The normal field, from(A6), is E=ssp E +ap,e cos ax and at ay<<I and on the crests, this is E,= E+ ao,. The perturbation of electric traction is then(per unit area)d sae)=6,E,ae,. Instabilit will ocer f this excees the restoring surfac tension effect. E。E2Qφ1>y E (A9) The quantity a is 2/2, where A is the wavelength of the ripple. Thus, if long-wave ripples are possible, a small field is sufficient to produce instability. We will later b interested in drawing liquid from small capillaries; if the capillary diameter is D, the largest wavelength will be 2D, or o-I which gives the instability condition E> (A10) E。D For example, say D-0 1mm, and y=0.05N/m(Formamide, CHBON) The minimum 丌×0.05 field to produce an instability is then V885×10-2×10 1.33x10'V/m. This is high but since the capillary tip is thin(say, about twice its inner diameter, or 0. 2mm), it may take only about 1.33 x10x2x10-=2660 Volt to generate it. a more nearly correct estimate for this will be given next 16.522, Space Pre Lecture 23-25 Prof. Manuel martinez.The normal field, from (A6), is Ey = −∂φ ∂y = E∞ +αφ1e−αy cosαx and at αy << 1 and on the crests, this is Ey = E∞ + αφ1 . The perturbation of electric traction is then (per unit area) δ ε o 2 Ey ⎛ 2 ⎝ ⎞ ⎠ = ε oE∞αφ1 . Instability will occur if this exceeds the restoring surface tension effect: ε oE∞αφ1 > γ φ1α2 E∞ or E∞ > γα εo (A9) The quantity α is 2π /λ , where λ is the wavelength of the ripple. Thus, if long-wave ripples are possible, a small field is sufficient to produce instability. We will later be interested in drawing liquid from small capillaries; if the capillary diameter is D, the largest wavelength will be 2D , or α = π D , which gives the instability condition E∞ > πγ ε oD (A10) For example, say D=0.1mm, and γ = 0.05N /m (Formamide, CH3ON). The minimum field to produce an instability is then π × 0.05 8.85 ×10−12 ×10−4 = 1.33 ×107 V / m. This is high, but since the capillary tip is thin (say, about twice its inner diameter, or 0.2mm), it may take only about 1.33 ×107 × 2 ×10−4 = 2660 Volt to generate it. A more nearly correct estimate for this will be given next. 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 7 of 36