We study the streaming flow past a
rapidly rotating circular cylinder (figure 1)1. The starting point is the full continuity and
momentum equations without any approximations. We assume that the solution is a boundary layer flow near the
cylinder surface with the potential flow outside the boundary layer. The order
of magnitude of the terms in the continuity and momentum equations can be estimated inside the boundary layer. When terms
in the order of 
and higher are
dropped, where 
is the boundary layer thickness and 
is the radius of the cylinder, the equations used
by Glauert (1957)
[2] are recovered. Glauert's solution ignores the irrotational rotary component of the flow inside the
boundary layer, which is consistent with dropping terms in the
governing equations.
We propose a new solution to this
problem, in which the velocity field is decomposed into two parts. Outside the
boundary layer, the flow is irrotational and can be
decomposed into a purely rotary flow and a potential flow past a fixed
cylinder. Inside the boundary layer, the velocity is decomposed into an irrotational purely rotary flow and a boundary
layer flow. Inserting this decomposition of the velocity field inside the
boundary layer into the governing equations, we obtain a new set of equations
for the boundary layer flow, in which we do not drop the terms in the order of or higher. The pressure can no longer be
assumed to be a constant across the boundary layer, and the continuity of shear
stress at the outer edge of the boundary layer is enforced. We
solve this new set of equations using Glauert's
method, i.e. to expand the solutions as a power series of 
, where 
is the uniform
stream velocity and 
is the circulatory velocity at the
outer edge of the boundary layer. The pressure from this boundary layer
solution has two parts, an inertia part and a viscous part. The inertial part
comes from the inertia terms in the momentum equations and is
in agreement with the irrotational pressure; the viscous part
comes from the viscous stress terms in the momentum equations and may be viewed
as a viscous pressure correction, which contributes to both drag and lift. Our
boundary layer solution is in reasonable to excellent agreement with the
numerical simulation of Padrino and Joseph (2004) [13].