  We present an analytical model for jets in Fanaroff & Riley Class I (FRI)
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 FRI source 3C31 and show that there is a
self-consistent solution. We derive the jet power, together with the variations
of mass flux and 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 VLA observations. Our prediction
for the shape of the laminar core can be tested with higher-resolution imaging.
