  The fate of small particles in turbulent flows depends strongly on the
surrounding fluid's velocity gradient properties such as rotation and
strain-rates. For non-inertial (fluid) particles, the Restricted Euler model
provides a simple, low-dimensional dynamical system representation of
Lagrangian evolution of velocity gradients in fluid turbulence, at least for
short times. Here we derive a new restricted Euler dynamical system for the
velocity gradient evolution of inertial particles such as solid particles in a
gas or droplets and bubbles in turbulent liquid flows. The model is derived in
the limit of small (sub Kolmogorov scale) particles and low Stokes number. The
system exhibits interesting fixed points, stability and invariant properties.
Comparisons with data from Direct Numerical Simulations show that the model
predicts realistic trends such as the tendency of increased straining over
rotation along heavy particle trajectories and, for light particles such as
bubbles, the tendency of reduced self-stretching of strain-rate.
