The pressure-driven growth model has been employed to study a propagating foam front in the foam-improved oil recovery process. A first-order solution of the model proves the existence of a concave corner on the front, which initially migrates downwards at a well defined speed that differs from the speed of front material points. At later times, however, it remains unclear how the concave corner moves and interacts with points on the front either side of it, specifically whether material points are extracted from the corner or consumed by it. To address these questions, a second-order solution is proposed, perturbing the aforementioned first-order solution. However, the perturbation is challenging to develop, owing to the nature of the first-order solution, which is a similarity solution that exhibits strong spatio-temporal non-uniformities. The second-order solution indicates that the corner’s vertical velocity component decreases as the front migrates and that points initially extracted from the front are subsequently consumed by it. Overall, the perturbation approach developed herein demonstrates how early-time similarity solutions exhibiting strong spatio-temporal non-uniformities break down as time proceeds.

中文翻译：

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相似解的分解：泡沫改进采油过程中前沿传播的扰动方法

压力驱动增长模型已被用于研究泡沫改进采油过程中的泡沫前沿。该模型的一阶解证明了前端存在一个凹角，它最初以与前端物质点速度不同的明确定义的速度向下迁移。然而，在以后的时间里，仍然不清楚凹角如何移动并与它前面任一侧的点相互作用，特别是物质点是从角中提取的还是被它消耗的。为了解决这些问题，提出了一个二阶解，扰乱了前面提到的一阶解。然而，由于一阶解的性质，扰动的发展具有挑战性，一阶解是一种表现出强烈时空非均匀性的相似解。二阶解表明拐角的垂直速度分量随着锋面的迁移而减小，最初从锋面提取的点随后被它消耗。总体而言，本文开发的扰动方法展示了表现出强烈时空非均匀性的早期相似性解决方案如何随着时间的推移而分解。