NONLINEAR WAVES IN CAPILLARIES

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NONLINEAR WAVES IN CAPILLARIES

L. Smirnova, I. Shreiber

To cite this version:

L. Smirnova, I. Shreiber. NONLINEAR WAVES IN CAPILLARIES. Journal de Physique Colloques,

1990, 51 (C2), pp.C2-93-C2-96. �10.1051/jphyscol:1990223�. �jpa-00230557�

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ler Congres Fran~ais d'Acoustique 1990

NONLINEAR WAVES IN CAPILLARIES

L.P. SMIRNOVA and I.R. SHREIBER

Institute of North Problems Development, 625003, Tyumen

-

^{3 ,}^{~ / b}^2774.

U.R.S.S.

A b s t r a c t

-

'Phe paper c o n s i d e r s t h e wave propagation i n e l a s t i c e n c l o s u r e s f i l l e d i n w i t h v i s c o u s f l u i d . It is shown t h a t i f hydrodynamic n o n l i n e a r i t y i s t a k e n i n t o account, t h e phenome- non can be d e s c r i b e d by a "multiwavem equation. Noncorrectness of reducing t h e e q u a t i o n s t o t h e e v o l u t i o n a l ones is shown f o r t h e case when f l u i d i s viscous. The case o f n o n l i n e a r wave propagation i n c a p i l l a r i e s i s t r e a t e d . The model e q u a t i o n is o b t a i n e d and t h e wave s t r u c t u r e i s c a l c u l a t e d .

While c o n s i d e r i n g t h e wave propagation i n t u b e s f i l l e d i n w i t h l i q u i d , t h e models t a k i n g i n t o account o n l y two t y p e s o f n o n l i n e a r i t y , i.e. "hydrodynamic" o r H d i f f u s i o n a l u ones a r e g e n e r a l l y used though t h e y should allow f o r t h e n o n l i n e a r i t y of e n c l o s u r e deformations, l i q u i d s t a t e ( l i q u i d rheology) e q u a t i o n s and t h e reology of t h e w a l l m a t e r i a l .

'Phus i n / I / i t h a s been shown f o r t h e case of i n v i s o i d f l u i d w i t h due

account o f hydrodynamic n o n l i n e a r i t y and enclosure t o r q u e t h a t t h e wave pro- p a g a t i o n i n t u b e s f i l l e d i n w i t h l i q u i d can be d e s c r i b e d by t h e e q u a t i o n o f t h e form

which i s c a l l e d t h e slmultiwavew equation. Here ^*)o ⁼

E ~ W "

_ist h e resonance frequency, ^C^, i s t h e sound v e l o c i t y i n a l i q u i d ,

424X,&d ,

G % ~ , ~ 4 * / / 4 ~ + 4 7 ;

.L?

, d

i s t h e Young module, Poisson c o e f f i c i e n t and d e n s i t y of t h e w a l l materiai/:espectively,

d? , h

a r e i t s r a d i u s and t h i c k n e s s ;

4

i s t h e

l i q u i d density.

P o r s m a l l f r e q u e n c i e s Eq. ( I ) can be reduced t o t h e modified Boussinesq equa- t i o n of t h e form

o r ^fort h e waves propagating t o one side-to t h e modified Korteweg-de-Vries e q u a t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990223

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CZ-94 COLLOQUE DE PHYSIQUE

I f k o r q u e l e s s e n c l o s u r e s a r e considered, t h e n t h e c l a s s i c a l Korteweg-de- Vries e q u a t i o n follows from Eq. (1 )

but f o r h i g h f r e q u e n c i e s ( a t A 7 % ,

1 -

t h e wavelength) t h e Klein-Gordon e q u a t i o n f o l l o w s

For g e n e r a l i z a t k o n , in t h e case o f v i s c o u s f l u i d one should t.ake i n t o

account 2w

!:

L,

-

t h e boundary s h e a r s t r e s s ) . 'Po determine Z, one can use t h e s o l u t l o n of t h e problem on n o n s t a t i o n a r y s h e a r s t r e s s on t h e p l a t e s u r f a c e at unsteady-state flow n o t i o n /2/

8

is t h e c o e f f i c i e n t of dynamic v i s c o s i t y ,

Having r e s t r i c t e d o u r s e l v e s t o t h e problems, i n which t h e e f f e c t s of non- s t a t i o n a r y deformations a r e not taken i n t o account, l e t us c o n s i d e r two ca;

s e s .

p 4 d A (

6

i s t h e depth of v i s c o u s l a y e r development i n The case when &I

a wave, $ i s t h e kinematic v i s c o s i t y of l i q u i d , L3 is t h e s i g n a l frequency) and t h e model e q u a t i o n i s of t h e form

P i s pressure.

Note t h a t one of t h e b a s i c q u e s t i o n s a r i s i n g i n t h i s problem concerns t h e v a l i d i t y of passing t o e v o l u t i o n a l equations.

A s i s known, t h e wave o p e r a t o r can be f a c t o r i z e d , a s a r e s u l t Eq. ( 6 ) i s reduced t o t h e e v o l u t i o n a l e q u a t i o n of t h e form

which i n t h e l i n e a r case at i n i t i a l "stepwise" p e r t u r b a t i o n h a s an e x a c t s o l u t i o n

Pig. 1 shows t h e s o l u t i o n s : curve I

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Eq, ( 6 ) , curve 2

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^Eq.⁽⁷⁾ i n t h e form of Eq.(8), curve 3

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c a l c u l a t i o n by t h e method /3/,

A l l t h e s e curves converge, if t h e l i q u i d v i s c o s i t y decreases. Zq.(6) i s t h e r e s u l t o f t h e s o l u t i o n o f two p r o b l e m , i,e. the nhypexbolicn one p r e s e n t i n g t h e wave propagat i o n and t h e "parabolic -one p r e s e n t i n g pulse d i f f w i o n i n

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curve 3

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s o l u t i o n by t h e method (Holmboe

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^Rouleau, ¹⁹⁶⁷⁾

a wave, t h e r e f o r e a t l a r g e v a l u e s o f t h e c o e f f i c i e n t of i n t e g r a l i n t h e right-hand p a r t , i.e. when v i s c o u s s p r e a d i n g dominates o v e r t h e wave r u n n i n g p a s s i n g from Eq. ( 6 ) t o ~ q . (7 ) i s i n c o r r e c t .

Pig. 2 F i n i t e s i g n a l e v o l u t i o n a t d i f f e r e n t v a l u e s o f t h e c o e f f i c i e n t Pig. 2 shows t h e c a s e s of t h e e q u a t i o n s o l u t i o n when t h e i n i t i a l perturba- t l o n i s of t h e t * b e l l l l shape, One can s e e t h a t when t h e c o e f f i c i e n t e q u a l s 0.002 t h e 'IbellH i s d i v i d e d i n t o two p e r t u r b a t i o n s evolving t o t h e " r i g h t u and t o t h e ' * l e f t H and a t t h e v a l u e 0.02 t h e " b e l l t * i s n o t s p l i t t e d and evolves a s a whole. This proves a d d i t i o n a l l y t h a t p a s s i n g t o e v o l u t i o n a l e q u a t i o n s w h i l e consideQng v i s c o u s f l u i d s i s i n v a l i d .

The second l i m i t i n g c a s g w h e n t h e c o n d i t i n n

g > d

i s f u l f i l l e d f o r p e r t u r - b a t i o n , i,e. t h e v i s c o u s s u b l a y e r h a s time t o develop up t o t h e tube r a d i u s and more during the time l e s s t h a n t h e p u l s e d u r a t i o n . I n t h i s c a s e one can c o n s i d e r t h a t t h e v e l o c i t y p e r t u r b a t i o n s i n a wave a r e described by t h e Poise u i l l e e q u a t i o n , i.e.

, -

s e c t i o n average v e l o c i t y

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C2-96 COLLOQUE DE PHYSIQUE

Having w r i t t e n t h e d i s c o n t i n u i t y e q u a t i o n i n t h e form

and preserved t h e assumptions, we s h a l l o b t a i n t h e e q u a t i o n of t h e form

With i n i t i a l c o n d i t i o n P / G ~ ~ ' P % - Eq. (10) h a s a s t a t i o n a r y s o l u t i o n i n Pig.3, i.e. t h e s t a t i o n a r y p r e s s u r e wave w i t h a s t e e p p r o f i l e propagates i n a c a p i l l a r y .

I , Orenbakh, Z .Me

, -

S&xnova L. P.

and S h r e i b e r I.R., Acous. J.

4 , ~ 0 1 . 3 3 (1987).

2. Landau L.D., L i v s h i t s E.M..

Continuum ~ G c h a n i c s

,

GITL,

-

(19541, 788.

3. Holmboe E. L. , Rouleau V.T.

m e o r e t i c a l fundamentals o f e n g i n e e r i n g calculations

.

vol. 32, (1967), 202-209.

~ i g . 3

W

v e r s u s 9