Numerical solution of the Giesekus model for incompressible free surface flows without solvent viscosity

We present a numerical method for solving the Giesekus model without solvent viscosity. This paper is concerned with incompressible two-dimensional free surface flows and employs the finite difference method to solve the governing equations. The methodology involves solving the momentum equation using the implicit Euler scheme and an implicit technique for computing the pressure condition on the free surface. The nonlinear Giesekus constitutive equation is computed by a second order Runge–Kutta method. A novel analytic solution for channel flow is developed and is used to verify the numerical technique presented herein. Mesh refinement studies establish the convergence of the method for complex free surface flows. To demonstrate that the technique can deal with complicated free surface flows, the time-dependent flow produced by a fluid jet flowing onto a rigid surface is simulated and the influence of the parameter α on the jet buckling phenomenon is investigated. In addition, the simulation of the extrudate swell of a Giesekus fluid was carried out and the effect of the parameter α on the flow was similarly examined.