  Observations point to old white dwarfs (WDs) accreting metals at a relatively
constant rate over 8~Gyrs. Exo-Oort clouds around WDs have been proposed as
potential reservoirs of materials, with galactic tide as a mechanism to deliver
distant comets to the WD's Roche limit. In this work, we characterise the
dynamics of comets around a WD with a companion having semi-major axes on the
orders of 10 - 100 AU. We develop simulation techniques capable of integrating
a large number ($10^8$) of objects over a 1 Gyr timescale. Our simulations
include galactic tide and are capable of resolving close-interactions with a
massive companion. Through simulations, we study the accretion rate of exo-Oort
cloud comets into a WD's Roche limit. We also characterise the dynamics of
precession and scattering induced on a comet by a massive companion. We find
that (i) WD pollution by an exo-Oort cloud can be sustained over a Gyr
timescale, (ii) an exo-Oort cloud with structure like our own Solar System's is
capable of delivering materials into an isolated WD with pollution rate $\sim
10^8 \mathrm{~g~s^{-1}}$, (iii) adding a planetary-mass companion reduces the
pollution rate to $\sim 10^7 \mathrm{~g~s^{-1}}$, and (iv) if the companion is
stellar-mass, with $M_p \gtrsim 0.1 M_\odot$, the pollution rate reduces to
$\sim 3 \times 10^5 \mathrm{~g~s^{-1}}$ due to a combination of precession
induced on a comet by the companion, a strong scattering barrier, and
low-likelihood of direct collisions of comets with the companion.
