F o r any t and z, e x p r e s s i o n s (2.4) s a t i s f y t h e l a s t equation of (2.3) and all the boundary conditions exactly, while the f i r s t equation of (2.3) is s a t i s f i e d a p p r o x i m a t e l y since as t - ~ and z -< h Ll(cl, c~) ~ coa(Ut - - h ) - l / ~ exp { - - 2 o U - l ( U t

- - h)l/~} + O.

It is seen f r o m (2.4) that the r e a g e n t concentrations in the p h a s e s n e a r the r e a c t o r e n t r a n c e equalize with the lapse of t i m e , r e a c h i n g the value e ~ c + ~ 0. Let us note that the a p p r o x i m a t e e x p r e s s i o n for the concentration c(x, t) in the continuous phase (2.4) yields the exact value at the r e a c t o r e n t r a n c e for z = 0. This is p r o v e d by d i r e c t integration of the f i r s t equation in (2.3) for z = 0 with the equality c+(t, 0) = 0 taken into account. LITERATURE 1.

2. 3.

4.

5.

CITED

Yu. P . Gupal0, A. D. Polyanin, and Yu. S. R y a z a n t s e v , "On diffusion to a chain of d r o p s (bubbles) for high P~elet n u m b e r s , " Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1 (1978). A. D. Polyanin, "On the diffusion i n t e r a c t i o n between d r o p s in a fluid," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2 (1978). Yu. P. Gupalo, A. D. Polyanin, and Yu. S. R y a z a n t s e v , "Moving p a r t i c l e interaction effects in the m a s s t r a n s f e r in r e a c t i n g d i s p e r s e d s y s t e m s , " in: Sixth Int. Colloq. on G a s d y n a m i e s of Explosions and R e active S y s t e m s , Stockholm Sweden, AIA (1977). S. Waslo and B. G a l - O r , " B o u n d a r y - l a y e r t h e o r y for m a s s and heat t r a n s f e r in clouds of moving drops, bubbles or solid p a r t i c l e s , " C h e m . Eng. Sci., 26, No. 6 (1971). B. P. L e c l a i r and A. E. H a m i e l e e , "Viscous flow through p a r t i c l e a s s e m b l a g e s at i n t e r m e d i a t e R e y nolds n u m b e r s . S t e a d y - s t a t e solutions for flow through a s s e m b l a g e s of s p h e r e s , " Ind. Eng. Chem. F u n d a m . , 7, No. 4 (1968).

LASERCAVITATION

IN

LIQUID

NITROGEN

P. I. Golubniehii, P. I. Dyadushkin, G . S. K a l y u z h n y i , S. D . K o r c h i k o v , and V. G. Kudlenko

UDC 620.193.6

Since the f i r s t d i s c o v e r y of l a s e r cavitation in liquids [1] a l a r g e n u m b e r of studies of this p h e n omenon have a p p e a r e d . The p r o b l e m is of i n t e r e s t b e c a u s e this is in p r a c t i c e the only way of :producing an isolated c a v i t a t i o n bubble within a liquid (with e l e c t r i c a l d i s c h a r g e s distortions a r e produced by the p r e s ence of the e l e c t r o d e s ) and also b e c a u s e of the uncertainty surrounding the s t a t e of the m a t e r i a l r e a l i z e d when such a c a v i t y c o l l a p s e s . Studies have been made of the dynamics of bubbles f o r m e d by l a s e r breakdown in a liquid using a technique b a s e d on r e c o r d i n g of a c o u s t i c a l and light i m p u l s e s produced during bubble f o r m a t i o n and c o l l a p s e [1], with h i g h - s p e e d photography [2], and by the shadow method with background illumination by a gas l a s e r [3]. The goal of the p r e s e n t study is to investigate the l a s e r cavitation in a m o s t - s i m p l e c r y o g e n i c liquid liquid nitrogen. Due to the c l o s e n e s s of the liquid nitrogen t e m p e r a t u r e to the boiling point, the p r e s s u r e within the cavitation c a v i t y at the l a t t e r ' s m a x i m u m d i m e n s i o n s , d e t e r m i n e d b a s i c a l l y by the s a t u r a t e d nit r o g e n v a p o r p r e s s u r e , will differ only insignificantly f r o m the e x t e r n a l p r e s s u r e and the d e g r e e of bubble c o m p r e s s i o n R / r will be s m a l l (here R is the m a x i m u m ; r, the m i n i m u m bubble radius, r e s p e c t i v e l y ) . The t e m p e r a t u r e T within the bubble at m a x i m u m bubble c o m p r e s s i o n can be written in the adiabatic a p p r o x i m a t i o n as

T = To(R/r)3~v-1), V o r o s h i l o v g r a d . T r a n s l a t e d f r o m Z h u r n a t P r i k l a d n o i Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 103106, S e p t e m b e r - O c t o b e r , 1979. Original a r t i c l e submitted October 10, 1978.

0021-8944/79/2005-0609507.50

9 1980 Plenum Publishing Corporation

609

Fig. 1

Fig. 2

8 [s

zl ~----

i

..Jr ~ P.atm 0

0,5

".L:

f,f z atm

3

Fig. 3

Fig. 4

61

65

69

Fig. 5

610

~

73

T , OK

2

w h e r e T o is the initial t e m p e r a t u r e ; ~/= 4 / 3 is the adiabatic index. As will be shown below, at a t m o s p h e r i c p r e s s u r e R / r - 3. In this c a s e the t e m p e r a t u r e within the bubble does not exceed 240~ Thus t h e r e will be no light radiation at the m o m e n t of bubble c o l l a p s e . R e g i s t r a t i o n of a c o u s t i c a l p u l s e s developed upon bubble collapse is c o m p l i c a t e d by s t r o n g r e f l e c t i o n s f r o m the c r y o s t a t walls of the a c o u s t i c a l radiation g e n e r a t e d during breakdown. Thus, it is d e s i r a b l e to use e i t h e r h i g h - s p e e d photography or the shadow method to study cavitation bubble d y n a m i c s . The l a t t e r technique is methodologically s i m p l e r and g e n e r a t e s sufficient information on the c a v i t y (value of the r a t i o R / r and cavity pulsation period), so it was chosen for the p r e s e n t study. A block d i a g r a m of the equipment used is shown in Fig. 1. The b e a m f r o m n e o d y m i u m l a s e r 1 is foc u s e d by lens 2 with focal length 25 m m to a point within the liquid nitrogen, located in optical c r y o s t a t 3. The l a s e r pulse length was 10 n s e c , with a m a x i m u m e n e r g y level of ~ 3 9 10 -2 J. T h e s y s t e m was adjusted s o that breakdown o c c u r r e d on the s t r a i g h t line connecting the h e l i u m - n e o n l a s e r 4, used for background illumination, and the FI~U-13 p h o t o m u l t i p l i e r 6, in front of which a KS-13 red filter 5 was installed. The light filter was needed to attenuate the light radiation f r o m the breakdown s p a r k incident on the photomultiplier. A t y p i c a l o s c i l l o g r a m of c a v i t y pulsations is shown in Fig. 2. Since the c a v i t y d i m e n s i o n s w e r e much l a r g e r than the gas l a s e r wavelength, the effective light s c a t t e r i n g section m a y be c o n s i d e r e d equal to ~l 2, w h e r e I is the bubble r a d i u s . Then the value of the r a t i o R / r m a y be obtained f r o m the e x p r e s s i o n tllr = (Ala)li 2,

w h e r e A is the m a x i m u m and a, the m i n i m u m , signal amplitude on the s h a d o w g r a m . This r e l a t i o n s h i p m a y be v e r i f i e d in the following m a n n e r . It is known that the bubble pulsation period T depends on the value of the m a x i m u m radius R as [4] v = t,83 R(o/P)'/2, w h e r e p is the liquid density and P is the s u r r o u n d i n g p r e s s u r e . f i r s t and second bubble p u l s a t i o n s we obtain

Then for the r a t i o of the p e r i o d s of the

T1/~. = (A1/A.)ll ~, w h e r e the index 1 r e f e r s to the f i r s t pulsation and 2, to the second. This r e l a t i o n s h i p p r o v e d valid for all shadowgrams. F i g u r e 3 shows the dependence of the period of the f i r s t pulsation upon the e x t e r n a l p r e s s u r e . The solid line shows the t h e o r e t i c a l function ~- ~ P - 5/6 , valid for the condition of constant e n e r g y expended by the c a v i t y [4]. It is evident that in the p r e s s u r e r a n g e above 1 a i m the e x p e r i m e n t a l data a g r e e well with theory. Below 1 a i m t h e r e is significant d i v e r g e n c e between t h e o r y and e x p e r i m e n t , which can be explained by the deviation of the c a v i t y f r o m a s p h e r i c a l shape. The amount of such distortion i n c r e a s e s with reduction in p r e s s u r e , due to i n c r e a s e in cavity d i m e n s i o n s , and consequent reduction in the stability of its f o r m . This conclusion m a y also be c o n f i r m e d by a n a l y s i s of s h a d o w g r a m s , which show significant distortions in pulsation f o r m at low p r e s s u r e s . F i g u r e s 4 and 5 show the dependence of the r a t i o R / r upon e x t e r n a l p r e s s u r e and t e m p e r a t u r e . A c c o r d ing to [5], the value of the r a t i o R / r is d e t e r m i n e d by the gas content p a r a m e t e r 75 = p / P , w h e r e p is the p r e s s u r e within the c a v i t y at m a x i m u m bubble r a d i u s , n a m e l y B/r = (1 + 38 - - 6~.6)/38.

(1)

In a f i r s t a p p r o x i m a t i o n one m a y c o n s i d e r the p r e s s u r e within the cavity at its m a x i m u m d i m e n s i o n s to be equal to the p r e s s u r e of s a t u r a t e d nitrogen v a p o r . R e s u l t s of calculations with Eq. (1) and this a s s u m p t i o n a r e shown by the lines in Figs. 3-5. It is evident that the e x p e r i m e n t a l data a g r e e well with Eq. (1). T h u s , we m a y conclude that the d y n a m i c s of a cavitation cavity in liquid nitrogen a r e d e s c r i b e d well by the known e x p r e s s i o n s for a conventional liquid, with the exception of the l o w - p r e s s u r e r a n g e , in which nons p h e r i c a l c a v i t y f o r m begins to play a r o l e . The d e g r e e of c o m p r e s s i o n (ratio R / r ) in liquid nitrogen is low, not exceeding a value of six (at an e x c e s s p r e s s u r e of 2 a i m and t e m p e r a t u r e of 65~ as a consequence of

611

which bubble c o m p r e s s i o n is not of a s e v e r e c h a r a c t e r (collapse), leading to the e m i s s i o n of an acoustical pulse, c o m p a r a b l e in magnitude to the pulse produced at breakdown. To i n c r e a s e the d e g r e e of c o m p r e s s i o n it is n e c e s s a r y to d e c r e a s e the value of the gas content p a r a m e t e r , which can be done by increasing the e x ternal pressure. LITERATURE 1. 2. 3. 4. 5.

CITED

A . A . Buzukov and V. S. Teslenko, "Sonoluminescence upon focusing of l a s e r radiation within a liquid, ~ P t s m a Zh. Eksp. T e o r . F i z . , 14, No. 5 (1971). A . A . Buzukov, Yu. A. Popov, and V. S. Teslenko, WExperimental study of the explosive p r o c e s s g e n e r a t e d by focusing a l a s e r monopulse in w a t e r , ~ Zh. P r i k l . Mekh. Tekh. Fiz., No. 5 (1969). A . G . Akmanov, V. G. BenTkovskii, P. I. GolubnichU, S. I. Maslennikov, and V. G. Shemanin, n L a s e r sonoluminescence in liquids, n Akust. Zh., 19, No. 5 (1973). K . A . Naugol~nykhand N. A. Roi, E l e c t r i c a l D i s c h a r g e s in W a t e r [in Russian], Nauka, Moscow (1971). M . G . Sirotyuk, ~Experimental studies of u l t r a s o n i c cavitation, Win: H i g h - P o w e r Ultrasonic Fields [in Russian], Nauka, Moscow (1968).

STRUCTURE IN A POROUS

OF

A SHOCK

WAVE

FRONT

SOLID

S. Z. Dunin

and

V. V. Surkov

UDC 534.222.2

The investigation of the n a t u r e of wave propagation in substances with a disruption of continuity is i m portant for s e v e r a l reasons: The study of shock heating of a porous m a t e r i a l in high-intensity waves m a k e s it possible to deduce the equation of state of the continuous m a t e r i a l under anomalous conditions (megabar p r e s s u r e s and t e m p e r a t u r e s of the o r d e r of the melting point) [1]; the m a j o r i t y of m a t e r i a l s a r e not continuous in nature, and the wave propagation p r o c e s s is l a r g e l y d e t e r m i n e d by the actnal s t r u c t u r e of the solid. In the investigation of shock waves in solids with a disruption of continuity it is e s s e n t i a l to take the following considerations into account. F i r s t , as in the analysis of a shock front in gases with r e t a r d e d excitation of c e r l a i n d e g r e e s of f r e e d o m [1], the s t r u c t u r e of the shock t r a n s i t i o n in porous solids must be investigated with r e g a r d for the i n e r t i a l p r o p e r t i e s of the m e d i u m [2-4]. Thus, in the shock loading of a porous solid to p r e s s u r e s in the tens of kilobars (such that the influence of heating of the substance can be neglected), the s t r u c t u r e of the wave front is affected by the p o r e - s e l e c t i o n dynamics [2-4]. An investigation of this type indicates that the p r e s s u r e in the substance depends not only on the density of the substance, but also on its d e r i v a t i v e s . Second, a n u m b e r of t h e o r e t i c a l and e x p e r i m e n t a l studies [3-8] suggest an appreciable influence of the viscous p r o p e r t i e s of the p o r o u s m a t e r i a l on the nature of the propagation and attenuation of shock waves. T h i r d , e s t i m a t e s [2-4, 9] show that the p o r o s i t y changes significantly only when the e n t i r e m a s s of the solid substance e n t e r s into the ductile state. In the p r e s e n t study we discuss the c h a r a c t e r i s t i c s of low-intensity shock wave propagation, where the influence of heating of the s u b s t a n c e can be neglected (tens o f k i l o b a r s ) , but the actual nature of wave p r o p a gation is l a r g e l y d e t e r m i n e d by the b e h a v i o r of the p o r o u s solid in the ductile state, viz., the p o r e - s e l e c t i o n dynamics e x e r t s a strong influence on the wave s t r u c t u r e . 1. The shock profile is investigated in the example of a plane s t a t i o n a r y wave propagating with velocity D. In this c a s e all physical quantities (density, p a r t i c l e velocity, etc.) turn out to depend only on one v a r i a b l e = x - IX, and the equations of m a s s and m o m e n t u m conservation a r e e a s i l y integrated. Considering media of low p o r o s i t y , we can neglect the dependence of the s t r e s s deviator on the p o r o s i t y factor [10] and r e g a r d it as constant, with a value close to the yield point of the solid. Then in a coordinate s y s t e m attached to the shock wave the equations are written in the f o r m poD ----p(D

v), p -- Po ----povD,

(1.1)

Moscow. T r a n s l a t e d f r o m Z h u r n a l Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 106-114, S e p t e m b e r - O c t o b e r , 1979. Original a r t i c l e submitted S e p t e m b e r 27, 1978. 612

0021-8944/79/2005-0612 $07.50 9 1980 Plenum Publishing Corporation