Exact Solutions to the Navier-Stokes Equation

Oscillating Plate

Consider that case of a viscous fluid near an oscillating wall as shown in Figure 3. The unsteady Navier-Stokes reduces to

(25)

Figure 2. Schematics of flow near an oscillating wall.

The boundary conditions are:

,

(26)

.

(27)

Let

.

(28)

Then

,

(29)

,

(30)

.

(31)

Substituting (29)-(31) into Equation (25) it follows that

,

(32)

or

,

(33)

,

(34)

,

(35)

Thus, the velocity profile is given as

.

(36)