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The Maximum Regularity Property of the Steady Stokes Problem Associated with a Flow Through a Profile Cascade
Acta Applicandae Mathematicae ( IF 1.6 ) Pub Date : 2021-02-23 , DOI: 10.1007/s10440-021-00396-4
Tomáš Neustupa

We deal with a steady Stokes-type problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain \(\varOmega \). The corresponding Stokes–type problem is formulated by means of the Stokes equation, equation of continuity and three types of boundary conditions: the conditions of periodicity on the curves \(\varGamma _{\hspace {-1.1pt} 0}\) and \(\varGamma _{\hspace {-1.1pt} 1}\), the Dirichlet boundary conditions on \(\varGamma _{\hspace {-1.1pt} \mathrm {in}}\) and \(\varGamma _{\hspace {-1.1pt} p}\) and an artificial “do nothing”–type boundary condition on \(\varGamma _{\hspace {-1.1pt} \mathrm {out}}\). (See Fig. 1.) We explain on the level of weak solutions the sense in which the last condition is satisfied. We show that, although domain \(\varOmega \) is not smooth and different types of boundary conditions meet in the corners of \(\varOmega \), the considered problem has a strong solution with the so called maximum regularity property.



中文翻译:

与流经剖面级联的稳态斯托克斯问题的最大正则性质

我们处理一个稳定的斯托克斯型问题,该问题与牛顿不可压缩流体通过空间周期性剖面级联流动有关。使用的数学模型基于缩小到一个空间周期,该空间周期由有界的2D域 \(\ varOmega \)表示。相应的斯托克斯型问题是通过斯托克斯方程,连续性方程和三种边界条件来表达的:曲线\(\ varGamma _ {\ hspace {-1.1pt} 0} \)上的周期性条件和\(\ varGamma _ {\ hspace {-1.1pt} 1} \)\(\ varGamma _ {\ hspace {-1.1pt} \ mathrm {in}} \)\(\ varGamma _上的Dirichlet边界条件{\ hspace {-1.1pt} p} \)以及\(\ varGamma _ {\ hspace {-1.1pt} \ mathrm {out}} \)上的人工“不执行”类型边界条件 。(见图1。)我们在弱解的层次上解释满足最后一个条件的意义。我们表明,尽管域\(\ varOmega \)并不平滑,并且不同类型的边界条件在\(\ varOmega \)的拐角处均满足,但是所考虑的问题具有所谓的最大规则性属性的强大解决方案。

更新日期:2021-02-23
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