In this article we analyze a recently proposed model for boundary layer flow of a nanofluid past a permeable stretching/shrinking sheet. The boundary value problem (BVP) resulting from this model is governed by two physical parameters; $ {{\lambda}} $, which controls the stretching ($ {{\lambda}} >0 $) or shrinking ($ {{\lambda}} < 0 $) of the sheet, and $ S $, which controls the suction ($ S>0 $) or injection ($ S<0 $) of fluid through the sheet. For $ {{\lambda}} \ge 0 $ and $ S\in \mathbb{R} $, we present a closed-form solution to the BVP and prove that this solution is unique. For $ {{\lambda}} < 0 $ and $ S< 2\sqrt{-{{\lambda}}} $ we prove no solution exists. For $ {{\lambda}} < 0 $ and $ S = 2\sqrt{-{{\lambda}}} $ we present a closed-form solution to the BVP and prove that it is unique. For $ {{\lambda}} < 0 $ and $ S> 2\sqrt{-{{\lambda}}} $ we present two closed-form solutions to the BVP and prove the existence of an infinite number of solutions in this parameter range. The analytical results proved here differ from the numerical results reported in the literature. We discuss the mathematical aspects of the problem that lead to the difficulty in obtaining accurate numerical approximations to the solutions.

中文翻译：

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分析纳米流体通过可渗透拉伸/收缩片材的流动

在本文中，我们分析了最近提出的纳米流体通过可渗透拉伸/收缩片材的边界层流动的模型。该模型产生的边界值问题（BVP）由两个物理参数控制；$ {{\ lambda}} $，控制图纸的拉伸（$ {{\ lambda}}> 0 $）或收缩（$ {{\ lambda}} <0 $），以及$ S $，控制通过薄片抽吸（$ S> 0 $）或注入（$ S <0 $）流体。对于$ {{\ lambda}} \ ge 0 $和$ S \ in \ mathbb {R} $，我们为BVP提供了一种封闭形式的解决方案，并证明该解决方案是唯一的。对于$ {{\ lambda}} <0 $和$ S <2 \ sqrt {-{{\ lambda}}} $，我们证明不存在解决方案。对于$ {{\ lambda}} <0 $和$ S = 2 \ sqrt {-{{\ lambda}}} $，我们为BVP提供了一种封闭形式的解决方案，并证明了它是唯一的。对于$ {{\ lambda}} < 0 $和$ S> 2 \ sqrt {-{{\ lambda}}} $我们为BVP提供了两个封闭形式的解，并证明了在此参数范围内存在无数解。在此证明的分析结果与文献报道的数值结果不同。我们讨论了问题的数学方面，这些问题会导致难以获得解决方案的精确数值近似值。