  We present a deep, low-frequency radio continuum study of the nearby
Fanaroff--Riley class I (FR I) radio galaxy 3C 31 using a combination of LOw
Frequency ARray (LOFAR; 30--85 and 115--178 MHz), Very Large Array (VLA;
290--420 MHz), Westerbork Synthesis Radio Telescope (WSRT; 609 MHz) and Giant
Metre Radio Telescope (GMRT; 615 MHz) observations. Our new LOFAR 145-MHz map
shows that 3C 31 has a largest physical size of $1.1$ Mpc in projection, which
means 3C 31 now falls in the class of giant radio galaxies. We model the radio
continuum intensities with advective cosmic-ray transport, evolving the
cosmic-ray electron population and magnetic field strength in the tails as
functions of distance to the nucleus. We find that if there is no in-situ
particle acceleration in the tails, then decelerating flows are required that
depend on radius $r$ as $v\propto r^{\beta}$ ($\beta\approx -1$). This then
compensates for the strong adiabatic losses due to the lateral expansion of the
tails. We are able to find self-consistent solutions in agreement with the
entrainment model of Croston & Hardcastle, where the magnetic field provides
$\approx$$1/3$ of the pressure needed for equilibrium with the surrounding
intra-cluster medium (ICM). We obtain an advective time-scale of $\approx$$190$
Myr, which, if equated to the source age, would require an average expansion
Mach number ${\cal M} \approx 5$ over the source lifetime. Dynamical arguments
suggest that instead, either the outer tail material does not represent the
oldest jet plasma or else the particle ages are underestimated due to the
effects of particle acceleration on large scales.
