Page 398 - Introduction to Continuum Mechanics
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382 Flow Near an Oscillating Plate
Omitting body forces and assuming a constant pressure field, the only nontrivial Navier-Stokes
equation is
It can be easily verified that
satisfies the above equation if
From Eq. (6.16.2a), the fluid velocity at x 2 = 0 is
v = aco$(a)t+e)
Fig. 6.11
Thus, the solution Eq. (6.16.2) represents the velocity field of an infinite extent of liquid lying
in the region x^ 0 and bounded by a plate at KI — 0 which executes simple harmonic
oscillations of amplitude a and circular frequency o). It represents a transverse wave of
wavelength -gr, propagating inward from the boundary with a phase velocity -*-, but with
rapidly diminishing amplitude ( the falling off within a wavelength being in the ratio
e~ = 1/535). Thus, we see that the influence of viscosity extends only to a short distance
from the plate performing rapid oscillation of small amplitude a.