  Planet engulfment can be inferred from enhancement of refractory elements in
the photosphere of the engulfing star following accretion of rocky planetary
material. Such refractory enrichments are subject to stellar interior mixing
processes, namely thermohaline mixing induced by an inverse
mean-molecular-weight gradient between the convective envelope and radiative
core. Using MESA stellar models, we quantified the strength and duration of
engulfment signatures following planet engulfment. We found that thermohaline
mixing dominates during the first $\sim$5$-$45 Myr post-engulfment, weakening
signatures by a factor of $\sim$2 before giving way to depletion via
gravitational settling on longer timescales. Solar metallicity stars in the
0.5-1.2 $M_{\odot}$ mass range have observable signature timescales of $\sim$1
Myr$-$8 Gyr, depending on the engulfing star mass and amount of material
engulfed. Early type stars exhibit larger initial refractory enhancements but
more rapid depletion. Solar-like stars ($M$ = 0.9$-$1.1 $M_{\odot}$) maintain
observable signatures ($>$0.05 dex) over timescales of $\sim$20 Myr$-$1.7 Gyr
for nominal 10 $M_{\oplus}$ engulfment events, with longer-lived signatures
occurring for low-metallicity and/or hotter stars (1 $M_{\odot}$, $\sim$2$-$3
Gyr). Engulfment events occurring well after the zero-age main sequence produce
larger signals due to suppression of thermohaline mixing by gravitational
settling of helium (1 $M_{\odot}$, $\sim$1.5 Gyr). These results indicate that
it may be difficult to observe engulfment signatures in solar-like stars that
are several Gyr old.
