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.

Highlights

  • This study concerns the fluid motion induced by a rigid circular disk of radius a vibrating along its axis in the vicinity of a stationary parallel surface

  • Our asymptotic analysis: (i) reveals that the time-averaged pressure variations across the edge region are comparable in magnitude and opposite in sign to those found along the wall-bounded gas layer; (ii) leads to simplified expressions that expedite the evaluation of the steady squeeze-film force over a wide range of conditions of practical interest, facilitating the operation and control of high-frequency systems; (iii) unveils a boundary on the α2 − Λ parametric plane across which the force switches from repulsion to attraction; (iv) demonstrates numerically that the force is only weakly dependent on the specific geometrical configuration and (v) compares favourably with recently published computational results

  • The method of matched asymptotic expansions was used to relate the Navier–Stokes solutions describing the flow in two distinct regions – a slender region between the parallel surfaces featuring radial flow driven by the disk oscillations and an asymptotically smaller non-slender near-edge region where the motion is driven by said radial flow

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Summary

Introduction

This study concerns the fluid motion induced by a rigid circular disk (or piston) of radius a vibrating along its axis in the vicinity of a stationary parallel surface. Our asymptotic analysis: (i) reveals that the time-averaged pressure variations across the edge region are comparable in magnitude and opposite in sign to those found along the wall-bounded gas layer; (ii) leads to simplified expressions that expedite the evaluation of the steady squeeze-film force over a wide range of conditions of practical interest, facilitating the operation and control of high-frequency systems; (iii) unveils a boundary on the α2 − Λ parametric plane across which the force switches from repulsion to attraction; (iv) demonstrates numerically that the force is only weakly dependent on the specific geometrical configuration and (v) compares favourably with recently published computational results (see Andrade et al 2020)

Distinct regions and characteristic scales
Leading-order solution in the gaseous film
Time-averaged pressure distribution
An expression for the squeeze-film force
Limiting cases of interest
Problem formulation
Selected numerical results
Pressure drop across the edge region
Creeping flow for α2 1
Inviscid flow for α2 1
Time-averaged squeeze-film force
Findings
Conclusions
Full Text
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