  Fine particle suspensions (such as cornstarch mixed with water) exhibit
dramatic changes in viscosity when sheared, producing fascinating behaviors
that captivate children and rheologists alike. Recent examination of these
mixtures in simple flow geometries suggests inter-granular repulsion is central
to this effect --- for mixtures at rest or shearing slowly, repulsion prevents
frictional contacts from forming between particles, whereas, when sheared more
forcefully, granular stresses overcome the repulsion allowing particles to
interact frictionally and form microscopic structures that resist flow.
Previous constitutive studies of these mixtures have focused on particular
cases, typically limited to two-dimensional, steady, simple shearing flows. In
this work, we introduce a predictive and general, three-dimensional continuum
model for this material, using mixture theory to couple the fluid and particle
phases. Playing a central role in the model, we introduce a micro-structural
state variable, whose evolution is deduced from small-scale physical arguments
and checked with existing data. Our space- and time-dependent model is
implemented numerically in a variety of unsteady, non-uniform flow
configurations where it is shown to accurately capture a variety of key
behaviors: (i) the continuous shear thickening (CST) and discontinuous shear
thickening (DST) behavior observed in steady flows, (ii) the time-dependent
propagation of `shear jamming fronts', (iii) the time-dependent propagation of
`impact activated jamming fronts', and (iv) the non-Newtonian, `running on
oobleck' effect wherein fast locomotors stay afloat while slow ones sink.
