A Quasilinear Parabolic Equation Describing the Elongation of Thin Filaments of Polymeric Liquids

摘要

The equation under study - derived from physical principles in this paper - describes the elongation of a filament of a polymeric liquid subjected to a force f at both ends. The liquid is assumed to satisfy certain accepted 'rubberlike liquid' constitutive relations, and the filament is assumed to be thin, which permits reduction of the problem to one space dimension. The unknown variable u denotes the position of a fluid particle at time t, which was at position x at t = - infinity, i.e., before the deformation started, we have u(x, - infinity) = x. In this paper the equation under study is transformed in such a way that it fits into the framework of the general mathematical theory for quasilinear parabolic equations. This makes it possible to prove that for any given 'initial condition' a solution exists at least on a certain time interval. (it is a part of the analysis to discover what is an appropriate meaning of 'initial condition' to be associated with the problem under study). Moreover, we shall prove that for forces f(t), which approach zero exponentially for t approaches + or - infinity and are small in a suitable sense, there is a solution for all times t, - infinity < t < + infinity, and this solution approaches a stationary limit as t approaches + infinity.