Order-of-magnitude scaling of the cathode region in an axisymmetric transferred electric arc
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I. INTRODUCTION
THE main goal of this work is to demonstrate a new technique of order-of-magnitude scaling (OMS)[1] to a complex engineering problem: the welding arc. This technique allows the simultaneous use of dimensional analysis and asymptotic considerations such as dominant balance.[2] The main difference between OMS and previous work in the field of dimensional analysis and asymptotic considerations[3,4,5] is that in OMS, the unknown functions are required to vary smoothly between the edges of the domain. This way, these functions and their derivatives up to a second order have special properties and can be characterized by a single characteristic value. Dimensional analysis provides an exact description of the functional dependence in the problem, based on a complete set of dimensionless groups. The asymptotic considerations determine the functional expression of the dimensionless groups and describe their relative importance. This way, dimensionless groups of little influence can be discarded without significant loss in accuracy. The asymptotic considerations also provide scaling for the problem. The scaling factors and relevant dimensionless groups obtained can be combined with numerical calculations in order to express these results in a general, dimensionless form. The OMS technique is a useful tool for a wide variety of problems, with many driving forces described by a set of differential equations. For the sake of simplicity using this new tool, the present analysis focuses on the fluid motion of the plasma in the cathode region of the arc, where P.F. MENDEZ, Postdoctoral Associate, and T.W. EAGAR, Professor of Materials Engineering, are with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. M.A. RAMIREZ, formerly Graduate Student, Department of Materials Science and Engineering, Massachusetts Institute of Technology, is Professor, Instituto Tecnologico de Morelia, 58120 Morelia, Mich., Mexico. G. TRAPAGA, formerly Principal Research Associate, Department of Materials Science and Engineering, Massachusetts Institute of Technology, is Professor, Laboratorio de Investigacion en Materiales, CINVESTAVIPN, 76230 Queretaro, Qro., Mexico. Manuscript submitted May 18, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS B
the electromagnetic forces are dominant.[6] Due to its small size relative to the rest of the arc, the cathode region can be assumed to be isothermal, in contrast to other regions in the domain, where significant temperature gradients are present. Other regions of the arc cannot be assumed to be isothermal. In these cases, the OMS technique would still be useful, but new equations (e.g., conservation of energy) must be considered. In such cases, the complexity of the problem would increase substantially, with many more parameters necessary to describe the problem and many more balances and asymptotic regimes to analyze. Previous attempts have been made to provide general and simple expressions that capture the behavior of the arc. So
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