Narrow Results

We study a system of nonlinear partial differential equations governing the motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain, with a non-Newtonian viscosity depending on the second invariant of the rate of deformation tensor. Considering the equations in a suitably decomposed form, we establish, for small and suitably regular data, existence of a unique solution using a fixed point argument in an appropriate functional setting. This model includes the classical Oldroyd-B fluid as a particular case.

The traditional weakly compressible SPH (WCSPH) method has been reported to suffer from large density variations, resulting in non-physical pressure fluctuations. In this article, a truly incompressible SPH (ISPH) method is developed and extended to free surface flows of shear-thinning fluids, in which the viscosity is modeled by the Cross model. The ISPH method employs a pressure Poisson equation to satisfy the incompressibility constraints, and the Navier-Stokes equations are solved in a Lagrangian form using a two-step fractional method. The method is firstly verified by solving the impact of a Newtonian droplet with a horizontal rigid plate in comparison with the available literature data. Then, ISPH is extended to simulate the phenomena of a Newtonian and a Cross droplet impact and spreading over an inclined rigid plate. In particular, the different flow features between Newtonian and Cross droplets after the impact are discussed. All numerical results show that ISPH method can not only extend to free surface flows of shear-thinning fluids, but also reduce the pressure noise in comparison with WCSPH.