On The Lubrication of a Rotating Shaft with Incompressible Micropolar Fluid

Abstract

In this work, we investigate the lubrication process of a slipper bearing. The slipper bearing consists of two coaxial cylinders in relative motion, where an incompressible micropolar fluid (lubricant) is injected in a thin gap between them. We compute the asymptotic approximation of the solution to the governing micropolar system as a power series in terms of the small parameter ε representing the thickness of the shaft. The proposed approximation is given in the explicit form, allowing us to clearly observe the effects of the micropolar nature of the fluid.

Justification of the Higher Order Effective Model Describing the Lubrication of a Rotating Shaft with Micropolar Fluid

“…We plug the asymptotic expansion (6) into the micropolar system of Equations (4) and (5) and collect terms by the same powers of ε, leading to a recursive sequence of problems. The computations presented in Section 3.2, which are related to the regular part of the expansion, were presented by Marušić-Paloka, Pažanin, and Radulović in [30], and we recall them for the sake of the readers' convenience.…”

“…where u 0 , w 0 , u 1 , and w 1 are given by (10), (16), (19), and (26), p 0 and p 1 are the solutions of the Reynolds problems (12)- (14) and (21)- (24), while B 0 , b 0 , W 0 , B 1 , b 1 , W 1 are the solutions of the problems (27), (28), (29), (30), and H 0 , h 0 , Y 0 , H 1 , h 1 , Y 1 are solutions of the analogous problems posed on the opposite side z = l. Although we have corrected the boundary layer effects by constructing the appropriate boundary layer correctors in Section 3.3, the residual in the divergence equation is not small enough to obtain satisfactory error estimates. In order to correct this, we construct the divergence corrector in the forthcoming section.…”

“…The main advantage of considering the micropolar fluid is that it takes into account the microstructure of the fluid. Although a lower-dimensional model for the considered problem was recently formally derived in [30] by the authors of this paper, the justification of its usage is still missing in the literature. The justification of the derived model represents the main novelty of this work.…”