  We investigated the linear stability of pipe flow with anisotropic slip
length at the wall by considering streamwise and azimuthal slip separately as
the limiting cases. Our numerical analysis shows that streamwise slip renders
the flow less stable but does not cause instability. The exponential decay rate
of the least stable mode appears to be $\propto Re^{-1}$ when the Reynolds
number is sufficiently large. Azimuthal slip can cause linear instability if
the slip length is sufficiently large. The critical Reynolds number can be
reduced to a few hundred given large slip lengths. Besides numerical
calculations, we present a mathematical proof of the linear stability of the
flow to three-dimensional yet streamwise-independent disturbances for arbitrary
Reynolds number and slip length, as an alternative to the usual energy
analysis. Meanwhile we derived analytical solutions to the eigenvalue and
eigenvector, and explained the structure of the spectrum and the dependence of
the leading eigenvalue on the slip length. The scaling of the exponential decay
rate of streamwise independent modes is shown to be rigorously $\propto
Re^{-1}$. Our non-modal analysis shows that overall streamwise slip reduces the
non-modal growth and azimuthal slip has the opposite effect. Nevertheless, both
slip cases still give the $Re^2$-scaling of the maximum non-modal growth and
the most amplified disturbances are still streamwise rolls, which are
qualitatively the same as in the no-slip case.
