**3. Stokes' second problem**

Fundamental studies of fully-developed and periodic pipe and duct flows with pressure gradients that vary sinusoidally have been done (Sexl, 1930). From such studies, we know that, when an incompressible and viscous fluid is forced to move under a pulsating pressure difference in a pipe or a duct, some characteristic features are always observed. Some of these features are similar to those that are observed to occur in the boundary layer adjacent to a body that is performing reciprocating harmonic oscillations. These features are related to the results of a classic problem solved by Stokes, known as Stokes' second problem, which gives details of the behavior of the boundary layer in a viscous fluid of kinematic vis‐ cosity, *v* , that is bounded by an infinite plane surface that moves back and forth in its own plane with a simple harmonic oscillation that has a circular frequency, *ω* .

Stokes solution shows that, for this type of flow, 1) transverse waves propagate away from the oscillating surface and into the viscous fluid; 2) the direction of the velocity of these waves is perpendicular to the direction of propagation; 3) the oscillating fluid layer so gen‐ erated has a phase lag, *φ* , with respect to the motion of the wall; and 4) that phase lag, which varies with y, the distance from the wall, is given by, *<sup>φ</sup>* <sup>=</sup> *<sup>y</sup> <sup>δ</sup>* , where *δ* represents a length scale introduced by Stokes; that length, called the depth of penetration of the wave into the fluid, is given by *<sup>δ</sup>* =( <sup>2</sup>*<sup>v</sup> <sup>ω</sup>* )1/2 . The thickness of the boundary layer is proportional to the penetration depth, *δ* . The value of the constant of proportionality varies with the point that one designates to be the edge of the boundary layer. Thus, For example, if one defines the edge of the boundary layer to be the point in the flow where the speed inside the boun‐ dary layer become equal to 99% of the speed of flow outside the boundary layer, the con‐ stant of proportionality is 4.6. Then, the thickness of the boundary layer at that point is equal to 4.6 *δ*.
