DOUBLE-DIFFUSIVE CONVECTION WITH FINE DUST

R. C. Sharma*), Neela Rani

Department of Mathematics, Himachai Pradesh University, Shimla-171 005, India

The effect of fine dust (suspended particles) is considered on the thermosolutal convection. The principle of exchange of stabilities is satisfied for thermal Rayleigh number greater than or equal to solute Rayleigh number. If the solute Rayleigh number exceeds the thermal Rayleigh number, the oscillatory modes may come into play. The effect of stable solute gradient has a stabil- izing effect whereas the effect of suspended particles is to destabilize the layer. The effect of a uniform rotation is also considered and is found to have a stabilizing effect in the presence of suspended particles on the thermosolutal convection.

I. I N T R O D U C T I O N

The problem of thermohaline convection in a layer of fluid heated from below and subjected to a stable salinity gradient has been investigated by Veronis [1]. The physics is quite similar to Veronis' [1] thermohaline configuration in the stellar case in that helium acts like salt in raising the density and in diffusing more slowly than heat. This problem of the onset of thermal instability in the presence of a solute gradient is of great importance because of its application to oceanography, atmospheric physics and astrophysics. The heat and solute being two diffusing components, double-diffusive convection is the general term used in dealing with such phenomena.

The onset of B6nard convection (thermal instability) has been discussed in detail by Chandrasekhar [2]. Chandra [3] performed careful experiments in an air layer and found a contradiction between the theory and the experiment. Chandra [3] found that the instability depended on the depth of the layer. A B6nard-type cellular convection with fluid descending at the cell centre was observed when the predicted gradients were imposed, if the layer depth was more than 10 mm. But if the depth o f the layer was less than 7 mm, convection occurred at much lower gradients than predicted and appeared as irregular strips of elongated cells with fluid rising at the centre. Chandra called this motion "columnar instability"i The effect of particle mass and heat capacity on the onset of B6nard convection has been considered-by Scanlon and Segel [4]. Palaniswamy and Purushotham [5] have considered the stability of shear flow of stratified fluids with fine dust and have found the effect of fine dust to increase the region of instability.

Motivated by interest in fluid-particle mixtures, columnar instability and keeping in mind the physical situations occurring in atmospheric physics, oceanography and astrophysics, we set out to study the effect of suspended particles on the double- diffusive convection. Owing to the importance of the Coriolis force, the effect of a uniform rotation on the problem has also been considered.

*) Present adress: Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1.

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R. C. Sharma et al.: Double-diffusive convection with fine dust

2. FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS

.Consider a horizontal fluid-particle layer of depth d unbounded in the horizontal (x, y) directions and bounded by the planes z = 0 and z = d. This layer is heated and .soluted from below such that an adverse linear temperature gradient - f l and an adverse linear solute gradient - f l ' are maintained. The equations of motion and continuity for the fluid are

+ i . I-- ..I

(2) V . u = 0.

Here ,o, # (= Qv), p and u(u, v, w) denote respectively the density, the viscosity, the pressure and the velocity of the pure fluid, v(1, r, s) and N(x, t) denote the velocity and number density of the particles. K = 6rc/~q, ~1 being the particle radius, is the Stokes' drag, 9 is the acceleration due to gravity, x = (x, y, z) and ~, --= (0, 0, 1). Assuming uniform particle size, spherical shape and small relative velocities between the fluid and particles, the net effect of the particles on the fluid is equivalent to an extra body force per unit volume KN(v - u).

The force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid. The buoyancy force on the particles is neglected. The interparticle reactions are ignored for we assume that the distance between particles are quite large compared with their diameter. I f N m is the mass of particles per unit volume, then the equations of motion and continuity for the particles are

(3) mN t + (v. V) v = - m N g ~ + KN(u - v)

(4) + V . (Nv) = 0 . ~t

Let C, Cp, T, S, q and q' denote respectively the heat capacity of fluid at constant pressure, the heat capacity of particles, the temperature, the solute concentration, the "effective" thermal conductivity is the conductivity of the clean fluid and an analogous "effective" solute conductivity. If the particles and the fluid are assumed to be in thermal and solute equilibrium, then the heat and solute conduction equa- tions give

The equation of state for the fluid is

(7) ~ - ~Oo[1 - a ( V - T o ) § a'(S - So)],

Czech. J. Phys. B 39 (1989) 71 1

R. C. Sharma et al.: Double,diffusive convection with fine dust

where 9o is the mean density of the clean fluid, a is the coefficient of thermal ex- pansion and a' is the analogous coefficient of solvent expansion.

The initial state of the system, denoted by subscript 0, is taken to be a quiescent layer with a uniform particle distribution No. The initial state Uo--0, v 0 = 0, To = - f l z , So = - f l ' z , No = constant is an exact solution to the governing equa- tions.

Let 59, 5p, N, @, F, u and v denote respectively the perturbations in density Qo, pressure Po, number density No, temperature T, solute concentration S, fluid velo- city (zero initially) and particle velocity (zero initially). Then the linearized dimen- sionless perturbation equations are

N~_ ~ au _

0 t

(8)

(9)

(10)

. . . . VSp + V2u + to(v - u) + NaO~ -- NsF2 ,

9u " C - - ..~- U - - V ,

8t

8M V . u = 0 , - - - + V . v = 0 ,

Ot

(11) (1 + h) ? 69 = (w + hs) + V20 -ff

(12) (1 + h) OF (w + hs) + V2F ~ - =

Physical variables have been scaled using d, d2/k, k/d, 9vk/d 2, Bd and fl'd as the length, time, velocity, pressure, temperature and solute concentration scale factors respectively, w and s are the vertical fluid and particle velocities, v is the kinematic viscosity of the fluid, k is the thermal diffusivity, k' is the solute diffusivity, Np = v/k is the Prandtl number, NR = g~xfld4/(vk)is the Rayleigh number, Us = go:'fl'd4/(vk ') is the analogous solute Rayleigh number, f = Nom/Qo = "ro~Nr, is the mass fraction,

co = KNod2/(90v), "r = mk/ (Kd2) , h = fCp/C and M = N / N o .

In writing equation (8), use has been made of the Boussinesq equation of state

5o = - 9 o ( a O - a ' r ) .

Consider the case of two free surfaces having uniform temperature and solute concentration. The case of two free surfaces is most appropriate for stellar atmo- spheres (Spiegel [6]). The boundary conditions appropriate for the problem are

O2w (13) w - - 69 = F = 0 at z = 0 and 1.

az 2

Eliminating v and 5p, the fluid, heat and solute equations become

(14) (L, - L2 V 2) V2w = L2(N. V20 - Ns V~F),

7 1 2 Czech. J. Phys. B 39 (1989)

R. C. Sharma et al.: Double-diffusive convection with fine dust

( (15) L 2 H--0 _ V 2 0 = z - - + H w, 0t 0t

(16)

where

L2 ~ - - V 2 F = z - ~ + H w,

L1 = N ~ 1 z - - + F , L 2 = z - - + l , F = f + 1, H = h + 1, Ot 2 Ot] Ot

02 02 92 92 92 V 2 = ~ + + - - and V~ = - - + - - . 0X 2 0y 2 922 0X 2 0y 2

Eliminating O, F between equations (14)-(16) , we obtain

(17) ( L 1 - L 2 V 2) (H ~t-V2)V2w=(NR-Ns)(z O+Ot H ) V 2 w .

We analyze the perturbations in terms of normal modes by seeking solutions whose dependence on x, y and t is given by

(18) W(z) exp(i~xX + i~yy + nt),

where ~ = (~x 2 + ~2)1/2 is the wavenumber of the disturbance and n is the growth rate. Using (18), equation (17) becomes

(19) (D 2 - - ~ 2 _ Hn) [L 1 - L2(D 2 - 0~2)] (D 2 - az) W = ~2(TF/ -]- H) (NR -- Us) W,

where d

Lt = N;l(zn z + Fn), L2 = (~n + 1) and D = - - dz

3. PRINCIPLE OF EXCHANGE OF STABILITIES AND OSCILLATORY MODES

Here we determine the conditions under which the principle of exchange of stab- ilities is satisfied and the oscillatory modes are introduced.

Let

U = ( D 2 - - t~ 2) W, (20) and

(21) X = (L I + L2tx z - L2 D2) U.

The equation satisfied by W, in terms of X, is

(22) (D 2 _ g2 _ Hn)X = o~2(zn + H)(Nrt - Ns) W.

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R. C. Sharma et al.: Double-diffusive convection with fine dust

Multiply equation (22) by X*, the complex conjugate of X and integrate over the range of z. We have

(23) f ~ X . ( D 2 _ a2 _ H n ) X dz = cz2(zn + H)(NR - Ns) S ~ X * W d z .

Integrating by parts and using equations (20), (21) and boundary conditions (13), equation (23) gives

I , + (a2 + Hn)I2 = o~2(NR -- Ns) (zn + H)(L~I3 + L*I,,), (24)

where

11 = So 1 IDxl dz, (25) 12 = S$ Ixl 2 d z ,

I3 = ~1 (IDWI 2 + ~21WI 2) dz,

1, = Yo' IuI dz

which are all positive definite. Putting n = ino, where n o is real, into equation (24) and equating the imaginary parts, we obtain

2 HI2 + Cr -- Ns ) (N ; 'HFI3 + "rhI,,) (26) "o = -

Np lz2~Z(Ne. - Ns) 13

or

(27) n o = 0 .

Since n o is real and integrals 11 - 14 are positive definite, it follows that no = 0 for all NR > hrs. This establishes that n is real for NR > Ns and that the principle of exchange of stabilities is valid for the problem under consideration. For NR < Ns, n 2 may be positive meaning thereby that the stable solute gradient may bring in oscillatory modes in the system.

4. THE STATIONARY CONVECTION

When instability sets in as stationary convection, the marginal state will be charac- terized by n = 0 and equation (19) reduces to

(28) (D 2 _ ~2)3 W = -~2H(N R - - Ns) W.

Here we consider the case of two free boundaries. It can be shown that all the even order derivatives of W vanish on the boundaries and hence the proper solution of equation (28) characterizing the lowest mode is

(29) W = W0 sin nz ,

where I41o is a constant.

Substituting the solution (29) in equation (28) we obtain

7 1 4 Czech. J. Phys. B 39 (1989)

R. C. Sharma et al.: Double.diffusive convection with fine dust

(30)

Equation (30) yields

(31)

and

(32)

NR + = - + ] V s . H~ 2

dNR = _ (7~ 2 -~- 32) 3

d H H20~ 2

dNR 2 + 1 .

dNs

The effect of suspended particles is to destabilize the layer. The stable solute gradient has stabilizing effect on the system.

5. EFFECT OF ROTATION

Here we consider the same problem as described above except that the system is in a state of uniform rotation t2(0, 0, O). The Coriolis force on the particles is neglected. The linearized nondimensional perturbation equations of motion for the fluid are

( 3 3 ) N~. 1 ?u _ 0 8p + V2u ~- O) ( l - ll) ~- T1/zv, St ~x

(34) Np_ 1 8v _ St ~y

(35) N~- 1 dw _ St ~z

+ + ,o (r - v) - T 2 ' : u ,

8p + V2w + co(s - w) + NR6) -- Nsr

where TA = 4 0 2 d4/v 2 is the Taylor number and equations ( 9 ) - (12) remain unaltered. Eliminating v and 6p between equations (33)-(35) and (9), we obtain

- - = L2(Li - L2 V 2) V2(NR O -- N s r ) , (36) (L1 - Lz V2) 2 V2w + L2zTA 632W

dz 2

(37) L2 , - V z 6 ) = z - - + H w , St

(38) L2 V 2 F z + H w ~t St

Eliminating O, F between equations (36)- (38) and nsing expression (18), we obtain

(39) (D z - a z - H n ) [ ( L 1 + L 2 ~ z - L 2 D z ) z ( D z - ~ z ) +L22TAD 2] W =

= o~2(L1 + Lz~z 2 - L2 D 2) (zn + H) (NR - Ns) W.

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R. C. Sharma et al.: Double-diffusive convection with fine dust

For the stationary convection, n = 0 and equation (39) reduces to

(40) [(D 2 _ ~2)3 + TAD 2] W = -~2H(NR - Ns) W.

Again considering the case of two free boundaries with constant temperature and concentration and using the proper solution (29), equation (40) gives

U, = (n2 + ~2)a + TAnZ + U s . o~2(1 + h)

(41)

Equation (41) yields

(42) dNR "/1;2 m m . - -

dT A ~2(1 + h) '

(43) dNR = + 1 , dNs

and

(44) dNR _ (re2 + ~2)3 + TAre2

dh 0~2(1 + h) 2

The effect of suspended particles is to destabilize the layer even in the presence of rotation. The rotation and stable solute gradient have stabilizing effects on the system.

Received 12 January 1988

References

[1] Veronis G.: J. Marine Res. 23 (1965) 1. [2] Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford,

1961. [3] Chandra K.: Proc. Roy. Soc. (London) A 164 (1938) 231. [4] Scanlon J. W., Segel L. A.: Phys. Fluids 16 (1973) 1573. 15] Palaniswamy V. L, Purushotham C. M.: Phys. Fluids 24 (1981) 1224. [6] Spiegel E. A.: Astrophys. J. 141 (1965) 1068.

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