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We developed a lattice Boltzmann method coupled with the Oldroyd-B constitutive equation to simulate a viscoelastic fluid. In this work, the flow field of the solvent is solved using an incompressible lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model, while the advection operator of the polymer stress tensor is directly calculated with the help of the particle distribution functions. Specifically, we present a numerical scheme for the advection of the polymer stress tensor through the truncation of second-order Taylor expansion, which does not need to introduce the extra distribution functions and has better numerical accuracy. We consider two types of numerical tests to examine the performance of the presented method, including a two-dimensional (2D) channel flow and the 4:1 contraction problem. Our numerical results for the 2D channel flow agree well with the analytical results and some experimental results reported in the previous studies. Moreover, the numerical results also indicate that the current method can capture some complex rheological behaviors of the 4:1 contraction flow.

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In the present work, the phenomenon of an Oldroyd-B drop impact and spreading on an inclined rigid plate at low impact angles is simulated numerically using the smoothed particle hydrodynamics (SPH) method. In order to remove the unphysical phenomenon of fracture and particle clustering in fluid stretching which is the so-called tensile instability, an artificial stress term is employed which has been successfully proposed in simulations of elastic solids. Particularly, the effects of surface inclination and the different regimes of drop impact and spreading on an inclined surface are investigated. The numerical results show the capability of the proposed scheme in handing the unsteady viscoelastic free surface flows. All numerical results of using the SPH method are in agreement with the available data.