Granular Matter 1, 163–181 c  Springer-Verlag 1999 A constitutive model of multiphase mixtures and its application in shearing ﬂows of saturated solid-ﬂuid mixtures Yongqi Wang, Kolumban Hutter Abstract 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 [11] and Passman et al. [23] for each constituent. Based on the M¨ uller-Liu form of the second law of thermodynam- ics 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 ordi- nary diﬀerential equations describing a steady ﬂow of the solid-ﬂuid mixture between two horizontal plates. The re- sulting boundary value problem is solved numerically and results are presented for various values of parameters and boundary conditions. It is shown that simple shearing gen- erally 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 be- tween 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 par- ticles to accumulate in regions of low shear rate. Key words Solid-ﬂuid mixture, Constitutive equations, Shearing ﬂow Received: 30 June 1998 Y. Wang, K. Hutter Institute of Mechanics, Darmstadt University of Technology Hochschulstr. 1, D-64289 Darmstadt, Germany e-mail: wang@mechanik.tu-darmstadt.de, e-mail: hutter@mechanik.tu-darmstadt.de Correspondence to : K. Hutter The authors want to thank for ﬁnancial support by the Deutsche Forschungagemeinschaft within the Special Collab- orative Program (SFB) 298. Prof. Dr. K. Wilmanski’s review of the manscript is appreciated. 1 Introduction The mechanics of multiphase mixtures is of fundamen- tal importance in several ﬁelds of engineering, i.e., debris ﬂows, soil mechanics, ground water engineering, sediment transport as well as many other ﬁelds in mechanical and chemical engineering. In this work our major interest is in granular-ﬂuid mixtures. Such a material is a collection of a large number of discrete solid particles with interstices ﬁlled with a ﬂuid or a gas. In most ﬂows involving gran- ular materials, the interstitial ﬂuid plays an insigniﬁcant role in the transportation of momentum, and thus ﬂows of such materials can be considered dispersed single phase rather than multiphase ﬂows. A detailed review of ﬂows of single granular materials has been presented by Hutter & Rajagopal [14]. It is widely known today that granular media exhibit microstructural eﬀects on their macroscale, which is ac- counted for, in general, by adding an additional dynam- ical equation for the solid volume fraction ν s . Diﬀerent authors do not unanimously agree upon the form of this equation. Svendsen & Hutter [26] treated the solid-volume fraction as an internal variable and write an evolution equation balancing its time rate of change with its pro- duction. Wilmanski [32] on the other hand, using statis- tical arguments on the microscale demonstrated that the Svendsen-Hutter equation needed to be complemented by a ﬂux term, thus arriving at a complete balance law. On the other hand, almost 25 years ago, Goodman & Cowin [11] were, based on the theory of structured media, intro- ducing a balance law of equilibrated forces in which second time derivatives of ν s , i.e., ¨ ν s were balanced with a ﬂux, a production and supply term. In some occasions or when the mass of the intersti- tial ﬂuid is comparable to that of the solids the interac- tions between the ﬂuid and solid phases are signiﬁcant, one should study these ﬂow problems by employing the theory of multiphase ﬂows. Study of multiphase ﬂow has attracted considerable attention in the past thirty years. Some detailed reviews of solid-ﬂuid mixtures have been presented by Hutter et al. [15] and Takahashi [28]. The large number of articles published on ﬂuid-solid ﬂows typ- ically employ one of two theories, (i) averaging or (ii) mixture theory. In the averaging approach, equations of motion, valid for a single constituent, are modiﬁed to account for the presence of the other components and