  Particles moving inside a fluid near, and interacting with, invariant
manifolds is a common phenomenon in a wide variety of applications. One
elementary question is whether we can determine once a particle has entered a
neighbourhood of an invariant manifold, when it leaves again. Here we approach
this problem mathematically by introducing balance functions, which relate the
entry and exit points of a particle by an integral variational formula. We
define, study, and compare different natural choices for balance functions and
conclude that an efficient compromise is to employ normal infinitesimal
Lyapunov exponents. We apply our results to two different model flows: a
regularized solid-body rotational flow and the asymmetric Kuhlmann--Muldoon
model developed in the context of liquid bridges. Furthermore, we employ full
numerical simulations of the Navier-Stokes equations of a two-way coupled
particle in a shear--stress-driven cavity to test balance functions for a
particle moving near an invariant wall. In conclusion, our
theoretically-developed framework seems to be applicable to models as well as
data to understand particle motion near invariant manifolds.
