In a recent paper, Al-Housseiny and Stone (J Fluid Mech 706:597–606, 2012) considered the dynamics of a stretching surface and how this interacts with the boundary-layer flow it generates. These authors discussed the cases $$c= -3$$ c = - 3 for an elastic sheet and $$c= -1$$ c = - 1 for the viscous fluid, c being representative for the stretching velocity of the sheet. The aim of the present paper is to extend the analysis of Al-Housseiny and Stone (2012) to the general values of c , to allow for both a stretching and a shrinking sheet and for the surface to be permeable through the parameter S , where $$S>0$$ S > 0 for the fluid withdrawal and $$S < 0$$ S < 0 for fluid injection. Both the cases $$S=0$$ S = 0 (impermeable surface) and $$S \ne 0$$ S ≠ 0 (permeable surface) are considered for both stretching surfaces and shrinking surfaces. In all these cases, asymptotic solutions are presented for large values c and S (both withdrawal and injection).

中文翻译：

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关于伸缩渗透面边界层流动中的方程

在最近的一篇论文中，Al-Housseiny 和 Stone (J Fluid Mech 706:597–606, 2012) 考虑了拉伸表面的动力学以及它如何与它产生的边界层流相互作用。这些作者讨论了弹性片的 $$c= -3$$ c = - 3 和粘性流体的 $$c= -1$$ c = - 1 的情况，c 代表片的拉伸速度。本论文的目的是将 Al-Housseiny 和 Stone (2012) 的分析扩展到 c 的一般值，以允许拉伸和收缩片材以及表面可渗透参数 S，其中$$S>0$$ S > 0 用于流体抽取，$$S < 0$$ S < 0 用于流体注入。$$S=0$$ S = 0（不可渗透表面）和 $$S \ne 0$$ S ≠ 0（可渗透表面）两种情况都被考虑用于拉伸表面和收缩表面。在所有这些情况下，都会为大值 c 和 S（撤出和注入）提供渐近解。