Viscoacoustic squeeze-film force on a rigid disk undergoing small axial oscillations

Abstract

This paper investigates the air flow induced by a rigid circular disk or piston vibrating harmonically along its axis of symmetry in the immediate vicinity of a parallel surface. Previous attempts to characterize these so-called ‘squeeze-film’ systems largely relied on simplifications afforded by neglecting either fluid acceleration or viscous forces inside the thin enclosed gas layer. The present viscoacoustic analysis employs the asymptotic limit of small vibration amplitudes to investigate the flow by systematic reduction of the Navier–Stokes equations in two distinct flow regions, namely, the inner gaseous film where streamlines are nearly parallel to the confining walls and the near-edge region of non-slender flow that features gas exchange with the surrounding stagnant atmosphere. The flow in the gaseous film depends on the relevant Stokes number, defined as the ratio of the characteristic viscous time across the film to the characteristic oscillation time, and on a compressibility parameter, defined as the square of the ratio of the acoustic time for radial pressure equilibration to the oscillation time. A Strouhal number based on the local residence time emerges as an additional governing parameter for the near-edge region, which is incompressible at leading order. The method of matched asymptotic expansions is used to describe the solution in both regions, across which the time-averaged pressure exhibits comparable variations that give opposing contributions to the resulting time-averaged force experienced by the disk or piston. A diagram structured with the Stokes number and compressibility parameter as coordinates reveals that this steady squeeze-film force, typically repulsive for small values of the Stokes number, alternates to attraction across a critical separation contour in the parametric domain that exists for all Strouhal numbers. This analysis provides, for the first time, a unifying viscoacoustic theory of axisymmetric squeeze films, which yields a reduced parametric description for the time-averaged repulsion/attraction force that is potentially useful in applications including non-contact fluid bearings and robot locomotion.

Benefits of controlled inclination for contactless transport by squeeze-film levitation

“…For typical SFL systems (Matsuo et al., 2000; Yoshimoto et al., 2013; Zhao, 2010), the squeeze film is slender, , whereby the tilt angle is small, ; the characteristic wavelength of the flexural oscillations is comparable to , whereby any lateral displacement of points on the deforming plate surface is negligible; and the transport speed is negligibly small (Ide et al., 2007) compared with the characteristic speed of steady gaseous streaming in the film: (Ramanarayanan, Coenen, & Sánchez, 2022). …”

“…the transport speed is negligibly small (Ide et al., 2007) compared with the characteristic speed of steady gaseous streaming in the film: (Ramanarayanan, Coenen, & Sánchez, 2022).…”

“…Assumptions (i) and (ii) allow application of the slender-flow approximation (Ramanarayanan et al., 2022) to model gas flow in the film. Due to (iii) and (iv), the only non-trivial boundary condition to be imposed for the flow velocity stems from the transverse motion of the plate surface, which can now be expressed with use of the simplified equation where denotes a coordinate system whose origin travels along the wall (see figure 3), with coincident to the wall and denoting the distance to the plate centre.…”

“…As shown by Melikhov, Chivilikhin, Amosov, and Jeanson (2016) and Ramanarayanan et al. (2022), the flow dynamics in the thin air layer of a squeeze-film system is characterized by three principal time scales – that of the driving oscillations , viscous diffusion across the film and acoustic pressure equilibration along the film – which enter in the theoretical description through two non-dimensional parameters, the relevant Stokes number (or, equivalently, the associated Womersley number ) and a compressibility number .…”

Incompressible Squeeze-Film Levitation

“…Melikhov et al [19] developed a viscoacoustic model and identified the different operating regimes for air squeeze films as a function of the levitation height, confirming a purely viscous regime for typical squeeze film levitation systems. Ramanarayanan et al [20] proposed another unified theory which described critical parametric conditions that causes levitation forces to switch to adhesion forces in air squeeze film systems. Remarkably, in the incompressible limit, their formulation predicted only weak adhesive squeeze film forces, adding even more uncertainty around the behaviour of incompressible squeeze films.…”