Stokes' 1st

Stokes' first problem, also erroneously called Rayleigh flow:

Continuity:

y-momentum:

x-momentum:

The x-momentum equation becomes:

where is the dynamic viscosity.

Exercise:

How would you normally find u?

A simpler way to solve is to guess that the solution is similar: after rescaling u and y, all velocity profiles look the same.

Original profiles:

Supposed shape after scaling u with V₀, and y with a characteristic boundary layer thickness that increases with time:

Mathematical form of the similarity assumption:

The proof is in the pudding; if it satisfies the P.D.E., I.C., and B.C., it is OK.

Put :

Separate into terms depending only on and terms depending only on t:

It does not make a difference what you take the constant; this merely changes the value of , not the physical solution.

Solving the O.D.E.s for and f, we solve the P.D.E. For the boundary layer thickness so

For the velocity profile hence

where erfc is the complementary error function defined as

Exercise:

Derive the expressions for and f.

Total:

You should now be able to do 7.14, 16, 17