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4.3 The capillary stability of a plane jet. An incompresible inviscid liquid with density p and surface tension y is at rest between the free plane surfaces z = £a. Show that the stability of the liquid is governed by the equation Ap' = 0 for - < x, y < 0,-a < z <a, 0 00 and the boundary conditions a2p' a2 a2 ap' р #y + at2 ax2 at z = fa. ду2 ) дz Taking normal modes with p' a exp[st +i(kx + ly)], show that a’ps2 = -ãồ tanh ã or -ãcoth ã, Y Exercises 67 = where a = a(k2 + 12)1/2. Deduce that this plane jet is stable, unlike the round jet, which is unstable to axisymmetric disturbances. [After Drazin & Reid (1981, Problem 1.10). Hint: pðu'/at = -Vp', V. u'=0; and n=+(-a5'/ax, -a5'lay, 1), p' = yv n, w' = -as'/at, at z = ta. Note that s2 ~ -ya}/ap as ã → for short capillary waves at a free surface, in common with the round jet.] = .