The Unsteady Laminar Flow between Two Parallel Discs with Arbitrarily Varying Gap Width

Abstract

A theoretical analysis is presented for the unsteady laminar flow of an incompressible fluid in a narrow gap between two parallel discs of which gap-width h (t) varies arbitrarily with time. An infinite set of the non-dimensional time-dependent parameters [numerical formula], …… is introduced providing that the function h (t) is continuously differentiable, and the exact solutions of the Navier-Stokes equations and the thermal energy equation are obtained as the multifoldseries of these non-dimensional parameters. As an application, a detailed numerical study has been made of the fundamental case when the walls perform reciprocating harmonic oscillations with finite amplitudes {h (t)=h0 (1+asinωt)}. The flow characteristics are governed by the two non-dimensional parameters a and Rω=(h02ω)/υ. As compared with the sinusoidal velocity of the gap-width change, (dh)/(dt)=aωh0cosωt, the varying hydrodynamical force acting on the wall surface becomes more distorted in wave form as a increases, and becomes more advanced in phase as Rω increases. Heat is generated in the fluid between the walls through viscous friction : when the fluid is cooled by only one wall surface the fluid temperature becomes very much higher than when the cooling is done by both the two walls.