A22 SURFACE STABILITY E: If the liquid surface deforms slightly, ↑↑↑↑↑↑ the field becomes stronger on the protruding parts, and more charge concentrates there. The traction of the surface field on this charge (e)=o En for a conductor(the 1/2 accounts for the variation of En from its outside value toO inside the liquid ). This traction then intensifies on the protruding parts, and the process can become unstable if surface tension, r, is not strong enough to counteract the traction. In that case, the protuberance will grow rapidly into some sort of large-scale deformation, the shape of which depends on field shape, container size, etc potental, which ple is assumed sinusoidal, and small (initially at least), then the outside If the surface ripp ch obeys Vo=o with p=o on the surface, can be represented approximately by the superposition of that due to the applied field E, plus a small perturbation. Using the fact that Re(e")is a harmonic function(z-X+iy), 中≡-Ey+ The surface is where d=0, and this, when ay<< l, is approximately given by O≡-Ey+中cosa,or (A7) The surface has a curvature 1/R cosax. which is maximum at crests R=2 and gives rise to a surface tension restoring force(perpendicular to the surface)of (cylindrical surface) 16.522 spel m artipezssanch LectureA2.2 SURFACE STABILITY If the liquid surface deforms slightly, the field becomes stronger on the protruding parts, and more charge concentrates there. The traction of the surface field on this charge is ρs ( )En 2 = ε o 2 En 2 for a conductor (the 1/2 accounts for the variation of En from its outside value to 0 inside the liquid). This traction then intensifies on the protruding parts, and the process can become unstable if surface tension, γ , is not strong enough to counteract the traction. In that case, the protuberance will grow rapidly into some sort of large-scale deformation, the shape of which depends on field shape, container size, etc. If the surface ripple is assumed sinusoidal, and small (initially at least), then the outside potential, which obeys ∇2 φ = o with φ = o on the surface, can be represented approximately by the superposition of that due to the applied field E∞ , plus a small perturbation. Using the fact that Re e iαz ( ) is a harmonic function (z=x+iy), φ ≅ −E∞ y + φ1e−αy cosαx (A6) The surface is where φ = o , and this, when αy << 1, is approximately given by o ≅ −E∞y +φ1 cosαx , or y ≅ φ1 En cosαx (A7) The surface has a curvature 1/ Rc ≅ d2 y dx2 = φ1 α2 E∞ cosαx , which is maximum at crests (cosα x=1): Rc = E∞ φ1α 2 (A8) and gives rise to a surface tension restoring force (perpendicular to the surface) of γ Rc (cylindrical surface). 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 6 of 36