  The unsteady, lineal translation of a solid spherical particle through
viscoelastic fluids described by the Johnson-Segalman and Giesekus models is
studied analytically. Solutions for the pressure and velocity fields as well as
the force on the particle are expanded as a power series in the Weissenberg
number. The momentum balance and constitutive equation are solved
asymptotically for a steadily translating particle up to second order in the
particle velocity, and rescaling of the pressure and velocity in the frequency
domain is used to relate the solutions for steady lineal translation to those
for unsteady lineal translation. The unsteady force at third order in the
particle velocity is then calculated through application of the Lorentz
reciprocal theorem, and it is shown that this weakly nonlinear contribution to
the force can be expressed as part of a Volterra series. Through a series of
examples, it is shown that a truncated representation of this Volterra series,
which can be manipulated to describe the velocity in terms of an imposed force,
is useful for analyzing specific time-dependent particle motions. Two examples
studied using this relationship are the force on a particle suddenly set into
motion and the velocity of a particle in response to a suddenly imposed steady
force. Additionally, the weakly nonlinear response of particle captured by a
harmonic trap moving lineally through the fluid is computed. This is an analog
to active microrheology experiments, and can be used to explain how weakly
nonlinear responses manifest in active microrheology experiments with spherical
probes.
