Viscoplastic fluids can hold bubbles/particles stationary by balancing the buoyancy stress with the yield stress – the key parameter here is the yield number $Y$ , the ratio of the yield stress to the buoyancy stress. In the present study, we investigate a suspension of bubbles in a yield-stress fluid. More precisely, we compute how much is the gas fraction $\phi$ that could be held trapped in a yield-stress fluid without motion. Here the goal is to shed light on how the bubbles feel their neighbours through the stress field and to compute the critical yield number for a bubble cloud beyond which the flow is suppressed. We perform two-dimensional computations in a full periodic box with randomized positions of the monosized circular bubbles. A large number of configurations are investigated to obtain statistically converged results. We intuitively expect that for higher volume fractions, the critical yield number is larger. Not only here do we establish that this is the case, but also we show that short-range interactions of bubbles increase the critical yield number even more dramatically for bubble clouds. The results show that the critical yield number is a linear function of volume fraction in the dilute regime. An algebraic expression model is given to approximate the critical yield number (semi-empirically) based on the numerical experiment in the studied range of $0\le \phi \le 0.31$ , together with lower and upper estimates.

中文翻译：

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粘塑性流体中的气泡云

粘塑性流体可以通过平衡浮力和屈服应力来保持气泡/颗粒静止——这里的关键参数是屈服数 $Y$ , 屈服应力与浮力应力的比值。在本研究中，我们研究了屈服应力流体中的气泡悬浮液。更准确地说，我们计算气体分数是多少 $\phi$ 可以在没有运动的情况下被困在屈服应力流体中。这里的目标是阐明气泡如何通过应力场感受它们的邻居，并计算气泡云的临界屈服值，超过该值的流动被抑制。我们在具有单尺寸圆形气泡的随机位置的完整周期框中执行二维计算。研究了大量的配置以获得统计上收敛的结果。我们直观地预计，对于更高的体积分数，临界产量数更大。我们不仅在这里确定了这种情况，而且我们还表明，气泡的短程相互作用增加了气泡云的临界产量数，甚至更加显着。结果表明，临界产量数是稀释状态下体积分数的线性函数。在研究范围内的数值实验的基础上，给出了一个代数表达式模型来近似临界产量数（半经验） $0\le \phi \le 0.31$ ，以及下限和上限估计。