The flow of an incompressible viscous dusty gas, induced by two infinite flat plates oscillating in their own planes, is studied. Expressions for both the gas and the dust velocities are derived using the Laplace transform. The velocity profiles are drawn for different for different configurations of the dust particles and the results are compared with those of the clean gas

Vimala, C. S.

English

Defence Science Journal

FLOW OF A DUSTY GAS BETWEEN TWO .OBCILLATING PLATES Indian Institute of Science, Bangalore (Eeeeived 1 ~ d n e  1971 ; revised 4 - September 1971) I The flow of an incompressible viscous dusty gas, induced by two infihite flat labs oscillating in their own planes, is studied. Expressions for both the gas and the dust velocities are derive$using the Laplace tran8fom. The velocity profiles are drawn for different configurations of the dust particles aqd the results are compared with , those of the clean gas. . The contamination of air in cities by dust particles has necessitated the study of the flow of dusty gases. Saffman1. formulated the basic equations for the flow of a dusty gas. Since then, there have been several papers in this field. Regarding the plate problems, Liu2 and Michael & Miller8 have studied the flow produced by the motion of an infinite plane in a dusty fluid occupying the semi-infinite space above it. Recently Khan4 has discussed the flow of a viscbus liquid between two harmonically oscillating inhi te  plates, when constant body force is applied initially in the direction of motion of the plates. In this paper, the flow of a dusty gas between two infinite flat plates oscillating in their own planes is investigated The Laplace transform teohnique is used in solving the problem. e G O V E R N I N G  E Q U A T I O N S  A viscous, inoompessible, dusty gas bounded By two infinite flat plates executing simple harmonic oscillations with a frequency 'o' in their own planes y = a is considered. Both the gas and the particle cloud are supposed to be static3 at  the Beginning. The dust particles are assumed to Be spherical in shape and uniform in size so that the conservation equations given by Saffisbnl are applicable. The number density of the particles is taken as a constant throughout the flow. Under' these assumptions, the flow will be a pardlel flow in which t;ho streamlines are along the x-axis and the velocities are functions of the distance y and time t. 1 Now let y and t denote the non-dimensionalised distance and time, with reference to a and - respec- t W tively. Let Re be the Reyrzolds' number. Let f be the mass concentration of the dust particles and let 7 be 1 the relaxation time of the dust particles, non-dimensionalised with respect to - . If zc and v are the velooi- W ties of the gas and the dust respectively, non-dimensionalised with respect to aw , then Sakm's equations1 reduce to the following simultaneous equations : which are to be solved under the following initial and boundary ctonditions : Initial Conditions @ u = v = O a t t = O f o r a U y  (8) Boundpry Conditions u = u , s i n t a t y =  A 1. (4) Since the flow is symmetrical about the plane y=O, only the flow in the region 0 < 5 4 1 is consi- dered and accordingly the boandary conditions (4) are equivalent to the foUotv;ng : u = u,, sin t at 9 = 1 . . VIMALA :Qusty Gas Between Two Os$illating Plates 2 t Q, n=O for 0 2 y.< 1, t - 7 COB t + 72 e -t/7 ., at y r= 1, (13) where . u E = E O S ~  (dg cas ((z/G, 3 = sini &in (a, . ,  (14) E, = wsh (y2/X) cos (ydY), P, = sinh (ydx) sin fg2/fl, where X and Ylare given by with and . h p1 = 4 + B, B = A - B  (17) where < 8Re  B = - -  16 Re2 (1 + f p + 8 Ro T (2 + 1)2 Z (f - 1) + (2 (2 + r4 P}' (1s) I 8 R e o  1 For 8 clean gas ( i . ~ . ,  for a gas without dust partidesk the velocity of the gas ia given b ~ 4  * - ( 2 n + l ) a Z  (- 'p  ( 2 n + 1 ) e ~ [  * 4Re u, = 16 uo n Re [ l 6 ~ & +  ( 2 n + l y #  J + [ %  / { ( . . B ~ ~ R T ~ ~ R - ) ~  + ( a i n h d ~ ~ d ~ ) 2 ) ] .  + sinh (y @$) sinb ( 4 R q )  sin (y1/Re/2) sin (dm)] + -+ cos t ( $ i d  ( y 4 ~ ~ )  oosh (4-1 sin (y 4 ~ 2 )  Go8 ( 4 ~ 2 )  -DEB. 801. J., VOL. 22, OOTOBER 1972 As t + co, eilt a d  e ~ d  vanish (For, p, and fi are the roots of the qudtatic equation with no change of sign and are obviously negative) so that there is no cont,ribution to the velmities from ' the summation terms. Hen& the solutions for the asymptotic case t -t co are ; 26 = Ey (E sin t - F cod-t) + Pg(E COB t-4- B sin t )  I for o < g < I, (20) + F~{(F + ET) sin t + (E -FT; cos t 13 v = sint-T cost at  y = 1, I which are periodic functions. D I S G U S B I O N  A N D  C O N C L U B I O N S  The velocities 4, u and v are ca1c;lated - - For times t = 50, 100,500 taking the parameters characterizing' the flow as follows : Re = 10" ; u, = 0.1 ; f = 0.1, 0.2 ; and T = 0.05,O.l. ' Rg3. 1-3 represent the variations of u,, u and v with y at  different times. The velocity profiles for zc are similar to that of the clean gas showing that the presence of dust does not have any effect on the manner in which u varies with y . Furthermore, the dust velocity v varies in the same fashion as u does (except for t = 100, for wbich there is a discrepancy between the shapes of the u and v profiles which may be due to the VIMALA :Dusty Gas 'getween Two Oscilfating Plates J Fig. %-Velocity distributiolls at t=100. .:: - Fig. 3-Velocity distributions at t=6001 capability, of the scale chosen there to represent velocities mrreot ta five decimal places so that the change in the behaviours of zc and v is seen only afi,er the fourth decimal place). I t  is evident$ from the graphs that, as we go away from the plate, the ahange in f has an appreoiaMe effect on u compared to the very smdl effect due to a change in r,  whereas a change T hae much more influence on v than a change in f has. On the other hand, very near the Boundmy, the gas velocity is un- sffeoted by the presence of dust while the dust velocity is the same for all f but depends on I. ' Henue it may be concluded that the gas velooity depends more on the mass ooncentration of the dust p&icles than on their size. On the contrary, the sire of the dust particles has more influence on the velocity of the dust particles than their concentration itself has. Aftbr s long time, the velocities become periodic but are still dependeqt on the siee and oonoe~tr~tion f dust particles in the gas. 2% 4 BPP.EN.DIX The inverse Laplam t r ~ f o r m a , ~ ,  v of i res- b pectively are given by the h t e w  r + i w  r + 1 1 - 1 eP" ii dp and - 2 ~ i  2-w 8' I Bi dp r- ioa v i w  I which oan be evaluated by means of contour integra- tion. Since there is no branch point, theaontow C chosen is the closed curve ABC formed by the line s=r  and a semicircle C with origin as centre and radius R (Fig. 4) so that . ' .  r + i m  S e @ i i  dp = ,R -P ~o ep' ii dp FI-- iQa A - [ f ept ii dp -J 8 u dp] -R?c Oo A - A;BC 0 I Fig. Montour  f o q e d  by line x=r and a semi oirole U ~ k g  Cauchy's theorem of residues and Jordan's C with orihn as ~entre and radins'R. lemma, we have 1 . = -  2 a i  e"* E dp = sum of the residues of ( ept ii ii) at its poles r -ioa Similar1 y, r + i m  1 .=- J 2 ~ i  ept Gdp = sum of the residues of ( ept i ) at its poles r-ioa Calmdating the residues andsimplifying further, we obtain expressions (11) to (13) for u and u. A C K N O W L E D G E M E N T S  The author wishes to thank Dr. Girishwm Nath for suggesting the problem and for his eeteemed guidance and encouragement throughovt the preparation of this paper. The author is also grateful to the Indian Institute of Science, Bangalare for giving the scholarship and for providing all the facilities. 6 ,  E E P E B E N C E S  f.- Sp~maa, P.  G., J .  Fluid Meclr., 18 (1982), 120. . . 2. LID, J. T. C., P?q. Fluid#, 9 (1966), 1716. . e . p. 'bmum, a. H., & ~ Q ~ L L w ,  D. $., Y9he7nato'kai 18 (1968), 97, 4. K w ,  M .  A,.&., Dqf.'&i. J. ,  19 /19691,13!3, A I 1 .  

Flow of a Dusty Gas Between Two Oscillating Plates

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Flow of a Dusty Gas Between Two Oscillating Plates

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