  We perform $3$D numerical simulations to investigate the sedimentation of a
single sphere in the absence and presence of a simple cross shear flow in a
yield stress fluid with weak inertia. In our simulations, the settling flow is
considered to be the primary flow, whereas the linear cross shear flow is a
secondary flow with amplitude $10\%$ of the primary flow. To study the effects
of elasticity and plasticity of the carrying fluid on the sphere drag as well
as the flow dynamics, the fluid is modeled using the elastovisco-plastic (EVP)
constitutive laws proposed by \cite{saramito2009new}. The extra non-Newtonian
stress tensor is fully coupled with the flow equation and the solid particle is
represented by an immersed boundary (IB) method. Our results show that the
fore-aft asymmetry in the velocity is less pronounced and the negative wake
disappears when a linear cross shear flow is applied. We find that the drag on
a sphere settling in a sheared yield stress fluid is reduced significantly as
compared to an otherwise quiescent fluid. More importantly, the sphere drag in
the presence of a secondary cross shear flow cannot be derived from the pure
sedimentation drag law owing to the non-linear coupling between the simple
shear flow and the uniform flow. Finally, we show that the drag on the sphere
settling in a sheared yield-stress fluid is reduced at higher material
elasticity mainly due to the form and viscous drag reduction.
