INDUCED DUSTY FLOW DUE TO NORMAL OSCILLATION OF WAVY WALL

A two-dimensional viscous dusty flow induced by normal oscillation of a wavy wall formoderately large Reynolds number is studied on the basis of boundary layer theory in the case where the thickness of the boundary layer is larger than the amplitude of the wavy wall. Solutions are obtained in terms of a series expansion with respect to small amplitude by a regular perturbation method. Graphs of velocity components, both for outer flow and inner flow for various values of mass concentration of dust particles are drawn. The inner and outer solutions are matched by the matching process. An interested application of present result to mechanical engineering may be the possibility of the fluid and dust transportation without an external pressure. 2000 Mathematics Subject Classification. 76Txx.

the entire flow field has been classified more elaborately by taking up the fourth-order solutions for discussion. Solutions are obtained in terms of series expansion with respect to the small amplitude by regular perturbation method. The inner (boundary layer flow) and the outer (flow beyond the boundary layer), solutions are matched by a matching process given by Kevorkian and Cole [3]. Graphs of the velocity components, both for the outer and the inner flow for various values of mass-concentration of the dust particles are drawn.

Formulation of the problem.
In order to formulate the fundamental equations of motion of the two-phase fluid flows and to bring out the essential features, certain basic assumptions are made. They are as follows: (1) The fluid is an incompressible Newtonian fluid.
(2) Dust particles are assumed to be spherical in shape, all having the same radius and mass and undeformable.
(3) The bulk concentration (i.e., concentration by volume) of the dust is very small so that the net effect of the dust on the fluid particles is equivalent to an extra force αλ( q p − q) per unit volume, (given by P. G. Saffman), where q(x, t) is the velocity vector of the fluid, q p (x, t) is the velocity vector of the dust particles.
(4) It is also assumed that the Reynolds number of the relative motion of the dust and the fluid is small compared with unity.
The effect of the dust enters through the two parameters α and λ.
We consider a two-dimensional flow of an incompressible viscous dusty fluid due to an infinite sinusoidal wavy wall of amplitude a and wave length L, which is oscillating vertically with a frequency ω/2π and an amplitude a cos(2π x/L), x being the coordinate in the down stream direction of the flow and y, the coordinate perpendicular to it. The motion of the wall is described by where a is the amplitude of the wavy wall, L is the wave length, and ω is the frequency. The equations of conservation of momentum for the fluid and the dust are given by where p is fluid pressure, K is Stokes resistant coefficient, m is the mass of a particle N o is the number density and µ is the coefficient of viscosity. Here we assume that (a/L) 1. We normalize all lengths by characteristic length (L/2π), all velocities { q and q p } by characteristic speed (Lω/2π), the fluid pressure p by (ρL 2 ω 2 /4π 2 ), the time by characteristic time (1/ω). The above equations of motion of the fluid and dust become q t + ( q ·∇) q = −grad ·p + 1 R ∇ 2 q + αλ q p − q , (2.3) q p t + q p ·∇ q p = α q − q p , (2.4) div q = 0, div q p = 0, (2.5) where the nondimensional parameters concerning the dust are , (2.6) and Reynolds number R = (L 2 ω/4π 2 ν), K = 6πµa, where c is the characteristic speed, ν is the kinematic viscosity and α, λ are the dust parameters. The boundary conditions are

7)
|u|, |v|, |up|, |vp| < ∞ as y → ∞, (2.8) where h = ε cos x sin t and ε = (2π α/L) 1. Equation (2.7) represents no slip condition of the fluid on the wall. By introducing the stream function Ψ (x,y,t) and Ψ p (x,y,t) for the fluid dust, respectively, the governing equation (2.3), (2.4), and the boundary condition (2.7), (2.8) become 3. Solution of the problem. When Reynolds number becomes large, the boundary layer is formed. As we have assumed that the thickness of the boundary layer is larger than the wave amplitude, following Tanaka [5], regular perturbation technique can be applied to the present problem. If δ is the thickness of the boundary layer, the nondimensional may be defined asȳ = (y/δ) andΨ = (Ψ /δ). When the viscous term is supposed to be of the same order as the inertia terms, we have that δ 2 R is 0(1) as usual. The boundary conditions at y = h are expanded into Taylor series around h = 0 in terms of the inner variablesΨ andȳ as In order that Taylor series converges, 0(δ) must be larger than 0(h), that is, 0(ε) < 0(δ). Following Tanaka [5], we take δ = r ε 1/2 , r being an arbitrary constant of 0(1).

(3.17)
A series of the inner solutions should satisfy the boundary conditions on the wall, while the outer solution are restricted to be bounded as y increases, that is, It is necessary to match the inner and outer solutions. Following Keverkian and Cole [3], the matching is carried out for both x and y components of velocity of fluid and dust by the following principles: up to Nth order of magnitude. We seek solutions of first order in the following form: where * denotes complex conjugation. By substituting (3.26) in the first-order differential equations (3.4), (3.6), and the boundary condition (3.7), we obtain the following system of equations: and their solutions are When (3.32) are substituted in (3.29) it turns out to be a 1 = a 2 , where where c.c. stands for the corresponding complex conjugate. Taking into account that y = r ε 1/2ȳ expanding the exponential as decays very rapidly as ε → 0 (which is called transcendentally small term (T.S.T.) and is neglected in the matching process). We have Thus the matching condition is satisfied only if when similar process is carried for (3.16) we get The first-order solution is obtained as (3.42) Next we seek second-order solution Ψ 2 , Ψ p 2 ,Ψ 2 , andΨ p 2 in the following form: Substituting (3.42) into (3.8), (3.9), and (3.11), we get after calculations, We seek third-order solutions in the form Ψ 3 = sin 3xF 3 e 3it + sin 2xF 32 e 2it + sin xF 31 e it + c.c. + F 3s sin 2x, It is to be noted that the third-order inner solutions both for the fluid and dust have a steady streaming components F 3s and F p 3s . However, they attenuate very rapidly as y increases and are confined only in the boundary layer while no steady streaming is induced in the outer layer up to this order of approximation. Next we seek the fourth-order solution in the following form: where

Results and discussion.
We have seen that the third-and fourth-order solutions consist of the steady part in addition to the periodic one. As the contribution of steady term in the fourth-order is more significant to the solution, we take up the fourth-order solution for discussion.
We observe that the normal oscillation of the wall causes at first the periodic flow in the boundary layer having the same frequency as that of the wall oscillation and then it causes the flows of higher harmonics in the boundary layer and induces periodic flow in the outer flow successively. The components of the velocity of the fluid both for outer and inner flow have been plotted against y andȳ in Figures 4.1 to 4.13, respectively, for various values of x and t, taking R = 500, ε = 0.05.
The inner steady streaming parts of both fluid and dust are plotted againstȳ for various concentration parameter λ (Figures 4.1 and 4.2). We find that both in the case of fluid and dust the inner steady streaming parts approach a constant value in the form of the oscillation with respect to the distance from the wall.
The various values of the dust parameter λ make the velocity profiles of both the fluid and the dust well separated for small values of t. For a given x, they approach a constant value as y increases.
We observe from                 The behaviour of velocity components U i and U p i of the fluid at dust, can be studied from Figures 4.8 and 4.9, respectively. Here we see that U i is more oscillatory than U p i . But both become steady as y increases. The separation of profiles is felt more in a particular range of values of y. The presence of dust seems to have more impact on the fluid than on the dust as seen from the oscillatory nature of the fluid motion.
We study V i and V p i from Figures 4.10 and 4.11. It is very much interesting to note that with the increase of y while the dust parameter λ makes the profiles of the dust more separated it makes the profiles of the fluid more closer.
From Figures 4.12 and 4.13 we study the behaviour of inner transverse steady velocity components V is and V p is of the fluid and the dust. Unlike in other cases the presence of dust has got more impact on the motion of the dust than on the fluid.