KCL - A mathematical model to describe evolution of curves and surfaces
Montag, 15. Dezember 2014 14:15 Uhr;
Sprecher: Phoolan Prasad; Indian Institute of Science, Bangalore
In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. d-D kinematical conservation laws (KCL) are equations of evolution of a moving surface Ωt in d-dimensional (x1 , x2 , . . . , xd )-space Rd . The KCL are derived in a specially defined ray coordinates (ξ1 , ξ2 , . . . , ξd−1 , t), where ξ1 , ξ2 , . . . , ξd−1 are surface coordinates on Ωt and t is time. KCL are the most general equations in conservation form, governing the evolution of Ωt with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ωt when Ωt is a curve in R2 and is a curve on Ωt when it is a surface in R3 . Across a kink the normal n to Ωt and normal velocity m on Ωt are discontinuous. Since the KCL system contains only kinematical relations, it is an under-determined system of equations. In order to complete the system, we need to find additional equations representing the dynamics of Ωt from the governing equations of the medium in which Ωt propagates. The mathematical analysis of 3-D KCL system and computation with its help present a challenge since the eigenspace is not complete and there are geometric solenoidal constrains. We present a few examples of Ωt and numerical results.
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