Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
May 23-26, 2011

Van Vleck Hall
Department of Mathematics
University of Wisconsin -Madison
Madison, WI 53706

Beyond the Child-Langmuir Limit

Prof. Russel E. Caflisch

University of California, Los Angeles

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.