НЕЛИНЕЙНЫЕ СВОБОДНЫЕ ПУЛЬСАЦИИ ГАЗОВОГО ПУЗЫРЬКА В НЕСЖИМАЕМОЙ ЖИДКОСТИ

Ա ԱՍ Ա Ի Ի Ո Թ Ո Ի Ա Ա Ի Ա Ա ԻԱ Ի Ղ Ա Ի

59, 1_{2, 2006}

534.14.014

. .

. . Օ

Ա

և ք :

ք

, ք

: և

:

, :

G.G. Oganyan

The Nonlinear Free Oscillations of Gas–Bubble in a Noncompressible Fluid

The behavior of single spherically bubble in viscous fluid is investigation. The simplified equation for description of motion of rarefied mixture of bubble structure are used. Nonolinear equation of vibrations with quadratic nonlinearity is obtained. The exact particular it solutions are constracted.

.

,

,

. ,

, . ,

,

- - ,

.

,

, [1].

[2,3], – [1,4].

, , [3,4]. 1. .

,

.

, ,

[1,2] 2 2

2 1 1 2 1 1

3

4

,

const

2

d R

dR

dR

p

p

R

dt

R dt

dt

µ

⎛

⎞

−

= ρ

+ ρ

_⎜

_⎟

+

ρ =

2 3

2 2

2

20 20 2

,

const,

p

r

c

p

R

p

c

γ

⎛

ρ

⎞

=

_⎜

_⎟

ρ

=

γ =

ρ

⎝

⎠

(1.2)

1, 2 0 , ( ),

p

– ,

ρ

– ,

R

– ,

c

_p

c

v– ,

γ

– ,

µ

–

,

t

– . ,

-

(

p

)

p

p

(

p

)

(

)

R

R

(

R

)

p

p

₂

=

₀

1

+

′

₂

,

₁

=

₀

1

+

₁

′

,

ρ

₂

=

ρ

₂₀

1

+

ρ′

₂

,

=

₀

1

+

′

(1.3) . (1.1), (1.2)

, . . , .

(1.3) (1.2)

(

)

2

2 2 2

2

1

3

3

3

,

6

3

R

+

R

p

=

−

γ

R

+

γ

γ

+

R

−

=

ρ

(1.4)

- (1.1) ,

2 0 1

0 2

1 2 0

2 2

2

3

,

0

4

3

R

p

p

p

dt

dR

p

dt

R

d

ar

ar

ρ

γ

=

ω

=

+

−

µ

+

ω

γ

(1.5)

ar

ω

– .

(1.4) (1.5) 2

p

( )

2 2

0 1

2 2 2

2

1

3

,

3

4

,

0

3

ar ar

ar ar

p

t

p

R

R

R

R

µ

ω

α

=

γ

+

ω

γ

=

δ

=

γ

ω

+

α

−

ω

+

δ

+

&

&

&

_(1.6)

t

. (1.6)

p

₁

=

0

,

, . .

p

₁

( )

t

–

. ,

. 2. .

δ

=

0

. ,

,

:

p

₁

=

0

.

(1.6)

R

&

,

( )

R

f

c

R

R

R

2 _ar2 3 2 _ar2

3

1

3

1

3

2

1

3

3

3

1

3

ω

+

γ

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

γ

γ

+

+

γ

−

ω

+

γ

=

c

– ,

( )

R

f

, , (2.1). ,

( )

R

f

.

f

( )

R

,

c

, .1.

( )

R

f

b

3

<

b

2

<

b

1, ( .1,

2,3),

4 5

b

b

<

( .1, 1). ,

(

)

(

)

⎥

_⎦

⎤

⎢

⎣

⎡

+

γ

γ

−

+

γ

γ

=

_{2 2}

2

1

3

2

1

3

c

c

D

,

f

( )

R

, , . .

D

≤

0

.

(

)

2

1

3

2

0

+

γ

γ

≤

<

c

(2.2)

c

. ( .1, 1)

(2.1)

(

)

2 2

2 2

1

3

2

,

1

3

2

1

3

1

3

1

3

+

γ

γ

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

γ

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

γ

+

ω

+

γ

=

R

R

c

R

&

_ar (2.3)

.1 f

( )

R

c

(

γ=1,4

)

. : 1−c=2γ

(

3γ+1

)

−2,

(

)

2

[

(

)

]

2

1 3 216 5 , 53 3 , 1 3

2−c=γ γ+ − −c= γ γ+ − .

, (2.3)

R

4 5

1

3

2

1

3

1

b

R

b

=

+

γ

<

<

+

γ

−

=

b5

-0.2 0.2 0.4 0.6

-0.08 -0.06 -0.04 -0.02 0.02

b₁ b₂

b₃ b₄

f(R)

(2.3)

( )

2

3

2

ch

3

1

3

1

2

ar

R t

=

−

−

⎛

_⎜

ω

t

⎞

_⎟

γ +

γ +

_⎝

_⎠

(2.4)

,

t

=

0

5

b

R

=

.

R

( ) ( )

0

<

R

∞

, (2.4) ,

.

(2.2) (2.1)

(

1

)(

2

)(

3

)

2

2

3

1

3

b

R

R

b

R

b

R

&

=

γ

+

ω

_ar

−

−

−

(2.5)

3 2 1

,

b

,

b

b

–

f

( )

R

.

(2.5)

:

b

₃

≤

R

≤

b

₂.

z

b

z

b

R

=

₂

sin

2

+

₃

cos

2

(2.5)

(

)

1

sin

,

1

12

1

3

3 1

3 2 2 2 2 2

3

1

₋

<

−

=

−

ω

−

+

γ

=

b

b

b

b

s

z

s

b

b

dt

dz

ar (2.6)

(2.6) 0

z

, , ,

,

t

→

0

R

→

b

₃

,

, . .

z

→

0

.

, (2.6) [5]

( )

(

2 3

)

2 3

2

2

,

3

1

3

,

,

2

t

s

b

b

s

a

b

b

acn

b

t

R

⎟

Ω

=

γ

+

−

ω

ar

=

−

⎠

⎞

⎜

⎝

⎛ Ω

−

=

(2.7)

a

– ,

Ω

– ,

s

–

, . ,

Ω

. (2.7)

,

( )

( )

ar

s

K

b

b

s

K

T

ω

−

+

γ

=

Ω

=

1

4

1

3

3

4

3 1

( )

s

K

– .

ω

( )

s

(

b

b

) ( )

K

s

K

T

ar

ω

−

+

γ

π

=

Ω

π

=

π

=

ω

_{1 3}

3

1

3

2

2

,

b

₁

→

b

₂,

s

=

1

f

( )

R

5 3

b

b

≡

b

₂

≡

b

₄ .

, [5]

(

t

)

ch

(

t

)

ar

cn

Ω

2

,

1

=

−1

Ω

2

,

Ω

=

ω

(2.7) (2.4).

, ,

b

₂

→

b

₃,

b

₂

≠

b

₃

1

<<

s

, . . .

.

c

<<

1

2

c

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

γ

+

+

+

γ

−

+

γ

=

γ

=

±

=

γ

±

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

γ

+

−

+

γ

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

γ

+

γ

−

+

γ

=

2

3

1

3

2

3

1

3

1

3

1

3

8

3

,

2

3

2

4

3

1

3

1

1

3

3

3

1

3

3

2

1

1

3

3

2 2 2

2 3

, 2

2 2 2

1

a

a

a

s

a

c

a

c

b

a

c

b

q

,

q

( )

s

K

( )

s

s

[5]. , (2.7)

.2. δ=0 (2.7)

(4.2) .

(

a=0.09243, s=0.38687

)

(

a=0.01064,s=0.13515

)

.2 1,4 5,6.

(

a=0.33308,s=0.7071

)

2,3 .2 .

2 4 6 8 10 12

-0.03 -0.02 -0.01 0.01 0.02 0.03

R

4

5

6

2 4 6 8 10 12

-0.1 -0.05 0.05 0.1 0.15 0.2

t

ar ω

R

2

3

t

( )

(

)

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

ω

−

ω

⎟

⎠

⎞

⎜

⎝

⎛

γ

+

+

ω

+

γ

+

ω

−

=

R

a

t

a

t

a

t

t

t

R

cos

3

cos

256

3

3

1

3

2

cos

8

3

1

3

cos

2

2 2

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

γ

+

−

ω

=

ω

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

γ

+

+

+

γ

=

2 2 2 2 2

3

1

3

64

11

1

,

32

3

1

3

1

16

3

1

3

a

a

a

R

_ar (2.8)

R

– ,

ω

–

, .

c

→

0

, ,

c

≠

0

(

3

1

)

3

1

→

γ

+

b

. 3

.1:

(

)

[

]

3

1

1

36

1

,

1

3

1

72

11887

107

,

1

3

36

72

107

2 3 , 1

2

_γ

₊

=

_γ

₊

±

=

+

γ

γ

=

b

b

c

(

γ

=

1

,

4

)

(2.7) (2.8)

a

=

0

.

01064

,

s

=

0

.

13515

.

, (2.7) (2.8), .2 5.

c

s

a

. 3. .

0

2

2

−

α

=

ω

+

δ

+

R

R

R

R

&

&

&

_ar (3.1)

, [6] [7,8] .

...

4 3 1 2 2

1

+

+

+

+

=

− −

F

r

r

F

r

F

r

R

(3.2)

( ) ( ) (

t

,

F

t

i

=

1

,

2

,...

)

r

_i

. (3.2) (3.1), 2 3 4

,

,

− − −

F

F

F

. ,

2 2 1 1 2 2 3 2 2 1

2

3

2

δ

5

3

δ

50

1

ω

2

1

,...

,

,

5

δ

6

,

6

tt t ttt t tt t ar tt t t tt t

F

F

F

F

F

F

r

F

F

F

F

F

F

r

F

r

− − −

α

−

α

+

α

+

α

−

α

=

=

=

⎟

⎠

⎞

⎜

⎝

⎛ +

α

−

=

α

=

&

&

&

(3.3)

( )

t r₇

2

F

. , (3.3)

r

₁

( )

t

,

( )

t

r

₇ ,

.

r

i

=

0

7

≥

0

4 5 6

=

r

=

r

=

r

. , (3.2) . 1

−

F

0

2

_{2 3}

2 2 2

2

+

δ

r

+

ω

r

−

α

r

r

=

r

&

&

ar

&

(3.4)

(3.3)

r

₃, ,

r

₃

, (3.1),

1

3

2

,

0

2

3

3

_α

=

_γ

₊

ω

=

=

ar

r

r

(3.5)

( )

t

F

.

r

₃

=

0

.

(3.4)

(

c c

1

,

2

=

const

)

2 2

2 , 1 2

2 2

4

2

,

2 1

ar t

t

e

c

e

c

r

=

ω

+

ω

ω

=

−

δ

±

δ

−

ω

(3.6) (3.3) (3.6)

( )

t

F

[

t t

]

t

tt

F

c

e

c

e

F

1 2

2 1

6

5

ω ω

₊

α

−

=

δ

+

(

B B

1

,

2

=

const

)

( )

t t t

e

c

e

c

e

B

B

t

F

1 2

5

1

6

5

1

6

_{2 2}

2

1 1 1 5

2 1

ω ω

δ −

δ

+

ω

ω

α

−

δ

+

ω

ω

α

−

+

=

(3.7)

(3.7) (3.3),

0

3

=

r

, (3.1)

2 1

,

c

c

0

,

0

,

25

6

2 2

1 2

2

=

δ

=

=

=

ω

_ar

c

c

r

(3.8) (3.8) (3.7),

( )

t

=

1

+

e

−δ 5

=

1

+

e

−ω 6

,

B

₁

=

B

₂

=

1

F

t art

(3.1),

r

₁ (3.3),

(

)

2

(

)

2 ₂

2 ₆

5

6

1

1

25

art

t ar

R

=

δ

+

e

δ −

=

ω

+

e

ω −

α

α

( )

( )

2

₍

1

₎

0

0

4

2 3

1

ar

R

∞ =

R

=

ω

=

α

γ +

(3.9)

, .

,

δ

=

0

α

ω

=

2 3 ar

r

( )

t

( )

t

F

=

1

+

exp

ω

_ar , (3.1) (2.4).

r

₃

=

0

, F

( )

t

( )

t

(

i

t

)

F

=

1

+

exp

±

ω

_ar

( )

⎟

⎠

⎞

⎜

⎝

⎛ ω

+

γ

=

⎟

⎠

⎞

⎜

⎝

⎛ ω

α

ω

=

− −

t

t

t

R

ar ar ar

2

cos

1

3

3

2

cos

2

3

2 2

2

.

4. . -

(1.6),

p

₁

=

0

[9].

( )

t

=

A

( )

t

ϕ

+

r

(

A

ϕ

) (

+

r

A

ϕ

)

ϕ

=

ω

t

+

β

R

cos

₂

,

₃

,

,

_ar (4.1)

ε

~

A

β

,

r

₂

r

₃–

ε

2

ε

3 ,

ε

– . (4.1)

2

r

r

₃. ( )

A

ϕ

(

)

⎟

⎠

⎞

⎜

⎝

⎛

₋

_γ

₊

ω

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

ω

α

−

ω

=

ϕ

δ

−

=

2 2 2

4 2

1

3

48

5

1

12

5

1

,

2

A

A

A

A

_ar

ar ar

&

&

2

r

r

₃

ϕ

ω

α

=

⎟

⎠

⎞

⎜

⎝

⎛

₋

_ϕ

ω

α

=

cos

3

48

,

2

cos

3

1

1

2

3

4 2

3 2

2 2

A

r

A

r

ar ar

(4.1)

( )

( )

( )

( )

( )

3

2

2

cos

2

0

6

cos

ϕ

+

ε

ω

α

−

ϕ

+

=

R

A

A

t

A

t

t

R

ar

(4.2)

( )

(

)

t

ar ar

ar

e

c

A

A

t

A

A

R

2

1 2

2 2

2

,

1

3

48

5

,

2

δ −

=

ω

δ

+

γ

+

ω

=

ϕ

ω

α

=

( )

A

R

– e , c1–

0

=

δ

(4.1) (4.2), , ,

(

)

2 2

1

5

1

3

1

,

const

48

ar

t

⎛

A

⎞

t

A

c

ϕ ≡ ω = ω

_⎜

−

γ +

_⎟

=

=

⎝

⎠

( )

A

ω

– , .

,

. c1

0

=

t (2.7) (4.2) – ,

(2.8) (4.2) – . .

1408

.

0

,

1923

.

0

,

5254

.

0

_{2 3}

1

=

b

=

b

=

−

b

, 2

.1. (4.2)

A

=

c

₁

=

−

0

.

16412

. 2,3, .2 , (2.7) (4.2)

. , (2.7) (4.2) .

a

s

<

1

.

038687

.

0

,

09243

.

0

=

=

s

a

(

)

[

]

2

1

3

8

5

.

5

γ

γ

+

−

=

c

f

( )

R

.

(4.2)

A

=

c

₁

=

−

0

.

04621

. (2.7) (4.2)

.2 1 4.

(2.7) (2.8), .2 5. (4.2)

A

=

c

₁

=

−

0

.

005372

, 6 .2 . 5 6 , (4.2) (2.7)

.

1. . . . .1. .: , 1987. 464 . 2. Chapman R.B., Plesset M.S. Thermal effects in the free oscillation of gas bubbles //

Trans. ASME, ser. D. J. Basic Eng. 1971. V.93, №3. . .: // . . . 1971. .93. №3. .37-40.

3. . . //

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