377.
ON THE STABILITY OF THE LAMINAR MOTION OF AN INVISCID FLUID.
[Philosophical Magazine, Vol. xxvi. pp. 1001—1010, 1913.]
THE equations of motion of an inviscid fluid are satisfied by a motion such that U, the velocity parallel to x, is an. arbitrary function of y only, while the other component velocities V and W vanish. The motion may be supposed to be limited by two fixed plane walls for each of which y has a constant value. In order to investigate the stability of the motion, we superpose upon it a two-dimensional disturbance u, v, where u and v are regarded as small. If the fluid is incompressible,
du dv _ „ , .
and if the squares and, products of small quantities are neglected, the hydro-dynamical equations give*
"~37av~®...................(2)
From (1) and (2), if we assume that as functions of t and on, u and v are proportional to ei(nt+kx}, where k is real and n may be real or complex,
(3)
^
In the paper quoted it was shown that under certain conditions n could not be complex; and it may be convenient to repeat the argument. Let
njJc = p + iq, v = a + i/3,
* Proceedings of London Mathematical Society, Vol. xi. p. 57 (1880); Scientific Papers, Vol. i. p. 485. Also Lamb's Hydrodynamics, § 345......... (38)