Flow of a fluid—solid mixture between flat plates

Chemzeal E”glnewmg science, Vol. 46, No. 7, pp. 1713-1723, 1991. cam9 *509,91 $3.00 + 0.00 Printed in Great Bniain. 0 1991 Pergamon Press plc

FLOW OF A FLUID-SOLID MIXTURE BETWEEN FLAT PLATES

G. JOHNSON+ Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A.

M. MASSOUDI U.S. Department of Energy, Pittsburgh Energy Technology Center, Pittsburgh, PA 15236X940, U.S.A.

and

K. R. RAJAGOPAL’ Department of Mechanical Engineermg, University of Pittsburgh, Pittsburgh, PA 15261,_U.S.A.

(First received 23 May 1990; accepted in revised form 15 August 1990)

Abstract-A mathematical description for a Aowing mixture of solid particulates and a fluid is developed within the context of mixture theory. Specifically, the equations governing the flow of a two-component mixture of a Newtonian fluid and a granular solid are derived. These relatively general equations are then reduced to a system of coupled ordinary differential equations describing a steady flow of the mixture between flat plates. The resulting boundary value problem is solved numerically and results are presented for cases in which drag and lift interactions are important.

INTRODUCTION

Multiphase flows have become the subject of con- siderable attention because of their increasing import- ance in many industrial applications. Flowing mixtures consisting of solid particles entrained in a fluid are relevant to a variety of applications such as pneumatic and hydraulic transport of solid particles and fluidized beds. The large number of articles pub- lished concerning fluid-solid flows typically employ one of two theories, averaging or mixture theory (theory of interacting continua). In the averaging approach [cf. Anderson and Jackson (1967), Drew and Segel (1971), and Drew (1983)] point-wise equations of motion, valid for a single fluid or a single particle, are modified to account for the presence of the other components and the interactions between components. These equations are then averaged over time or some suitable volume, which is large compared with a characteristic dimension (for example, particle spacing or the diameter of solid particles) but small compared to the dimensions of the whole system. From the mathematical manipulation of the averaged quantities, a number of terms arise. These terms are usually interpreted as some form of interaction between the constituents. Constitutive relations to represent these interactive forces, as well as the stress tensors for each constituent, are required to complete the description.

The second method of modeling multicomponent systems is mixture theory. This theory, which traces

+ Supported by an appointment to the U.S. Department of Energy Fossil Energy Laboratory Graduate Participation Program administered by Oak Ridge Associated Univer- sities.

XAuthor to whom correspondence should be addressed.

its origins to the work of Fick (1855), was first presented within the framework of continuum mechanics by Truesdell (1957). The equations and principles of the mechanics of a single continuum are generalized to include any number of superimposed continua. The fundamental assumption of the theory is that, at any instant of time, every point in space is occupied by one particle from each constituent, in a homogenized sense. The historical development and details of mixture theory are given in the review articles by Atkin and Craine (1976), Bedford and Drumheller (1983), Bowen (1976), and several appendices in the recent edition of Rational Thermodynamics (Truesdell, 1984). El-Kaissy (1975), Homsy et al. (19X0), Ahmadi (1980, 1982), Passman and Nunziato (1981), Passman et al.

(1986), and Massoudi (1986,1988) have used such an approach for modeling fluid-solid systems. Craine (1971), Beevers and Craine (1982), and Passman and Drew (1990) look at the flow of two Newtonian fluids within the context of mixture theory. Like averaging, mixture theory also requires constitutive relations for the stress tensor of each component of the mixture and for momentum exchange between the components.

This paper is concerned with applying mixture theory to a purely mechanical system. Many previous models of solid particles in a fluid, whether using averaging or mixture theory, have relied upon an assumption that the solid particles behave as a linearly viscous fluid (Anderson and Jackson, 1967; El-Kaissy, 1975; Murray, 1965) with a viscosity 1-1, and a pressure field ps. The meaning of pS in this context is unclear and it leads to an indeterminacy in the governing equations (Massoudi, 1986). This indeterminacy is overcome by assuming a relationship between the solid pressure p, and the fluid pressure p, in

1713

1714 G. JOHNSON et al.

order to reduce the number of unknowns. A typical assumption is that ps = pJ. This assumption may be justified if the mixture is composed of materials like water and steam, but is inappropriate when one component is a granular solid. Some of the two-fluid models are also inconsistent in that they fail to reduce to the appropriate single-constituent model in the two extreme limits.

Recently, Massoudi (1986) advocated modeling the stress in the solid constituent of the mixture by a constitutive expression appropriate for flowing granular solids. The model described herein incorporates constitutive equations for a linearly viscous fluid and a flowing granular solid. The model is formulated in a manner such that the mathematical equations reduce to those describing a linearly viscous fluid when the solid volume fraction goes to zero, and to those describing a flowing granular solid when the fluid volume fraction goes to zero. Note that physically the volume fraction of the solid particles can never be unity.

Passman et al. (1986) used mixture theory to study problems similar to the one analyzed here. Our approach differs significantly from theirs in that our constitutive expression for the solid stress tensor and the interaction forces are different. Moreover, we do not need to introduce concepts like that of equilib- rated stresses which introduce an additional balance equation. We find that the usual balance laws of continuum mechanics are sufficient to study the problem under consideration.

The purpose of this work is to study the effects of interactions and granular material properties on the fluid velocity, solid velocity, and volume fraction profiles determined by solving specific boundary value problems. First, we review briefly the basic principles of mixture theory and discuss constitutive equations for the mixture components and for the interactions between components. We then derive and noudimerr- sionalize the general equations governing a flowing mixture of a linearly viscous fluid and a granular solid. The ordinary differential equations describing steady flow of the mixture between plates are then presented along with appropriate boundary conditions. Finally, numerically calculated velocity and volume fraction profiles are presented and discussed.

The numerical results are interpreted in terms of five dimensionless parameters depending upon the granular material properties, the fluid density and viscosity, the drag coefficient, and the lift coefficient. The boundary value problem is solved using multiple- shooting, finite-difference, and collocation methods; we find the collocation method to be most efficient for a large range of the parameters. We see that the distribution of the solid particles depends strongly upon the relative densities of the components. When the density of the solid phase is increased we see the expected “settling” of the solids toward the lower plate. The influence of the volume fraction on the velocity profiles is to cause a relative decrease in fluid velocity in areas of higher solids concentration and an

increase in areas of lower solids concentration. The effect on the solid is, appropriately, just the opposite; the solid moves faster in areas of higher concentration and slower when the concentration is lower. Varying the magnitude of the drag also yields consistent results in that fluid velocities decrease and solid velocities increase with increasing drag coefficient. Lift effects represent the influence of the velocity profiles on the distribution of solid in the flow. The change in the distribution of the solid due to lift effects results in an increase in fluid velocity near the plates and a decrease in fluid velocity where solid concentration has increased (in the center of the flow).

PRELIMINARIES

A brief review of the notation and basic equations of mixture theory is presented in this section. More detailed information, including an account of the historical development, is available in the review articles by Atkin and Craine (1976) and Bowen (1976).

The mixture of fluid and solid is considered to be a purely mechanical system. That is, thermal effects and chemical reactions are ignored. The fluid in the mixture will be represented by S, and the granular solid by S,. At each instant of time, t, it is assumed that each point in space is occupied by particles belonging to both S, and S,. Let X, and X, denote the positions of particles of S, and S2 in the reference configuration. The motion of the constituents is represented by the mappings

x1 = a,(X,,t), and x2 = xZ(X2,t) (1)

where the subscripts 1 and 2 refer to the fluid and granular solid, respectively. These motions are assumed to be one-to-one, continuous and invertible. The kinematical quantities associated with these motions are

D,Sl DZl2 v1 =Dt> vz =y

DIVI a, =Dt,

D,v, a2 =x (3)

L 22 1 ax,’

L, =av, 3X,’

D, = ;(L, + L:), D, = ;(L, + L:) (5)

w, = $L, - LT), w, = $L, - L;) (6)

where v denotes velocity, a acceleration, L is the velocity gradient, D the stretching tensor, and W the

D, spin tensor, Dt denotes differentiation with respect to

D, t, holding X, fixed, and - denotes the same opera- Dt

tion holding X, fixed. Also, p1 and p2 are the bulk densities of the mixture

components given by

PI = #JPl, P2 = VPs (7)

Flow of a fluid-solid mixture between flat plates 1715

where pI is the density of the pure fluid, ps is the density of the solid grains, v is the volume fraction of the solid component, and 4 is the volume fraction of the fluid. For a saturated mixture 4 = 1 - Y. The mixture density p,,, is given by

PIP! = Pl + P2 (8)

and the mean velocity v of the mixture is defined by

Pin” = Pl”, + PzV2. (9)

Conservation of mass Assuming no interconversion of mass between the

two constituents, the equations of conservation of mass for the fluid and solid are

and

801 at + div(p, v,) = 0 (IO)

8PZ at + div(p,v,) = 0. (11)

Conservation of linear momentum Let T, and T, denote the partial stress tensors of

the fluid S, and solid S,, respectively. Then the balance of linear momentum for the fluid and solid are given by

p,T$ = div T, + p, b, + f, (12)

D2v2

” Dt ~ = div T, + pZ b, - f, (13)

where b represents the body force, and f, represents the mechanical interaction (local exchange of momentum) between the components.

Conservation of moment of momentum This principle implies that

T, +T,=T;+T;. (14)

The partial stresses need not be symmetric, however.

CONSTlTUTIVE EQUATIONS

We shall assume that the fluid and solid phases are dense enough to be modeled as homogeneous continuous media so that we may exploit the theory of interacting continua. Based on our knowledge of modeling in the theory of granular materials and a linearly viscous fluid, it would be natural to assume all the constitutive functions depend on [cf. Rajagopal et al. (1990)]

~1, pz. VP,, VP,, VVP,, VVP,, vi - vz. D,, D,.

(15)

Then, using methods that are now standard in continuum mechanics [cf. Shi et al. (1981). and Atkin and Craine (1976)], we can obtain restrictions and forms for such constitutive expressions. Here, we discuss an alternative approach which is to postulate the consti-

tutive expressions by simply generalizing the structure of the constitutive relations from a single-constituent theory. In general, the constitutive expressions for T, and T, depend on the kinematical quantities associated with both the constituents. However, we assume that T, and T, depend only on the kinematical quantities associated with the solid and fluid, respectively. This assumption is sometimes called “the principle of phase separation” [cf. Drew (1976)] and was first put forward by Adkins (1963a, b).

In the majority of fluid-solid mixtures, the fluid is either a gas or water. Therefore, it is appropriate to assume that the fluid behaves as a linearly viscous fluid, whose constitutive equation is

T, = C - P(P, I+ A,fo, 1 tr D,lI + b,-b, PI (16) where p is the fluid pressure, I, and pL, are the viscosities, D, is the stretching tensor for the fluid defined in eq. (5), and I is the identity tensor. If the fluid is incompressible, then p is one of the unknown quantities in the problem that would have to be calculated. If the fluid is compressible, an equation of state is needed. In general, p, I, and pL, ate functions

of P,. There are basically two different ways of deriving a

constitutive relation for the stress tensor of granular materials-the continuum approach and the statist- ical approach. We use the continuum approach in our analysis. Goodman and Cowin (197 1, 1972) develop a continuum theory for the stress tensor of a granular media. Cowin (1974) shows that by including effects due to the gradient of the volume fraction, the Mohr-Coulomb yield criterion for granular materials can be modeled. Savage (1979) uses standard repres- entation theorems to derive an expression for T, similar to that of Goodman and Cowin (1972). Savage (1984) gives a review of the expressions proposed for T,. Massoudi (1988), in modeling the solid particles in a fluidized bed, uses an expression similar to that derived by Savage (1979). In this study, we assume that the stress tensor for a granular material is given by (Rajagopal and Massoudi, 1990)

T, = C&(PZ) + 6L(~z) grad pz l grad pz

•t &2(p2) tr D211 + 6S(~2)D2

+ h(pJ grad p2 0 gradp, (17)

where l denotes the scalar product of two vectors, and @I denotes the outer, or tensor, product of two vectors. The spherical part of the stress in eq. (17) can be interpreted a; the solid pressure P,. The material moduli /?I and fls are material parameters that reflect the distribution of the granular particles, and 8, plays a role akin to pressure in a compressible fluid and is @en by an equation of state. The material modulus & is a viscosity akin to the second coefficient of viscosity in a compressible fluid, and j3 denotes the viscosity (i.e. the resistance of the material to flow) of the granular solids. Recently, Rajagopal and Massoudi (1990) have outlined an experimental/ theoretical approach to determine these material


moduli. Based on the available experimental measurements of Savage (1979), Savage and Sayed (I 984), and Hanes and Inman (1985) and the computer simu- lations of Walton and Braun (1986a, b), it is clear that granular materials exhibit normal stress effects. The above model [es. (17)] is a simplified version of the model proposed by Rajagopal and Massoudi (1990) which predicts the possibility of both the normal stress differences. Furthermore, Boyle and Massoudi (1990), using Enskog’s dense gas theory, have ob- t_aine_d explicit expressions for the material moduli

PO-84. A mixture stress tensor is defined as [cf. Green and

Naghdi (1969)]

where

T, = T, + T, (18)

T, = (1 - v)T,, and T, = T, (19)

so that the mixture stress tensor reduces to that of a pure fluid as v -+ 0 and to that of a granular-material as 4 -+ O.f T, may also be written as T, = vT,, where ?s may be thought of as representing the stress tensor for some (quite densely packed) reference configuration of the granular material.

The mechanical interaction between the mixture components, fI, is written as (Johnson et al., 1990)

f, = A,gradv + A,S(v)(v, - vI)

+ A,v(2 tr D:)-‘/4D1(~2 - vL)

+ A,v(W, ~ W,)(v, - v,) + Asa,, (20)

where a,, is a properly frame invariant measure of the relative acceleration between the mixture components and F(Y) represents the dependence of the drag coefficient on the volume fraction. The terms in eq. (20)

reflect the presence of density gradients,t drag, “slipshear” lift, “spin” lift, and virtual mass, respectively. Miiller’s (1968) work indicates that a term of the form A, grad Y must be included in the interactions in order to get well-posed problems. The term multiply- ing A, is a generalization of Saffman’s (1965, 1968)

‘Note that qb -+ 0 is equivalent to v + 1 in a saturated mixture. Thus the theory allows for the case of the mixture tending to a pure granular material without v + 1 but to some value Y,, strictly less than unity, usually referred to as the maximum packing fraction. We are interested, here, however in the case when there is a sticient amount of both the constituents, and hence we are not close to either of the limiting cases. Further, in keeping with the usual weighting procedures in multiphase Row, we shall represent T, as

T, = (1 - v)T, + v?;

where TA is discussed above. ‘The actual form of this interaction should include the

terms cl,gradp, + cr,gradp, where a, and G(* are constants. If we assume that the system is a saturated mixture with incompressible components, this expression simplifies to A, grad”, where A, = a, - a,. Since no information concerning the coefficients a, and a1 is available and a term of the same form arises from the granular solid stress tensor, this term will be neglected in the present work.

single-particle result first proposed in this form by McTigue et al. (1986).

GOVERNING EQUATIONS

Combining eqs (12), (13), (16), (17) and (20) yields the equations describing the flow of a mixture of a Navier-Stokes fluid and a granular solid:

av, pI

[ - dt + (gradv,)v, 1 = (grad v)p - (1 - v) grad p

+ ,I,[( - gradv) div v1 + (1 - v) grad (divv,)]

+ (1 - v)(grad;l,) div v, + 2y,[( - gradv)D,

+ (1 - v) divD,] + 2(1 - v)(gradp,)D, + p1 b,

+ A,grad v + A,g(v)(v, - v,)

+ A3v(? tr D~)-1’4D,(v2 - vl)

+ A,v(W, - WI)@, - v,) + Asa,, (21)

p2 av2 dt + (gradvh2 1 = grad/%

+ gradh(gradp, l aad p2)

+ 2j,{grad(gradpZ)jTgradp2 + grad&(divv,)

+ 8, grad (divv,) + fi3 div D, + (grad &)D2

+ 8,Llgrad (grad P,)I grad p2

+ kwb)div(grad~2H

+ grad&Cgradp, @ wh21 + Pzb2 - A, gradv - A,~(v)(v~ - v,)

- A,v(2trD:)m”4D,(v2 - vI)

- A,v(W, - WI)@, - vl) - A5a,,. (22)

Assuming that p/ and ps are constants, the granular solid constitutive equation is rewritten-with new_coef- ficie_nts defined _as: PO = p^,, PI = of/?,, Pr = &, b3 = pJ, B4 = ~,‘a,. Equations (21) and (22) are non-

dimensionalized in terms of the following dimensionless variables:

“,=y’,&=!?yl+=~ (23) 11.2 uo

Pf =:A% b b *=fi,b+‘,b+A (241

PO 4 9

p=P PO6

div* = Ldiv, grad* = L grad,

L a* =--a

urn 2 um uo (25)

DT = :[grad*V, + (grad*V,)T]

DT = k[grad*V, + (grad*V2)T]

WT = $grad*V, - (grad*V,)‘]

(26)


W; = i[grad*Y, - (grad*V,)z] (27)

where L is some characteristic length, for instance half the space between flat plates or a tube radius, II, is a characteristic velocity, p0 is a characteristic density, and x is the position vector. Equations (21) and (22) are written in dimensionless form as (dropping the asterisks)

(1 - v)p, [

$ + (gradV,)V, I

= (gradv)P

- (1 - v)gradP + A[( - gradv)divV,

f (1 - v) grad (divv,)]

+ (1 - v)(gradh)divV, + &[( - gradv)D,

+ (1 - v)divD,] + 2(1 - v) grad; ( >

D,

+ $1 - v)b, + Crgradv + C,.F(v)(V, - V,)

+ C,v(2tr D:)-114D1(VZ - V,)

+ CLv(WI - W,XV, - Vr) + A5a,, (28)

vps + (gradV,)V, = grad & 1 + grad B^,(gradv l gradv)

+ 2i,{grad(grad v)}‘grad v + grad s^,(divV,)

t &grad(divV,) + &divD, + (grad&)D,

+ B^., [{grad (grad v)} grad v + (grad v) div (gradv)]

+ gradg&radv @ gradv] + gvb, - Crgradv

- C,F(v)(V, -V,) - CJv(2trD:)-“4D,

x (V, - V,) - c,vw, - WIXV, - V,)

- c5bn (29)

where the following dimensionless groups are identi- fied:

& _ P&.L --1

cc/ (30)

i, = Bo

--+, = I% -

PO% -, B, = -& POUb~2

(31)

83 B^, =- P.%A

& = ___ 84 C-A, - (32)

p,u,zL=’ 1 - p,u,’

C,=A2L,C,= P&J

(33)

FLOW BETWEEN PLATES

Consider the pressure driven flow of a mixture modeled by eqs (16) and (17) between two horizontal flat plates. Let X denote the direction of motion and

let the plates be located at Y = - 1 and Y = 1. If the flow is steady and laminar, the velocity profiles and solids distribution can be assumed to have the form

V, = V( Y )e,, V, = U( Y)e,. v = v( Y). (34)

The balance of mass equations are automatically satisfied by these assumptions. Virtual mass effects are not important for this steady problem and it is assumed that “spin” lift may be neglected. Gravity is assumed to be the only body force_pre>ent.

Assuming that the coefficients B,-B, are smooth functions of v, Rajagopal and Massoudi (1990) use a Taylor series:

B, x B,, + B,r v + B,, v=, a = 0, 4 (35)

and show that E,, = So2 = B,, = B,, = 0. Here, due to lack of experimental measurements and to reduce the total number of parameters, it is assumed that, for all nonzero coefficients:

B, = B,, = B,, = Bo2, a=0,4 (36)

1, and p, are assumed constant. The momentum balances given by eqs (28) and (29)

are greatly simplified by these assumptions and may be written in component form as

(1 - v)Y” - V’V’ - ; Re(1 - v) ( >

+ C,Re.F(v)(U - V) = 0 (37)

V’P-((1-v); +CJvJVII-L’*y’(U-y) C !

-$[l-vj=O (38)

B,{(v + v2)U’ + (2v + l)v’U’}

- 2C29(v)(U - V) = 0 (39)

B,v’ + (Bl + 8,)(2(1 + v f v’)v’v” + (2v + ~XV’)~}

- 2 - c,v]v’/- 1’2 V’(U - V) = 0 (40)

where the prime denotes a derivative with respect to Y.

Eliminating pressure from eqs (37) and (38) by cross differentiating yields

(1 - v) V”’ - 2v’V” - Y” V’ + C,Re[LF’(U - V)

+ S(U’ - V)] = 0. (41)

Consider eqs (39H41) in the limiting case as v approaches 0. It is assumed, based on physical consid- erations, that 9 (v) + 0 as v + 0. If v = 0, then eqs (39) and (40) vanish. Equation (41) becomes

I/“’ = 0 (42)

which is consistent with Poiseuille flow of pure fluid. If 4 = 0 (v = v,,,), there is no fluid, which means there is no interaction between the solid and fluid. Since the


driving force is the pressure gradient in the fluid, when @=O

u = 0. (43)

Equations (39H41) have to be solved numerically; in order to do so the following definitions are introduced:

9, = C2Re, LB2 = s, 649 = 3

& (44)

J= --* - C, Fr(B, + B4)’

y==. BI + B,

(45)

Though introduced primarily for numerical con- venience, these dimensionless parameters do have physical interpretations. g1 is the ratio of the drag force exerted on the fluid-by the solid to the viscous forces within the fluid phase. g32 is the ratio of the drag force exerted on the solid by the fluid to the viscous forces within the solid component. g is a combination of the material coefficients from the granular solid constitutive equation; it affects the distribution of solid in the flow but does not directly influence the velocity profiles of either fluid or solid. 9 incorporates the effect of gravity into the equations. ..F is the lift coefficient; the fluid and solid velocity fields exert an effect on the distribution of the granular solid only when _F # 0. With these definitions, eqs (39H41) are rewritten as

(1 - Y)V”‘_ 2v’V”- v”V’+ B1[F’(U - v)

+ F(U‘ - Y’)] = 0 (46)

(v + v2)u,’ + (2v + 1)v’U’ - 92F(v)(u - V) = 0

(47)

2(1 + v + v2)v”V’ + (2v + l)(V)3 + gv’ + Yv

+ ~vlV’~-~‘~V’(U - V) = 0. (48)

Notice that, when -k” = 0, eq. (48) is not coupled to eqs (46) and (47) through the velocity field, and thus eq. (48) can be solved independently to determine Y. Based on Batchelor’s (1972) result for sedimentation of a dilute suspension of spheres, Drew (1976) pro- poses the following approximation for F(v):

S(v) = v(1 + 6.55v). (49)

This expression is used here with the understanding that it is not valid as v + 1.

Boundary conditions

Adherence boundary conditions are imposed on both constituents at each plate:

U( - 1) = U(1) = V( - 1) = V(1) = 0. (50)

The flow rate of the mixture is also prescribed as a condition. The mass flow rate of a two-component mixture is defined as

[(I - v)p/ F’ + vp, U] d y. (51)

A volumetric flow rate may be defined as

Q=[;, K&Y=/;, [(l - v)V+ vU]dY. (52)

When the mixture is neutrally buoyant, i.e. pI = ps = p, these two quantities are simply related by

Ql. = PQ. The volumetric flow rate of the mixture is the quantity prescribed in the present work.

The appropriate conditions on v are the boundary value of v at a plate and a prescribed average volume fraction, defined through

s

1 N= vd Y. (53)

1

These conditions may be specified in two ways. Sym- metric solutions may be obtained by specifying that v( 1) = v( - 1) and giving a value for N. Solutions that are, in general, not symmetric may be obtained if N and a value for v on one boundary are given. This condition is expressed as

~-*a as Y+ --_I (54)

where u is a constant. Then v(1) is determined from the solution.

When gravity is included in the equations, but lift is ignored, the symmetric boundary condition, v( - 1) = v(I) is not physically possible since v’ cannot equal

zero anywhere in the domain. In fact, v( - 1) and N are specified such that v( - 1) > v(l). Numerical values for v on the boundary could be obtained from experiments such as those performed by Segre’ and Silberberg (1962).

Numerical results and conclusions Attempts to solve eqs (46H48) subject to the above

boundary conditions were made using multiple- shooting, finite-difference and collocation methods. While for some parameter values all three methods arrive at the same solution, the collocation code COLSYS (Ascher et al., 198 1) provides solutions for a larger range of parameters. For situations in which more than one method is effective, COLSYS is also invariably most efficient.

There are some restrictions on the values the parameters may take. Clearly, the drag coefficient, Reynolds number, and material viscosities must be greater than zero, and therefore 9, and aZ must also be greater than zero. Though mathematically the lift coefficient _IZ may have either sign, it must be greater than zero for the direction of lift to match experimental observations (Repetti and Leonard, 1964). 9 is also restricted to be positive, implying that B, + B, < 0. Since B, is also less than zero (Rajagopal

and Massoudi, 1990), 3 must be greater than zero when gravity is included. In summary, all five parameters, gl. g2, %+, 9 and 9, are generally assigned only positive values. In the absence of gravity, 9 = 0, P’ = 0, and there is no restriction on .@.

We have done an extensive parametric study of these equations (Johnson et al. 1990b). The para-


metric study of the numerical solutions is not presented in detail here. Instead, a few representative velocity profiles will be included with a discussion of

the effects of the parameters. The parameter 9, affects the fluid velocity profile

as shown in Fig. 1. Increasing values of ~9~ result in a decrease in the maximum fluid velocity and a “flattening” of the profile. Note that as GBI gets very large (i.e. z lOOO), there are local maxima in the Auid velocity profiles. Though it does not change shape, the solid velocity also decreases due to the coupling of the two equations (see Fig. 2). Physically, an increase in the value of gI corresponds to either an increase in the drag coefficient or an increase in the Reynolds number, or b&h, and thus the results obtained are in keeping with physics.

Increasing S1 results in a decrease in fluid velocity and a corresponding increase in solid velocity. $S2 does not affect the shape of either the fluid or solid velocity profile. Physically, an increase in the value of Bz corresponds to an increase in the drag coefficient or a decrease in the nondimensional number B, or

1 50

02 = : B=l N = 0.3 a = 5

Fluid Veiocity,Dimensionless

Fig. 1. Effect of g1 on fluid velocity.

1 50 7

02 = I q =I t4=a.3 a=5

100 \

Dl * 100

0 50

t

0.00

-0.50

-1.00 0.00 0.50 1 .OO 1.50 2.00 2.50

Solid Velociry, Dimensionless

Fig. 2. Effect of gl on solid velocity.

both. The drag of the fluid on the solid is the only force driving the motion of the granular material in this problem. Thus, if g2 = 0 then U is identically zero; if Qz > 100 then U % V everywhere [see Fig.

WNI. Figure 3(a) is a plot of volume fraction vs Y for

several values of SJ (with 5!? = 0 and 9 = 0). Note that an integral condition [cf. eq. (53)] is used and the areas under each of the six curves are the same, although the boundary condition at the plates is different for each 8. Recall when Y = 0 the equation for Y [eq. (48)] is not coupled to either the fluid or the solid velocity field. The distribution of the granular solid is therefore not influenced by 9, or 9*. The velocity profiles, however, are strongly influenced by the choice of W, as shown in Figs 4 and 5. Mathemat- ically, this effect is due to the appearance of v in the equations for U and Y [eqs (46) and (47)]. Physically, the influence of v on the velocity profiles is to cause a relative decrease in the fluid velocity in areas of higher solids concentration and an increase in areas of lower solids concentration. The effect on the solid is, appro-

1 .SD N = 0.3

B-2 1 0 I .oo

0.50

P-

0 DO ~

-1 0.00 0.10 0.20 0.30 0.4a D.50

Volume Fraction

1.50

D, I 1 D2=,00 E=I N=0.3 Q=5

0.00 1 .oo 2.00 3.00 4.00

Velocity, Dimensionless

Fig. 3. (a) Effect of @ on volume fraction. (b) Effect of 6Bz on velocities,


I .50

1

B = -2.2 N = 0.3

‘.““rn > o.5o 1 /

0.00

-0.50

I L

(a)

-1.00 m 0.00 0.10 0.20 0.50 0.40 0.50 0.60

Volume Froct;on

1.50 01 * 100 02 = 1 B = -2.2 N = 0.3 0 = 5

0.00 1 00 2.00 3.00 4.00 5.00 6.00

Velocity; Dimensionless

Fig. 4. (a) Volume fraction profile. (b) Velocity profiles.

priately, just the opposite; the solid moves faster in areas of higher concentration and slower when the concentration is lower. These results are consistent with the form of the drag interaction, which depends upon v (becoming greater as v increases). Physically, drag transfers momentum from the fluid to the solid. As drag becomes more important, we expect to see decreasing fluid velocities and increasing solid velocities.

Figure 6 shows volume fraction profiles for various values of the dimensionless gravity parameter, Y. Notice that the solution corresponding to 3 = 0 is

not symmetric with respect to the mid-plane due to our requiring that v( - l)#v(t). Figure 7 shows the same kind of plot for a much smaller value of a. The dotted lines indicate profiles in the absence of gravity for the same boundary conditions. Note that, when Q > 0, Y is a monotonically decreasing function of Y and boundary conditions must be chosen appropriately. Clearly, as B is increased, the solid moves away from the upper plate towards the lower plate. At a fixed Y, Y decreases as 3 increases. Figures 8 and 9 show some representative velocity and volume fraction profiles for 9 > 0. Though 3 does not directly

El = 2.2 N = 0.3

1.00 _

050:

>

0.m -

(a)

-: 00-~,,~I.~.I,,,.I.,.,,~,.I.~.,,*CmrrrrrrTnl,,.,,,I o.ao 0.10 0.20 0.30 0.40 0.50

Volume Fraction

3 / I

:Il! 2.00 3 00 4.00


Fig. 5. (a) Volume fraction profile. (b) Velocity profiles.

I 50

B = 2.0 G = 0.0. 0.05, 0.5. 0.9, 1.34

0.10 0.20 0.30 0.40 0.50 0.60 0.70 Volume Fraction

Fig. 6. Effect of g on volume fraction

influence the velocity profiles, like B, it affects the velocities through its influence on the volume fraction. Figures 8 and 9 again illustrate the influence of volume fraction on the velocity profiles. In obtaining the solutions depicted in Figs 6-9, values for v( - l), v(1) and v’( - 1) are assumed (leading to the correspond-


150 1.50

B = 0.1 G = 0.0. 0.1. 0.2, 0.5 8 - 0.1 C - 0.3 N - 0.3

0.10 0.20 0.30 0.40 0.50 0.60 0.70

Volume Fraction

1.00

(al

0.50

t

0.00

-1.00 ",.,'.,.,'...".",,.1.1.,Il,,.,"".., 0.00 0.20 0.40 0.60 0.80

Volume Fraction

Fig. 7. E&-ct of d on volume fraction.

150 1 01 G - = 0.3 100 0 02 = - 5 1 N B = - 0.3 0.1

1.50 B - I.0 c * 0.01 N - 0.4840

1 .oo

(a)

0.50

>-

0.00

-0.50

-1.00 - 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Volume Fraction

000 050 100 150 2.00 2.50


Fig. 8. (a) Volume fraction profile (with gravity). (b) Velocity profiles (with gravity).

‘““l--T--Y (b) 0.50

b-

0.00

-0.50

-1.00 m 0.00 1 00 2.00 3.00 4.00 500 6.00


Fig. 9. (a) Volume fraction profile (with gravity). (b) Velocity profiles (with gravity).

ing values for Q and N shown in the legends of Figs 6-9).

Figure 10 shows the effect of a non-zero lift coefficient, 2’. Increasing Y, and thus the significance of lift, causes the granular solid to move away from the walls and collect towards the center of the channel. The fluid and solid velocity fields do directly influence the volume fraction in this case. The resulting change in the distribution of the solid causes an increase in fluid velocity near the walls and a decrease in fluid velocity where solid concentration has increased (in the center of the flow).

Acknowledgements-This work was performed under the auspices of the Laboratory Graduate Participation Program (by G. Johnson) and the Faculty Research Participation Program (by K. R. Rajagopal) under contract number DE- AC05-76GROOO33 between the U.S. Department of Energy and Oak Ridge Associated Universities.


B = 2 G = 1 N = 0.457

0.20 0.40 0.60 Volume Fraction

1.50 m 0.00 1 .OD 1 so 2.00 2.50 3.00 Dimensionless Velocity

Fig. 10. (a) Volume fraction profile (with gravity and lift). (b) Velocity profiles (with gravity and lift).

NOTATION

acceleration vector frame-indifferent relative acceleration interaction coefficients (i = 1-5) body force vector dimensionless ps constant part of t?s dimensionless parameter dimensionless As stretching tensor dimensionless parameter dimensionless parameter interaction force vector Froude number volume fraction dependence of drag gravitational acceleration identity tensor gradient of a velocity vector dimensionless parameter average volume fraction fluid pressure

P

: Ri- T u V

v

V W x X

x

X Y Y

dimensionless fluid pressure volumetric flow rate of mixture mass flow rate of mixture Reynolds number stress tensor solid velocity between plates velocity vector fluid velocity between plates dimensionless velocity vector spin tensor direction of flow between the plates position vector dimensionless x dimensionless position vector direction normal to plates dimensionless y

Greek letters pi granular solid coefficients (i = &4)

4 second coefficient of viscosity, fluid A dimensionless il,

w fluid viscosity (first coefficient of viscosity) V volume fraction of the solid

P density r dimensionless time

Q volume fraction of fluid

Subscripts

Lf referring to the fluid phase

2,s referring to the solid phase m referring to mixture

Superscripts T transpose * dimensionless quantity

Other symbols div divergence operator grad gradient operator tr trace of a tensor V gradient symbol

0 outer (dyadic) product

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Flow of a fluid—solid mixture between flat plates

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