Figure 5: Photographs of electrified cones of ethylene gylcol at various extreme electrostatic conditions within which the emitted current depends only on flow rate and not on voltage difference V, or the distance l between needle and ground. In(a) and(b)v=5kV(la=41.0nA) and 3 kV(Ib=41. 6 nA)respectively; L=9mm. In(c)the liquid reaches the surface before opening into a spray. Fig 6: Structure of the cone jet for the same solution as figure 5 at higher magnification The dimensionless flow rate variable(1)is n=2 (Re=10.2) 16.522, Space Pre Lecture 23-25 Prof. Manuel mar ropelsinnchez Page 11 of 36
16.522, Spac e Propuls i o n Lecture 23- 2 5 Prof. Manuel M artinez-Sanchez Page 11 of 36
the projection on it of that of the circular section through the same point R Meusnier's theorem E 1_(1 R UR cosa cosa cot a (A20) This means that Fig. 7 EE y cot a E A21 The question then is to find an external electrostatic field such that the cone is an equipotential (say, o=o), with a normal field varying as in(A21),ie, proportional to 1/Nr. Notice that the spheroids of Sec. A2.3 do generate cones in the limit when r>>a (with no= cos a), but this type of electrostatic field has En 1/r, and cannot be the desired equilibrium solution If we adopt a spherical system of coordinates( Fig. 8), it is known that Laplace's equation admit axi-symmetric"product"solutions of the type P=AP(cos 9)r B=A@(cos 9) where P, Q, are Legendre functions of the 1 and nd, respectively. Of the two, P, has Fig. 8 ingularity when 9=180, and Q The latter is acceptable, because 9=0 is inside the liquid cone, and we only need the solution outside. The normal field, from(A22b)is 16.522, Space Pl Lecture 23-25 Prof. Manuel marti Page 12 of 36
the projection on it of that of the circular section through the same point (Meusnier’s theorem): 1 Rc = 1 R ⎛ ⎝ ⎞ ⎠ cosα = cosα rsinα = cotα r (A20) This means that 1 2 ε oEn 2 = γ cotα r En = 2γ cotα ε or (A21) The question then is to find an external electrostatic field such that the cone is an equipotential (say, φ = o ), with a normal field varying as in (A21), i.e., proportional to 1/ r . Notice that the spheroids of Sec. A2.3 do generate cones in the limit when r>>a (with ηo = cosα) , but this type of electrostatic field has En ≈ 1/r , and cannot be the desired equilibrium solution. If we adopt a spherical system of coordinates (Fig. 8), it is known that Laplace’s equation admit axi-symmetric “product” solutions of the type φ = APν (cosϑ)r ν (A22a) or φ = A Qν(cosϑ)r ν (A22b) where Pν,Qν are Legendre functions of the 1st and 2nd kind, respectively. Of the two, Pν has a singularity when ϑ = 180o , and Qν has one at ϑ = o. The latter is acceptable, because ϑ = o is inside the liquid cone, and we only need the solution outside. The normal field, from (A22b) is then 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 12 of 36