A relativistic mixing-layer model for jets in low-luminosity radio galaxies

We present an analytical model for jets in Fanaroff & Riley Class I (FR I) radio galaxies, in which an initially laminar, relativistic flow is surrounded by a shear layer. We apply the appropriate conservation laws to constrain the jet parameters, starting the model where the radio emission is observed to brighten abruptly. We assume that the laminar flow fills the jet there and that pressure balance with the surroundings is maintained from that point outwards. Entrainment continuously injects new material into the jet and forms a shear layer, which contains material from both the environment and the laminar core. The shear layer expands rapidly with distance until finally the core disappears, and all of the material is mixed into the shear layer. Beyond this point, the shear layer expands in a cone and decelerates smoothly. We apply our model to the well-observed FR I source 3C 31 and show that there is a self-consistent solution. We derive the jet power, together with the variations of mass flux and entrainment rate with distance from the nucleus. The predicted variation of bulk velocity with distance in the outer parts of the jets is in good agreement with model fits to Very Large Array observations. Our prediction for the shape of the laminar core can be tested with higher-resolution imaging.