Approximate solution of second-order boundary- layer equations

Abstract

necessary to eject electrons continuously during a solar flare in order to maintain the electric field. This could be done with an electron linear accelerator emitting a beam of electrons at the design energy. These electrons could be expected to escape through the magnetic field on account of the relativistic increase in their mass. An allowance for this linear accelerator has been made in the weights quoted in Fig. 1. However, because of the greater simplicity and negligible weight of the inductive charge ejection scheme, it is to be hoped that the accelerator would be unnecessary.It can be seen from the foregoing discussion that the magnitude of the losses that we can expect is at present unknown. Losses take the form of motion of electrons toward the space vehicle or positive ions away from the vehicle, in either case at the expense of the energy of the electric field. Taking first the losses due to positive ions, we note that, since there are no trapped ions in the system, and since ions coming from outside the system (including the solar flare ions) are reflected without loss, the only source is from the ionization of neutral atoms in the electric field region. Following such an ionization, the electron that is born is retained on the magnetic field line where it is, but the ion is simply ejected. The worst case is if ionization takes place at the surface of the vehicle, for then each ion ejected carries with it an energy eF 0 acquired from the electric field. Two possible sources of neutral atoms are outgassing from the surface and micrometeorites. Assuming pessimistically that each ionization is at the surface of the vehicle, Table 2 shows the permissible outgassing rates for two values of the power consumed by the ion current. The outgassing can be seen to represent a serious problem, but it is probably not insuperable since conditions in space are very favorable to achieving a good bakeout of exposed surfaces. The micrometeorite rate near the earth is given by Whipple 5 as about 10~6 g/cm 2 -yr, but evidence obtained in deep space by Alexander, 6 using an instrument aboard Mariner II, showed flux rates 10 4 times lower than corresponding rates near the earth.The remaining source of loss arises from the possibility of diffusive motion of the electrons toward the space vehicle. This source of loss is at present, by many orders of magnitude, the least certain aspect of the whole device. We are attempting here to confine a plasma with a magnetic field; experience gained in the field of controlled thermonuclear fusion prompts us to comment on this problem with extreme caution. We can, however, point out that our configuration having the magnetic field "inside" and the plasma "outside" does fulfill the so-called minimum B requirement presently thought to contribute to stability. 7 Furthermore, certain types of instabilities which might have been expected to contribute to substantial rates of diffusion across the magnetic field, 8 and which are thought basically to be a result of the difference between the mas...

Reply by Author to H.E. Bethel

“…2 The modifications given by Bethel evidently will improve first-order approximate boundarylayer computations in regions of retarded flows.…”

“…The latter form of the approximating function adequately handled mildly retarded flows but did not satisfactorily handle strongly retarded flows and is incapable of predicting separation. 2 A result obtained in the vicinity of the separation point is given below to illustrate the applicability of the modified form of the approximating function to regions of adverse pressure gradient. This result is for the first-order boundaryayer equations and w T as obtained with the form of the approximating function (6).…”

“…4 A form of the approximating function similar to (5), but without the logarithmic term, was employed to solve first-order problems by Neilson et al 5 and Bethel. 2 In Ref. 2 it was found that the form (6) could only be employed in the retarded flow region, and a form similar to (2) but without the logarithmic term was utilized in the accelerated flow region.…”

“…2 In Ref. 2 it was found that the form (6) could only be employed in the retarded flow region, and a form similar to (2) but without the logarithmic term was utilized in the accelerated flow region. The latter form of the approximating function adequately handled mildly retarded flows but did not satisfactorily handle strongly retarded flows and is incapable of predicting separation.…”

“…3, 2197-2202 (1965) 2. Bethel, H. E., "On the convergence and exactness of the solutions of the laminar boundary-layer equations using the A^-parameter integral method of Galerkin-Kantorovich-Dorodnitsyn," Ph.D. Thesis, Pnrdue Univ.…”