In this paper, the interaction between two initially circular in-line bubbles during their rise in a viscoelastic liquid under gravity is numerically simulated by weakly compressible smoothed particle hydrodynamics (WC-SPH). The background fluid is assumed to obey Oldroyd-B rheological model and the surface tension between two phases is evaluated by continuum surface tension (CST) method. First, it is shown that the obtained results for two benchmark problems, including two rising bubbles in a Newtonian fluid and a single rising bubble in a viscoelastic fluid, are in good agreement with previously published results. Then, together ascending of two unequal in-line bubbles in an Oldroyd-B viscoelastic liquid is simulated. Two different configurations are considered for the position of larger bubble and the center-of-mass of the bubbles and its velocity are presented for each case. Also, the differences of velocity and stress distributions in the background liquid are evaluated for two configurations. Based on the obtained results, it could be said that when the larger bubble is placed above the smaller one, merging does not occurred and the upper bubble has a dual cusped trailing edge. In addition, the effects of Deborah number, polymer concentration, density and viscosity ratios, Reynolds number, and Bond number are investigated on rising bubbles. To the best of authors’ knowledge, it is the first time the simultaneous rising of two bubbles in Oldroyd-B fluids is investigated.

在本文中，通过弱可压缩平滑粒子流体动力学（WC-SPH）数值模拟了两个初始圆形管道气泡在重力作用下在粘弹性液体中上升期间的相互作用。假定背景流体服从Oldroyd-B流变模型，并且通过连续表面张力（CST）方法评估两相之间的表面张力。首先，表明两个基准问题的结果，包括牛顿流体中的两个上升气泡和粘弹性流体中的一个上升气泡，与先前发表的结果非常吻合。然后，一起模拟了Oldroyd-B粘弹性液体中两个不相等的直列气泡的上升。对于较大气泡的位置，考虑了两种不同的配置，并针对每种情况提出了气泡的质心及其速度。同样，针对两种配置评估背景液体中速度和应力分布的差异。根据获得的结果，可以说，当较大的气泡置于较小的气泡上方时，不会发生合并，并且上方的气泡具有双重尖峰的后缘。此外，研究了黛博拉数，聚合物浓度，密度和粘度比，雷诺数和键数对上升气泡的影响。据作者所知，这是第一次研究Oldroyd-B流体中两个气泡同时上升的现象。针对两种配置评估了背景液体中速度和应力分布的差异。根据获得的结果，可以说，当较大的气泡置于较小的气泡上方时，不会发生合并，并且上方的气泡具有双重尖峰的后缘。此外，研究了黛博拉数，聚合物浓度，密度和粘度比，雷诺数和键数对上升气泡的影响。据作者所知，这是第一次研究Oldroyd-B流体中两个气泡同时上升的现象。针对两种配置评估了背景液体中速度和应力分布的差异。根据获得的结果，可以说，当较大的气泡置于较小的气泡上方时，不会发生合并，并且上方的气泡具有双重尖峰的后缘。此外，研究了黛博拉数，聚合物浓度，密度和粘度比，雷诺数和键数对上升气泡的影响。据作者所知，这是第一次研究Oldroyd-B流体中两个气泡同时上升的现象。此外，研究了黛博拉数，聚合物浓度，密度和粘度比，雷诺数和键数对上升气泡的影响。据作者所知，这是第一次研究Oldroyd-B流体中两个气泡同时上升的现象。此外，研究了黛博拉数，聚合物浓度，密度和粘度比，雷诺数和键数对上升气泡的影响。据作者所知，这是第一次研究Oldroyd-B流体中两个气泡同时上升的现象。