Splash singularity for a free-boundary incompressible viscoelastic fluid model

Numerical computations in viscoelasticity show the failure of many numerical schemes when the Weissenberg number is beyond a critical value Keunings (J Non-Newtonian Fluid Mech 20:209–226, 1986, [6]). The existence of singularities in the continuum model could be the way to explain instability appearing in numerical simulations. We consider here a 2D Oldroyd-B type model at high Weissenberg number, and we show the existence of the so-called splash singularities (namely, points where the free boundary remains smooth but self-intersects). In our case, we assume physically realistic boundary conditions given by the static equilibrium of all the force fields acting at the interface. Our strategy is based on local existence and stability results applied to a family of smooth suitable initial configurations, we show they will evolve into a self-intersecting configuration, and then necessarily there exists a positive time t=t*, where the configuration has a splash singularity. To prove local existence and stability, we first apply a conformal transformation to the 2D domain, in order to separate the contact point with splash, and then we pass into Lagrangian coordinates to fix our domain, inspired by a Thomas Beale’s paper on the initial value problem for the Navier–Stokes equations with a free surface.