Scaling Arguments for Boundary Layers

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations  

The reference scale for the axial coordinate is denoted by zs and for the radial coordinate is given by rs. However, both length scales are arbitrary at this point in the derivation. The scale for the axial velocity (in the principal flow direction) u is mo, a uniform inlet velocity. The density and viscosity can also be scaled by their values in the inlet flow stream, po and p,o, respectively. The scale for the v velocity, vs, is an unknown for now. The nondimensional variables can then be written as 

Using these nondimensional variables, the continuity equation transforms to 

For the boundary-layer equations, where two-dimensional flow is retained, the continuity equation must retain both terms as order-one terms. Otherwise, a purely onedimensional flow would result. Certainly there are situations where one-dimensional flow 

Using this scale, we can write the continuity equation in the nondimensional form as 

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