A microshaft may become rough due to corrosion, abrasion, and deposition when it has been operating in a viscous fluid. It is of importance to investigate the effects and to estimate the level of the shaft’s surface roughness. In this study, we consider a bumpy shaft with its shape modeled by the product of two cosinoidal functions; the roughness ε is defined to be the ratio of the amplitude of the product to the mean radius b of the shaft. First, we consider the Couette flow of the shaft in a viscous fluid enclosed by a rotating smooth cylinder. A perturbation analysis is carried out for the Stokes equation with respect to ε up to the second-order with the key parameters including the azimuthal wave number n and the axial wave number α of the roughness, as well as the mean radius b. In addition, a perturbation analysis is performed for the Poiseuille flow in the gap between the shaft and the shrouded cylinder so that we have complete information for estimating the mean roughness of the shaft. Moreover, numerical simulations are carried out for the torque acting on the shaft at selected b, ε, and wave numbers n, α for verifying the accuracy of the perturbation results. It is shown that the mean torque M acting on the unit area of the bumpy shaft and the total flow rate Q of the Poiseuille flow are both modified by a second-order term of roughness in ε, namely, M = M₀ + ε²η and Q = Q₀ − ε²2πχ, where M₀ and Q₀ denote the torque and the flow rate, respectively, for the smooth shaft. The net effects are conveniently written as η = η₁ + η₂ and χ = χ₁ + χ₂, both comprising two components: η₁ = η₁ (b) < 0 (pure deficit) increases with increasing b and χ₁ = χ₁ (b) first increases and then decreases again with increasing b, while η₂ and χ₂ are complex functions of b, n, and α. For a given density of roughness A_c = nα, there exists an intermediate n at which the mean torque M is minimized, while the total flow rate Q is maximized. The main results are thoroughly derived with all the steps of derivation explained physically, and their relationships to the various geometrical parameters are used to establish a simplified model for predicting the shaft roughness within the range of reasonable accuracy.

中文翻译：

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缸内颠簸轴的斯托克斯流和预测轴粗糙度的模型

当微轴在粘性流体中运行时，由于腐蚀，磨损和沉积，它可能会变得粗糙。重要的是要研究这些影响并估计轴的表面粗糙度水平。在这项研究中，我们考虑了颠簸的轴，其形状由两个余弦函数的乘积建模。粗糙度ε定义为产品振幅与轴的平均半径b之比。首先，我们考虑轴的库埃特流在由旋转的光滑圆柱体包围的粘性流体中的流动。针对Stokes方程对ε进行扰动分析，直到关键点包括方位波数n的二阶粗糙度的轴向波数α以及平均半径b。此外，对轴和带罩圆筒之间的间隙中的泊瓦流进行了摄动分析，因此我们可以获得用于估计轴平均粗糙度的完整信息。此外，对在选定的b，ε和波数n，α下作用在轴上的转矩进行了数值模拟，以验证扰动结果的准确性。结果表明，平均扭矩M作用于颠簸轴的单位面积和总流量Q泊肃叶流都通过在粗糙的第二阶项改性ε，即，中号=中号₀ + ε ² η和Q = Q ₀ - ε ² 2 πχ，其中中号₀和Q ₀表示转矩和分别为光滑轴的流量。的净效应可方便地写为η = η ₁ + η ₂和χ = χ ₁ + χ ₂，两者都包括两个组件：η ₁ = η ₁（b）<0（纯赤字）增加而增加的b和χ ₁ = χ ₁（b）先增大后随再次降低b，而η ₂和χ ₂是b，n和α的复函数。对于给定的粗糙度A _c = nα，存在一个中间n，在该中间n处平均转矩M总流量Q最大化的同时最小化了 通过对所有推导步骤进行了物理解释，可以彻底得出主要结果，并将其与各种几何参数的关系用于建立简化模型，以在合理精度范围内预测轴粗糙度。