F Start-Up of Simple Shear Flow

The problem is identical to that solved in the preceding section, (3-119) and (3-120b), with the exception that the boundary condition as v - oc is replaced with the boundary condition 

However, in this case, it is straightforward to nondimensionalize. The characteristic velocity scale is uc = U, and an appropriate characteristic length scale is lc = d. Because the velocity field is established by means of diffusion of momentum, the characteristic time scale is tc = d2 j v. Using these characteristic quantities, we find that the problem in dimensionless form is 

Because characteristic velocity, length, and time scales all exist, we do not expect a solution of self-similar form, but instead seek a solution by using separation of variables. 

At steady state, we have already noted that the solution of (3 144) and (3 145) is just 

Substituting into (3 147) and solving the resulting ODEs for H and F, we find that the general solution that vanishes as 7 oo is 

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