are of the same sign.
As an example in two dimensions of (30), (31), we might have in real
u — W cos x . sin z. enti.........................(32)
u = ~~ Wsmx.cosz.ent,   v=Q...................(33)
Hitherto we have supposed the .fluid to be destitute of viscosity. When 5 include viscosity, we must add v (V2u, V2*u, V2w) on the right of equations ), (8), and (11), v being the kinematic coefficient. Equations (12) and (13) main unaffected. And in (11)
e have 'also to reconsider the boundary conditions at z = 0 and z — £. e may still suppose d = 0 and w = 0; but for a further condition we should obably prefer dw/dz = 0, corresponding to a fixed solid wall*. But this tails much complication, and we may content ourselves with the sup-sition dzw/dz* = Q, which (with w — Q) is satisfied by taking as before w Dportional to sin sz with s — q7r/£. This is equivalent to the annulment of ;eral forces at the wall. For (Lamb's ffydrodynamics, §§ 323, 326) these
•ces are expressed in general by
• dw    du             _dw    dv                             • ,^K.
Pxz == ~T-----1—7~ >       Puz------T~~ ~i~ ~T~}     ..................\y&)
r       dsc     dz               dy    dz
* [It would appear that the immobility and solidity of the walls are sufficiently provided for the condition iv = 0, and that for " a fixed solid wall" there should he substituted " no slipping the walls." W. P. S.]as Boussinesq* h'as shown, in the class of problems under consideration the influence of pressure is