Splash Singularities for a General Oldroyd Model with Finite Weissenberg Number

Authors: Elena Di Iorio (ICMAT-CSIC), Pierangelo Marcati and Stefano Spirito

Source: Archive for Rational Mechanics and Analysis

Date of publication: 22 October 2019



The Oldroyd model, introduced by J. G. Oldroyd in 1950, describes the complex behavior of viscoelastic fluids. The study of viscoelastic materials started in the XIX century, when mathematical physicists such as Kelvin and Maxwell started investigating and making experiments with models made of springs and dashspots. They proposed models represented by first order linear differential equations. Later, when human-made polymers were engineered, the need of more accurate and complex mathematical models arose since the existing ones were not suitable to describe all the viscoelastic materials. For this reason, the Oldroyd model has been introduced. It is a frame-invariant model, due to the presence of the Oldroyd derivative instead of the material derivative and it is characterized by the presence of both elastic and viscous forces, whose ratio is called Weissenberg number (We).

The paper addresses to the study of a 2D incompressible, free-boundary Oldroyd model, with finite We. The authors focus on the existence of singularities for the domain, in particular on the splash singularity, given by a configuration where the domain self-intersects in one point. The equations describing this model are the conservation of mass and the conservation of momentum but they are not sufficient to provide complete description of the flow and a constitutive equation involving the elastic stress tensor is required, which is the Oldroyd equation. These equations establish the final system together with the initial data and the free-boundary conditions, given by the balance of the force fields at the interface.

The proof is based on a conformal-lagrangian change of variables in order to fix the moving domain and the use a fixed point argument to prove the local existence for the system. As a consequence of the estimates, the researchers get that the local existence time T can be estimated from above. Finally, they prove stability estimates, with respect to a suitable one parameter family of perturbated solutions, and they are able to show the existence of a splash singularity.