The dynamics of aerosol particles in a gradient acoustic field

INT. COMM. HEAT MASS TRANSFER 0735-1933/84 $3.00 + .00 Vol. ii, pp. 563-567, 1984 ~_rgamon Press Ltd. Printed in theUnitedStates

THE DYNAMICS OF A E R O S O L PARTICLES IN A GRADIENT ACOUSTIC FIELD S.T.Morozov and V.T.Morozov K u i b y s h e v State University,

443086, Kuibyshev, USSR

(C~t,~nicated by R.I. Soloukhin and O.G. Martynenko) ABST~C T

An a p p r o a o h i s s u g g e s t e d %o o b t a i n a p p r o x i m a t e a n a l y t i o a l solutions for the problems of aerosol partiole dynmaies in an o s c i l l a t i n g medium o a r r i e r . In the oase considered, the oscillations o f t h e medium a r e due %o t h e a o t i o n o f t h e a c o u s t i c f i e l d on t h e t w o - p h a s e s y s t e m . The p r i n c i p a l r o l e of the oscillating amplitude ~radient in the drift of par%ioles toward a deoreasing osoillating amplitude is reveale d . The a l g o r y ~ h m s u g g e s t e d oan b e a p p l i e d %o t h e a n a l y s i s of similar models desoribin 8 the transient dynamos of het e r o g e n e o u s s y s t e m s w i t h low o o n o e n t r a t i o n a n d p a r t i c u l a r l y to the s~udy of partiole transfer in turbulent flow. C o n s i d e r t h e m o t i o n o f a d i s o r e t e p h a s e i n an a c o u s t i o f i e l d when t h e s p e e d o f t h e v i b r a t i o n a l motion of the medi-~ oarrier is p r e s o r i b e d and i s d ~ t e r L t n e d b y t h e r e l a t i o n of the ~rpe

v(r,t)

=

ACt)- ?(t) ,

(I)

where A ( r ) t a k e s a c c o u n t o f t h e wave a m p l i ~ a d e v a r i a t i o n along the spaoe coordinate r and ~o(%) i s t h e p e r i o d l o i t y of oscillations, w i t h % b e i n g t h e t i m e . I n c o n t r a s t %o t h e w e l l - k n o w n w o r k s [ 1 , 2 ~ , where a similar problem has been solved for the oase of a plane undamped wave (A = o o n s % . ) , i n t h i s work t h e a o o u s % t c f i e l d is oharaoterised b y ~ v / ~ r ~ 0 due t o a n o n - p l a n e o h a r a o t e r o f t h e wave a n d d i s s i p a t i v e processes - t h e c a s e more c o n s i s t e n t w i t h real phenomena. 563

564

S.T. M~rozov and V.T. Morozov

Vol, ii, No. 6

The discrete phase volume concentration is assumed %o be rather low and this enables one to construct the model ~n a single particle approximation. For simplicity, but without the loss of generality, the discussion will be restricted to the Stokes flow condition. The dimensionless spherical particle motion equation and the initial conditions in Euler's notation have the form bu -bt

bu +u--= br

~(v-u)

U ( r o , t o) = u o where

(2)

,

,

~ = 1 8 ~ L ( d ~ ~p c ) - 1

is

the

dimensionless

relaxation

pa-

rameter. I% should be noted that the Inclusion, into the initial equation, o f t h e b o d y f o r c e s a n d o f t h e %erms~ %hmt a l l o w f o r the

pressuregrmdien%

ated the

mass effect, analysis

presence

does not

of the

in the

medium

contribute

and for

any special

the

features

associ%o

model.

Realizing the oone%Tuotion of the solution as euF~ested in ref.

[3~

, a new v a r i a b l e wCr,t)

Equation

(2)

is

introduced

= uCr,t)

takes

the

- vCr,t)

(3)

form

bw by bw ~w ~," by ~ + ~ + w ~ + 'v" ~ + w ~ + v ~ + ,'rw m Op ~% bt br br br br

WCro,t o) The s o l u t i o n is parameter determined

~ = w

o

.

wo •

sought in the form of the expansion from the initial conditions

w = ~ n=l where

=

C~.)

6n W n ( r , % , ~ )

,

in the

Vol. ii, NO. 6

DYNAMICS OF AEROSOL P A R P I C L ~

565

The r e p r e s e n t a t i o n o f t h e s o l u t i o n by s e r i e s ( 5 ) a l l o w s one %o r e d u o e t h e n o n - l i n e a r problem ( 4 ) %o an i n f i n i t e r e o u r r e n t system of l i n e a r e q u a t i o n s of the type

~w I

bw I

, . + v

bt

bv

+(--

br

I

by

+~)w~ . ~ ( - -

br

by

+v--

bt

) - o,

(6)

%r

Wl(ro,% o) = I , e o e e e e e e e e e e e e e e e e e e e e e e e e e e e

e e e e e

e o e e e o e e e e e e e e e o e e e e e e e e e e e o e e e e

n-~.

bwn

bwn +v

by

+(--+~)w n+ ~

br

b%

~---

br

we ~ ~ = o

~ ,m-l

•

,

(v)

br

(t+m.n)

.nCro,to) = o, (n > l ) o o e e e o o e e e o e e o e e e e e e e e e e e e o e e e e e e e e e e e e e e e e e e e e e e e e e e ~ e e e e e o e e e e e e

The s t r u o t u r e o f ? q u a t i o n s ( 6 ) and (7) s u g g e s t s t h a t t h e s o l u t i o n Of problem (4) in the form of expansion (5) is non-trlvial in the oase of ~ = 0 either. Let us r e s t r i o t ourselves %o t h e f i r s t a p p r o x i m a t i o n ( 6 ) f o r whioh t h e e q u i v a l e n t s y s t e m o f e q u a t i o n s i n t o t a l d e r i v a t i v e s i s dw I bT i by by

v

+ (--+~)Wl--

dr

br

~

(--+v--) bt

,

br

dr --m

V

WlCro,t o) = 1 .

p

dt

Upon infestation and substitution of the result into equation (4), the solution is obtained for the partiole speed oharao%eristios

=

ro

%o(t) at

to

with allowance for equations (I) and (3) in the form

(8)

566

S.T. Morozov and V.T. Morozov

l

A(r)

Vol. ii, No. 6

A(r.) exp(-~t) it A2Cr)exp(~t)I d ~o(t) + ~2(t) aA(r) .... I dr. .~to

dt

(9)

dr

I n t h e p a r t i c u l a r . c a s e o f A = c o n s t , and ~o = sinW%, e q u a t i o n ( 9 ) y i e l d s t h e well-known s o l u t i o n [ 1 ] o f t h e S t o k e s p a r t i c l e dynamics problem in a plane undamped monoharmonic wave. The second term on the RHS of equation (9) accounts for the non-periodic relaxation stage of the particle motion due 4 ; 0 the non-equilibrium character of the initial conditions. It is easy to verify that the integral te1~ of equation (9)

t o g e t h e r w i t h t h e p e r i o d i c m o t i o n s p e e d component, w h i c h i s c h a r a c t e r i z e d by a certain phase shift angle and a certain degree of particle entrainment by an oscillating medium~ also contains a n o n - p e r i o d i c s p e e d component which a c c o u n t s f o r t h e d i s c r e t e phase drift toward a decreasing acoustic wave amplitude - a prlnoipal consequence of the gradient c h a r a c t e r of the acoustic field. The n a t u r e of t h e above e f f e c t i s s i m i l a r t o t h e phenomenon of t u r b u l e n t t r a n s v e r s e m i g l ~ t i o n of a e r o s o l s [ 4 ] • Nomenclature A dp

r

= = = = = =

amplitude fluctuation particle diameter medium cattier speed particle speed time space coordinate

o

=

L

=

characteristic characteristic

v U

t

speed (scale) linear size of model (scale) @reek 331bols

= dens i~y = visoosi~r = pulsation frequency

Vol. ii, No. 6

DYNAMICS OF AEROSOL PARTICL~

567

Subsor!~ts = particle = initial values = variable integral and expansion numbers. References

1.

0 . B r a n d t , H.Freund and E.Hiedemaun, Zur Theorie der a k u s t i sohen Koagulation, Kolloido Z. 77,N I, 103-115 (1936).

2.

D.B.Dianov, A.A.Podolsky and V.I.Turubarov~ Osoillating mo~lon of aerosol partioles in aoous%io fleld~ Kolloid. Zh. 29, N I, 69-75 (1967).

3.

V.T.Morozov and S.T.Morozov~ Small parameter method in the problems of pa~tiole pulsational mig1~ation~ in Small Parameter Methods and Their Applioations:Abs~racts of Lectures and Talks at the All-Union Sohool-Seminar at the Institute for ~athematios of %he Byelorussian Academy of Sciences,

~ l n s k , pp.97 (1982). 4.

E.P.Mednikov, Turbulent T r a n s f e r and Aero s o l D e p o s i t i o n , Nauka Publishing House, Moscow (1981).