A variational formalism for steady flow of dusty fluid problems I
A general variational formalism for the solution of steady flow of dusty fluids is given. The associated boundary conditions are enforced by suitable terms in a functional which is stationary at the solution of the given problem and consequently the expansion functions used need not satisfy any of the boundary conditions. Introduction. Problems of mechanics of systems with more than one phase have been considered by many authors. (See Saffman [10], Rao [9], Didwania and Homsy [5] and references cited there.) Situations which occur frequently are concerned with motion of liquid or gas which contains a distribution of solid particles. Such situations occur, for example, in the movement of dusty laden air, in problems of fluidization, in the use of dust in gas-cooling systems to enhance heat transfer process, and in the process by which raindrops are formed by the coalescence of small droplets which might be considered as solid particles for the purpose of examining their movement prior to coalescence. In a number of papers, variational principles have been derived for the solution of various boundary value problems: Tani [13], Sloan [11], Barrett [1], Wenger [14], and Smith [12]. More recently, variational principles have been derived for stationary advective problems where the boundary conditions are incorporated via suitable functionals and the restriction that the expansion functions need to satisfy the boundary conditions has been removed. (See Haj et al [6].) In solving convective diffusion problems, Moult, Burley, and Rawson [8] produced a functional corresponding to a local potential approach, where the first derivative term is kept fixed during the variation. In the present paper we propose a suitable functional for the solution of steady flow of dusty fluid problems which contain first and second derivative terms by considering a general variational formulation rather than the local potential approach of [8], We extend the technique of [6] by enforcing the prescribed boundary conditions by the proposed functional for the solution of the two phase problems. * Received December 26, 1986. 275 ©1988 Brown University 276 ADNAN A. HAJJ AND ELSAYED F. ELSHEHAWEY 1. Basic equations. The equations of motion of a dusty, steady, viscous incompressible fluid are (see Saffman [10]): (w • V)w = -l//?gradp + i/V2u + (KN/p)(u u), (1.1) divw = 0, (1.2) m[(u • V)u] = K(u u), (1.3) div(TVu) = 0, (1.4) where w and u denote the local velocity vectors of the fluid and dust particles respectively, p the density, p the static fluid pressure, v the kinematic viscosity, N the number density of a dust particle, K the Stokes resistance coefficient (for spherical particles of radius 8, it is 6npS, p is the fluid viscosity) and m the mass of a dust particle. Consider the velocity distribution of fluid and dust particles defined in the Cartesian coordinates x = (x,y) respectively as: u = (0,0, co(x)), (1.5) u = (U0,Uhu(\)), (1.6) N = N0(a constant), (1.7) where (0,0, cu(x)) and (Uq, U\, m(x)) are the velocity components of the fluid and dust particles, respectively; t/0 and U\ are constants. The equations of motion then reduce to the following coupled equations in a region R with boundary dR: -AV2co(\) + Da>(\) Du(\) G, x in R, (1.8) C • V«(x) + Du(\) Doj(x) = 0, x in R, (1.9) subject to the following boundary conditions: a>(x) = gi, x on dR, (1-10) u(x) = g2, x on dR, (1-11) where A — p/(mNo), D — K/m, G = -^/(mNo) = constant, and C = (t/o, U\). (1.12) 2. A functional embodying the boundary conditions. In this section we introduce the following adjoint problem to problem (1.8)—(1.11): -AV2i//(x) -1Dy/(x) Z)v(x) = G, x in R, (2.1) -C-Vv(x) + Z)v(x)-Zty(x) = 0, x in R, (2.2) with the associated boundary conditions: y/(x) = g\, x on dR, (2.3) v(x) = gi, x on dR. (2.4) Next, we produce a functional which is stationary at the solution of (1.8)—(1.11) and (2.1)—(2.4) for a class of functions which do not necessarily satisfy any of the boundary conditions. STEADY FLOW OF DUSTY FLUID PROBLEMS 277 Theorem. The solutions to the system (1.8)—(1.11) and (2.1)—(2.4) render the following functional stationary at the solution W = co, U = u, ¥ = y/, and V = v: FIWW, V] = f {AV*¥ VW + DVW DWV DVU G{W + x¥) JR + VCVU +VDU}d\ + <f [(gl-W)AVx¥+(gx-y¥)AVW]-nds JdR + <f (g2-U)[(g2-V) + VC n]ds JdR where n is the unit outward normal, ds is the differential along dR with positive direction being counterclockwise. Proof. Let co and u be the solution to (1.8)—(1.11), y/ and v be the solution to (2.1)—(2.4); Si, i = 1, 2, 3, 4 arbitrary functions. Define Gi (e) = F[a> + e$i, u, y/, v], (72(e) = F[0), U + £52. V< v]. (73(e) = F[co, u,y/ + es3, v], (74(e) = F[co, u,y/,v + es4]. Then F is stationary at the solution of (1.8)—(1.11) and (2.1)-(2.4) if dGk{ 0) = 0, k= 1,2,3,4. de Now, ^de^ J '^Sl + ~ ®Sl v ~ + <f [-AsxVy/ + (gx y/)AVsx] ■ nds. (2.6) JdR Using Green's theorem and (2.1), the first integral in (2.6) reduces to <f> As 1 Vy/ nds. JdR This integral cancels the last integral in (2.6) taking into consideration the boundary conditions (2.3). Hence, dG i(0) de In the same manner, it can easily be proved that dG2{0) _ dGi{0) _ <*G4(0) de de de and hence the functional F is stationary at the solution. -0, 278 ADNAN A. HAJJ AND ELSAYED F. ELSHEHAWEY 3. Matrix set-up for the solution. Now we seek approximate solutions to (1.8)(1.11) and (2.1 )-(2.4) of the form: M M 0)~cdm = J2ail)hi{x). u — uM — ̂ aWfiiix), ,=1 ,=1 (3.1) M M v ' v~vm = J2 bll)h'(x)' v vm = bl2)h>(x)i=l ;=1 Substituting (3.1) into the functional (2.5) and taking its stationary value with respect to b\x\ b\2) leads to the defining equations for the coefficients af1' and aj2> (equations for b\x) and bj2) are not needed): a ?)$)-($) where the matrices L, H, and E are M x M matrices and a(1>, a'2', Q*1^, and Q(2) are M vectors. The various matrices and vectors of (3.2) are given by: L,j = [ (Vhj ■ AVhi + hjDhi) dx [ (hjAVhj + hjAVhj) ■ nds, J R JdR Hij = [ hjDhi dx, J R Ejj — [ (hjC -Vhi + hjDhj) dx+ f {hjhj + hjhjC ■ n) ds, Jr JdR Q{jV) = [ Ghjdxf AgiVhj ■ nds, Jr JdR and Qf = [ g2{hj hjC ■ n) JdR i ds. ' d It is desirable, for stability reasons (see Mikhlin [7]) to take the basis functions /z, (x) in (3.1) as a set of orthogonal polynomials. Without loss of generality, and to keep the exposition simple, we consider the one-dimensional basis h,(x) and, for programming conveniency, we will define h^, k = —2, -1, / as h-2(x)= 1, h-i(x) = x, hi{x) = -(1 -x2)Ti(x), (3.3) where -1 < x < 1, and i = 0,1, ■■ • , M 3; T,(x) is a Chebyshev polynomial of the first kind. The reason for this choice of basis is to avoid introducing artificial singularities in the matrix equation (3.2). In calculating the elements of the various matrices and vectors in (3.2) we need the following: 4X) = (2/tt) /> x2)Tn(x)/{l x2)1/2]dx, n> 0, (3.4) where 5 if n = 2, dP= | 1 if« = 0, (3.5) otherwise, STEADY FLOW OF DUSTY FLUID PROBLEMS 279 using the results, i-x2 = i[r0(x)-r2(x)] and (3-6) TnTr = j[Tn+r + Tn-r] and the orthogonality property of Chebyshev polynomials. We also need dTM = (2/n) j\(\ x2)2Tn(x)/(l x2)l'2]dx (3.7) which can be related to dJ*1' by the following: dn] = ?[-4+2 + 2dil) ~ 4!-2|]' (3-8) where we have used (3.7), (3.6), and the property T-n = Tn. Similarly, it can be shown that dn+l) = (2/n) J [(1 -x2)'+ir„(x)/(l x2)*]dx (3.9) can be related to d^ by the following: (3.io) 4'+1) = (2/1) m> 1 which follows from (3.6) and the well-known Chebyshev relation: "(1 _X2}1 = j _ ^[2/(4m2 1 )]T2m(x). (3.11) m> 1 At this stage we are able to relate the elements of the matrices and vectors in (3.2) to d^ and d„+2\ r = 1,2. For example, the elements of the matrix H: h"=-dL hj(x)hi{x) dx = -D J] [(1 -x2)5/2TiTj/(l -x2)$]dx = -~[^/2) + ^(V2)]i w>a (3. !2)