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.
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