Abstract
In this paper a slender jet model of viscoelastic fluids which is asympotically derived from the full free surface boundary-value problem. The model system consists of four coupled quasi-linear differential equations in one space dimensions, where the nonlinear characteristics are given in closed form. Two characteristics are always real, two others may be real or complex, leaving open the possibility for change-of-type from hyperbolic to mixed elliptic/hyperbolic type. We proceed to exhibit exact solutions (constant, steady time dependent and space-time dependent) along which this model system undergoes a variety of change-of-type phenomena. Viewed purely as a one-dimensional (1-D) model for change-of-type, we explain the significance of type for the stability of these solutions and describe the numerical implications for each type. We also explain the physical significance of these model phenomena with respect to the original 3-D system, since these asymptotic equations are no longer valid once small-scale instabilities develop. Remarkably, these special solutions of the model system that exhibit change-of-type correspond to exact solutions of the 3-D Maxwell model with cylindrical free surface. The 1-D model equations, however, are not an invariant reduced system of the full 3-D free surface Maxwell model, so that the change-of-type exhibited here in the 1-D model is not directly responsible for a 3-D free surface change-of-type. Regularizations of this model as a catastrophic change-of-type develops are suggested.
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Communicated by Daniel Joseph
Research support is gratefully acknowledged from the Air Force Office of Scientific Research, Grant No. 88-0164.
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Forest, M.G., Wang, Q. Change-of-type behavior in viscoelastic slender jet models. Theoret. Comput. Fluid Dynamics 2, 1–25 (1990). https://doi.org/10.1007/BF00271426
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DOI: https://doi.org/10.1007/BF00271426