Stokes’s flow of a bumpy shaft inside a cylinder and a model for predicting the roughness of the shaft

Abstract

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 Ac = 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.