Laminar flow of compressible Newtonian fluids in capillaries is analysed. A perturbation solution is obtained to the vorticity–stream function form of the hydrodynamics equations for weakly compressible flow, having a relatively simple form. In contrast with previous analyses of this flow, we find both a non-zero radial velocity and non-zero radial pressure gradient. Expressions for pressure drop and friction factor are also found that show significant deviations from the incompressible case (Hagen–Poiseuille equation) that arise from both fluid inertia and bulk viscosity.