Kinetic Description of Multiscale Phenomena
The Annual Kinetic FRG Meeting
May 23-26, 2011
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Beyond the Child-Langmuir Limit
Prof. Russel E. Caflisch
University of California, Los Angeles
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Abstract: This talk describes solution of the Child-Langmuir problem of the continuum flow of electrons from cathode to anode across a channel by two methods: The first is direct numerical simulation. The second is a new analytic method. When put into characteristic variables - i.e., variables that follow a particle path - the equations admit a simple analytic, but implicit, solution for any choice of boundary data for velocity v, electric field E and current J at the cathode. For steady solutions, this approach clarifies the origins of the maximal current J_{max} for a given potential jump across the channel, as derived by Child and Langmuir for incoming velocity v=0 and by Jaffe for v>0. Moreover, it leads to an approximate form for the time-dependent solution when J>J_{max}. The results are compared to those from direct numerical simulation. |
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