A finite difference technique for solving a time strain separable K-BKZ constitutive equation for two-dimensional moving free surface flows

Abstract

This work is concerned with the numerical solution of the K-BKZ integral constitutive equation for two-dimensional time-dependent free surface flows. The numerical method proposed herein is a finite difference technique for simulating flows possessing moving surfaces that can interact with solid walls. The main characteristics of the methodology employed are: the momentum and mass conservation equations are solved by an implicit method; the pressure boundary condition on the free surface is implicitly coupled with the Poisson equation for obtaining the pressure field from mass conservation; a novel scheme for defining the past times t is employed; the Finger tensor is calculated by the deformation fields method and is advanced in time by a second-order Runge-Kutta method. This new technique is verified by solving shear and uniaxial elongational flows. Furthermore, an analytic solution for fully developed channel flow is obtained that is employed in the verification and assessment of convergence with mesh refinement of the numerical solution. For free surface flows, the assessment of convergence with mesh refinement relies on a jet impinging on a rigid surface and a comparison of the simulation of a extrudate swell problem studied by Mitsoulis (2010) [44] was performed. Finally, the new code is used to investigate in detail the jet buckling phenomenon of K-BKZ fluids.

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

“…7(a) with the other numerical data available in the literature, as well as the approximate solution given by Eq. (41). First, we can notice that there is a certain discrepancy between the different data available in the literature.…”

“…the flow of the surrounding air is omitted). This method was used to simulate the extrudate swell and jet buckling phenomena for the generalized Newtonian fluid model [34,35], and for various viscoelastic constitutive models [36][37][38][39][40][41]. In the level-set method, the position of the interface (or the free surface) is represented with a level-set function that varies continuously across the interface [42].…”

Extrusion Blow Molding Simulation using Cross-Section Technique for Complex Shape Bottles

“…The FEA result of the cross-section shape bottle blow molding process was good agreement with the experimental data more than the full shape bottle blow molding simulation. The average The blow molding simulations can improve for more accuracy by the assignation of other material model such as the K-BKZ material model which was followed the works by [24][25][26]. However, the final thickness of bottle of the cross-section shape bottle blow molding simulation was found the good agreement with the experiment data more than the full shape bottle.…”

Simulação do modelo constitutivo integral KBKZ-PSM usando diferenças finitas em malhas hierárquicas

“…Os resultados obtidos e as respectivas soluções analíticas para a velocidade u(y) e para os tensores τ xx , τ xy e τ yy são apresentados na Figura 3. Pode-se observar que existe boa concordância entre os resultados numéricos e as soluções analíticas que são apresentadas em [12]. Observa-se ainda que na malha IV (com refinamento local) a solução numérica aproxima-se da solução da malha III (malha mais fina), embora na região central as malhas I e IV apresentem as mesmas células.…”