Exact Solutions to the Navier-Stokes Equation

Similarity Solution (Group Theory)

Let

(5)

Equation (1) implies that

(6)

Thus,

(7)

Now introducing the similarity variables

(8)

we find

(9)

(10)

Substituting (9) and (10) in Equation (1), we find

(11)

or

(12)

Boundary and initial conditions (2)-(4) in terms of the similarity variables become

(13)

From Equation (12), it follows that

(14)

or

(15)

where the first boundary condition in (13) is used. The second boundary condition implies that

(16)

Equation (15) then becomes

(17)

or

(18)

The time variations of the velocity profile as predicted by Equation (18) are shown in Figure 2.

Figure 2. Time variations of velocity profile.