A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures
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A continuum theory of a multiphase mixture is formulated. In the basic balance laws we introduce an additional balance of equilibrated forces to describe the microstructural response according to Goodman & Cowin  and Passman et al.  for each constituent. Based on the Müler-Liu form of the second law of thermodynamics a set of constitutive equations for a viscous solid-fluid mixture with microstructure is derived. These relatively general equations are then reduced to a system of ordinary differential equations describing a steady flow of the solid-fluid mixture between two horizontal plates. The resulting boundary value problem is solved numerically and results are presented for various values of parameters and boundary conditions. It is shown that simple shearing generally does not occur. Typically, for the solid phase, in the vicinity of a boundary, if the solid-volume fraction is low, a layer of high shear rate occurs, whose thickness is nearly between 5 and 15 grain diameters, while if the solid-volume fraction is high, an interlock phenomenon occurs. The fluid velocity depends largely on the drag force between the constituents. If the drag coefficient is sufficiently large, the fluid flow is nearly the same as that of the solid, while for a small drag coefficient, the fluid shearing flow largely decouples from that of the solid in the entire flow region. Apart from this, there is a tendency for solid particles to accumulate in regions of low shear rate.
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