Funtional derivatives and finite elements for the steady spinning of a viscoelastic filament

Abstract

In many cases the familiar equations for the steady spinning of a viscoelastic filament can be solved without resort to differential equations. The method of solution is illustrated here using the Maxwell fluid as a model. The integral form of the Maxwell constitutive equation is used directly. This suggests extension of the method to fluid models with constitutive equations of the BKZ type. Since these constitutive models may have no corresponding differential model, the solution technique presented here shows promise of permitting the approximation of solutions to problems which have not previously been amenable to numerical solution. The solution procedure is based on the development of relatively simple expression for the linearized operator of the steady spinning equation. The operator is linearized about an arbitrary admissible velocity field, and the iteration equations for the Newton—Raphson method are solved using a finite element collocation procedure. The success of the procedure in the Maxwell fluid model problem is demonstrated.