This paper considers the significant role of cross-sectional geometry on resistance in co-axial pipe flows. We consider an axially flowing viscous fluid in between two long and thin elliptical coaxial cylinders, one inside the other. The outer cylinder is stationary, while the inner cylinder (rod) is free to move. The rod poses a resistance to the axial flow, while the viscous fluid poses a resistance to any motion of the rod. We show that the equations for flow in the axial direction – driven by a prescribed flux – and for flow within the cross-section of the domain – driven by the motion of the rod – decouple in the asymptotic limit of small cylinder aspect ratio into axial Poiseuille flow and transverse Stokes flow, respectively. The objective of this paper is to calculate numerically the axial and cross-sectional resistances and to determine their dependence on cross-sectional geometry – i.e. rod position and the ellipticities of the rod and bounding cylinder. We characterise axial resistance, first for three reduced parameter spaces that have not been fully analysed in the literature: (i) a circle in an ellipse, (ii) an ellipse in a circle and (iii) an ellipse in an ellipse of equal eccentricity and orientation, before extending our geometric parameter space to determine the overall optimal geometry to minimise axial flow resistance for fixed cross-sectional area. Cross-sectional resistance is characterised via coefficients in a Stokes resistance matrix and we highlight the interdependent effects of cross-sectional ellipticity and boundary interactions.

中文翻译：

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几何形状对椭圆管流阻力的影响

本文考虑了横截面几何形状对同轴管流中阻力的重要作用。我们考虑在两个长而细的椭圆同轴圆柱体之间轴向流动的粘性流体，一个在另一个圆柱体内。外筒是静止的，而内筒（杆）可以自由移动。杆对轴向流动构成阻力，而粘性流体对杆的任何运动构成阻力。我们表明，轴向流动的方程——由规定的通量驱动——以及域截面内的流动——由杆的运动驱动——在小圆柱纵横比的渐近极限解耦为轴向分别为泊肃叶流和横向斯托克斯流。本文的目的是数值计算轴向和横截面阻力，并确定它们对横截面几何形状的依赖——即杆位置以及杆和边界圆柱的椭圆度。我们首先对文献中尚未完全分析的三个简化参数空间进行表征：（i）椭圆中的圆，（ii）圆中的椭圆和（iii）等偏心椭圆中的椭圆和方向，在扩展我们的几何参数空间以确定整体最佳几何形状以最小化固定横截面积的轴向流动阻力之前。横截面电阻通过斯托克斯电阻矩阵中的系数表征，我们强调了横截面椭圆度和边界相互作用的相互依赖影响。