Two-scale nonlocal shear rate formulation of Bingham plastic fluid

This paper deals with a phenomenological model based on a nonlocal Bingham constitutive law. This nonlocal viscoplastic law accounts for possible microstructured effects of a heterogeneous fluid. Thermodynamic arguments are presented for the justification of this two-scale nonlocal shear rate law, which can be specialized either as a purely nonlocal or a gradient-type Bingham law. Plane shearing and uniaxial flow assumptions between parallel walls are basic and tractable hypotheses for a first investigation of nonlocal and gradient shear rate law. Analytical solutions are presented for the plane Poiseuille stationary flow inside a straight smooth channel of infinite width. Two higher boundary conditions at the walls are assumed for this so-called micromorphic medium, a static higher-order boundary condition and a kinematic one. A boundary layer phenomenon can be observed for such nonlocal problems. Parametric studies show the key role of the characteristic lengths of the nonlocal model on the shear flow phenomena. Consequently, the heterogeneous nature of microstructured yield-stress fluids may significantly affect velocity profiles, maximum velocity and total flow rate, which have been recently reported in literature.