This work presents a numerical technique for solving two-dimensional non- Newtonian free surface flows. The governing equations are solved by VOF method on a staggered grid. It uses marker particles to describe the fluid. To predict the shape of the jet buckling, Navier-Stokes equations are used while The viscosity (υ (γ&) ) as a function of shear rate (γ&) which determined by ( : . A modified Volume-of-Fluid (VOF) technique based on Youngs’ algorithm is used to track the jet buckling. As currently implemented, the present method can simulate twodimensional non Newtonian flow in which the viscosity is modeled by using the Cross model. Buckling is a physical instability; it is the fluid equivalent of Euler’s slender rod subject to a compressive force .Like the Euler’s rod, the fluid may buckle one way or the other and indeed for a 2D fluid jet it may buckle in any direction. the simulation of jet buckling is approved by both experimental and theoretical results of Cruickshank and Munson [7] and Cruickshank [8]. this concurs with results obtained by Tom´e et al. [9] for the two-dimensional case.

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