When two parallel plates move normal to each other with a slow time-dependent speed, the velocity field developed in the intervening film of fluid is approximately that of plane Poiseuille flow, except that the magnitude of the velocity is dependent on time and on the coordinate parallel to the planes. This fact is intrinsic to Reynolds’ lubrication theory, and can be shown to follow from the Navier-Stokes equations when both the modified Reynolds number (
Re
M
) and an aspect ratio (
δ
) are small. The modified Reynolds number is the product of
δ
and an actual Reynolds number (
Re
), which is based on the gap between the planes and on a characteristic velocity. The occurrence of flow instability and of turbulence in the film depend on
Re
. Typical values of
Re
, which are known to be required for the linear instability of plane Poiseuille flow, are of order 6000. This condition can be achieved, even if
Re
M
is of order 1, provided that
δ
is of order 10
-4
. Such parameter values are typical of lubrication problems. The Orr-Sommerfeld equation governing flow instability is derived in this paper by use of the WKBJ technique,
δ
being the approximate small parameter to represent the small length-scale of the disturbance oscillations compared with the larger scale of the basic laminar flow. However, the coefficients in the Orr-Sommerfeld equation depend on slow space and time variables. Consequently the eigenrelation, derivable from the Orr-Sommerfeld equation and the associated boundary conditions, constitutes a nonlinear first-order partial differential equation for a phase function. This equation is solved by use of Charpit’s method for certain special forms of the time-dependent gap between the planes, followed by detailed numerical calculations. The relation between time-dependence and flow instability is delineated by the calculated results. In detail the nature of the instability can be described as follows. We consider a disturbance wave at or near a particular station, the initial distribution of amplitude being gaussian in the slow coordinate parallel to the planes. In the context of the Orr-Sommerfeld equation and its eigenrelation, the particular station implies an equivalent Reynolds number, while the initial distribution of the disturbance wave implies an equivalent wavenumber. As time increases, the disturbance wave can be considered to move in the instability diagram of equivalent wavenumber against Reynolds number, in the sense that these parameters are time- and space-dependent for the evolution of the disturbance-wave system. For our detailed calculations we use a quadratic approximation to the eigenrelation, an approximation which is quite accurate. If the initial distribution implies a point within the neutral curve, when the plates are squeezed together the equivalent wavenumber falls while the equivalent Reynolds number rises, and amplification takes place until the lower branch of the neutral curve is nearly crossed. If the plates are pulled apart (dilatation) the equivalent wavenumber rises, while the Reynolds number drops, and amplification takes place until the upper branch of the neutral curve has been just crossed. In the case of dilatation the transition from amplification to damping takes place more quickly than for the case of squeezing, in part due to the geometry of the neutral curve.
Calculation of the Orr-Sommerfeld stability equation for the plane Poiseuille flow