inviscid flow, P+pu remains constant along the streamlines, while p is itself independent of r. Thus pu develops gradually at the expense of p, and this should lead to a radius-independent pu. Our flow is not exactly parallel or inviscid, but the approximation(verified from numerical 2-D solutions) is good enough for the present poses We then have (17) RT showing that most of the mass flow must occur in the cool, outside gas, since the numerator in(17)is independent of r We then have (18) and the problem is now to calculate the gas flux cpu) in the buffer gas. The simplest possible approximation is to state that all of the gas flow is carried by this uniform buffer (19) (R2-R2 elds x(2-R)7 and, substituting into(16) -R)78a(-)=2(=Z) The quantity Tc-Te depends on current and arc radius through Eq (6) Substituting, 16.522 Space Propulsion Lecture 11-12inviscid flow, p + ρu 2 remains constant along the streamlines, while p is itself independent of r. Thus ρu 2 develops gradually at the expense of p, and this should lead to a radius-independent ρu 2 . Our flow is not exactly parallel or inviscid, but the approximation (verified from numerical 2-D solutions) is good enough for the present purposes. We then have, ρu = p ρu 2 ( ) = p ρu2 ( ) RgT (17) showing that most of the mass flow must occur in the cool, outside gas, since the numerator in (17) is independent of r. We then have, ( ) ρu e ( ) ρu out = Tout Te (18) and the problem is now to calculate the gas flux (ρu)out in the buffer gas. The simplest possible approximation is to state that all of the gas flow is carried by this uniform buffer flow: ( ) ρu out ≅ m Ý π R2 − Ra 2 ( ) (19) This yields ( ) ρu e ≅ m Ý π R2 − Ra 2 ( ) Tou t Te (20) and, substituting into (16), m Ý π R2 − Ra 2 ( ) Tout Te Ra dRa dx cp Te − Tout ( ) ≅ 2kc Tc − Te ( ) The quantity Tc-Te depends on current and arc radius through Eq. (6). Substituting, 16.522, Space Propulsion Lecture 11-12 Prof. Manuel Martinez-Sanchez Page 9 of 18