  Although the critical Reynolds number for linear instability of the laminar
flow in a straight pipe is infinite, we show that it is finite for a divergent
pipe, and approaches infinity as the inverse of the divergence angle. The
velocity profile at the threshold of inviscid stability is obtained. A
non-parallel analysis yields linear instability at surprisingly low Reynolds
numbers, of about 150 for a divergence of 3 degrees, which would suggest a role
for such instabilities in the transition to turbulence. A multigrid Poisson
equation solver is employed for the basic flow, and an extended eigenvalue
method for the partial differential equations describing the stability.
