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Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration
Applied Mathematics and Computation 217 (2010) 414–421
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration Nikolai A. Kudryashov *, Dmitry I. Sinelshchikov Department of Applied Mathematics, National Research Nuclear University MEPHI, 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation
a r t i c l e
i n f o
Keywords: Liquid with gas bubbles Nonlinear evolution equation Nonlinear ordinary differential equation Exact solution Simplest equation method
a b s t r a c t Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction It is well known that a mixture of liquid and gas bubbles can be considered as an example of a nonlinear medium. Van Wijngaarden [1] was first who derived that the famous Korteweg–de Vries equation [2–4] can be used for describing nonlinear waves in liquid with gas bubbles. Taking into account the viscosity of liquid, Nakoryakov et al. [5] obtained the Burgers equation [6,7] and the Korteweg–de Vries–Burgers equation to describe the pressure waves in bubbly liquids. The dynamic propagation of acoustic waves in a half – space filled with a viscous, bubbly liquid under Van Wijngaarden linear theory was considered in the recent work [8]. Oganyan in [9] investigated a quasi – isothermal wave propagation in a gas–liquid mixture, in which he take into consideration the thermodynamic behavior of the near – isothermal gas in the bubbles. Many authors studied nonlinear processes in a liquid with gas bubbles using the numerical methods. For an example Nigmatulin and Khabeev [10,11] considered the heat transfer between an gas bubble and a liquid and studied the structure of shock waves in a liquid with gas bubbles with consideration for the heat transfer. The purpose of this work is to obtain nonlinear evolution equations for describing the pressure waves of in a liquid with gas bubbles taking into account the viscosity of liquid and the heat transfer on boundary a bubble and a liquid. 2. System of equations for describing waves in a liquid with gas bubbles with consideration for heat transfer and viscosity Suppose that a mixture of a liquid and gas bubbles is homogeneous medium [12,13]. In this case for description of this mixture we use the averaged temperature, velocity, density and pressure. We also assume that, the gas bubbles has the same size and the amount of bubbles in the mass unit is constant. We take the processes of the heat transfer and viscosity on the boundary of bubble and liquid into account. We do not consider the processes of formation, destruction, and conglutination for the gas bubbles.
* Corresponding author. E-mail addresses: [email protected], [email protected] (N.A. Kudryashov). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.033
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
415
We have that the volume and the mass of gas in the unit of the mass mixture can be written as
V¼
4 3 pR N; 3
X ¼ V qg ;
where R = R(x, t) is bubble radius, N is number of bubbles in mass unit, qg = qg(x, t) is the gas density. Here and later we believe that the subscript g corresponds to the gas phase and subscript l corresponds to the liquid phase. We consider the long wavelength perturbations in a mixture of the liquid and the gas bubbles assuming that characteristic length of waves of perturbation more than distance between bubbles. We also assume that, distance between bubbles much more than the averaged radius of a bubble. We describe dynamics of a bubble using the Rayleigh–Lamb equation. We also take the equation of energy for a bubble and the state equation for the gas in a bubble into account. The system of equation for the description of the gas bubble takes the form [12–16]
4m Rt ¼ Pg P; 3R 3vg Nuðn 1Þ 3nP g Pg;t þ ðT g T l Þ ¼ 0; Rt þ R 2R2 3 T 0 Pg R ; Tg ¼ Pg;0 R0 3 2
ql RRtt þ R2t þ
ð2:1Þ ð2:2Þ ð2:3Þ
where P(x, t) is a pressure of a gas–liquid mixture, Pg is a gas pressure in a bubble, Tg and Tl are temperatures of liquid and gas accordingly, vg is a coefficient of the gas thermal conduction, Nu is the Nusselt number, n is a polytropic exponent, m is the viscosity of a liquid. The expression for the density of a mixture can be presented in the form [5]
1
q
¼
1X
ql
þV )q¼
ql 1 X þ V ql
ð2:4Þ
:
Considering the small deviation of the bubble radius in comparison with the averaged radius of bubble, we have
Rðx; tÞ ¼ R0 þ gðx; tÞ;
R0 ¼ const; kgk R0 ;
Rðx; 0Þ ¼ R0 :
ð2:5Þ
Assume that the liquid temperature is constant and equal to the initial value
T l ¼ Tjt¼0 ¼ T 0 ;
T 0 ¼ const:
At the initial moment, we also have
t ¼ 0 : P ¼ Pg ¼ P0 ;
P0 ¼ const; V ¼ V 0 ¼
4 3 pR N: 3 0
Substituting Pg and Tg from Eqs. (2.1) and (2.3) into Eq. (2.2) and taking relation (2.5) into account we have the pressure dependence of a mixture on the radius perturbation in the form
g 3n, Pþ P gt þ ,P t þ ,ql ðR0 þ gÞgttt þ ð3n þ 4Þ,ql gt gtt þ R0 R0 ql 3R20 þ 4m, ql 6R20 4m, ql 8m,ð3n 1Þ þ 9R20 4mql 2P 3P gtt þ ggtt þ g2t þ gt þ 0 g þ 20 g2 ¼ 0; þ 2 2 3R0 3R R 0 0 R0 3R0 6R0
P P0 þ
,¼
2R20 P0 : 3vNuðn 1ÞT 0
ð2:6Þ
From Eq. (2.4) we also have the dependence q on g by means of formula (2.5)
q ¼ q0 lg þ l1 g2 ; q0 ¼
l¼
3q
2 l V0
R0 ð1 X þ V 0 ql Þ
2
;
ql 1 X þ V 0 ql
l1 ¼
;
2 l V 0 ð2 l V 0 R20 ð1 X þ
6q
q
1 þ XÞ 3
ql V 0 Þ
ð2:7Þ :
We use the system of equations for description of the motion of a gas–liquid mixture flow in the form
@ q @ðquÞ ¼ 0; þ @x @t
@u @u @P þ þu ¼ 0; @t @x @x
q
ð2:8Þ
where u = u(x, t) is a velocity of a flow of a gas–liquid mixture. Eq. (2.8) together with Eqs. (2.6) and (2.7) can be applied for describing nonlinear waves in a gas–liquid medium. This system of equations can be written as
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gt þ ugx þ gux
q0 2l u 1 gg ¼ 0; l x l t
q0 1 ðu þ uux Þ þ gut P x ¼ 0; l t l g
ð2:9Þ
3n, P gt þ ,P t ¼ ,ql ðR0 þ gÞgttt ð3n þ 4Þ,ql gt gtt R0 R0 ql 3R20 þ 4m, ql 6R20 4m, q ð8m,ð3n 1Þ þ 9R20 Þ 2 4mql 2P 3P gtt ggtt l gt g 0 g þ 20 g2 þ P0 : 2 3R0 t R0 3R0 6R20 R0 3R0
Pþ
Pþ
Consider the linear case of the system of Eq. (2.9). Assuming, that pressure in a mixture is proportional to perturbation radius, we obtain the linear wave equation for the radius perturbations from Eq. (2.9)
gtt ¼
c20
sffiffiffiffiffiffiffiffiffi 3P0 : c0 ¼ lR0
gxx ;
ð2:10Þ
Let us introduce the following dimensionless variables
t¼
l 0 t; c0
0
x ¼ lx ;
u ¼ c0 u0 ;
g ¼ R0 g 0 ; P ¼ P 0 P 0 þ P 0 ;
where l is the characteristic length of wave. Using the dimensionless variables the system of Eq. (2.9) can be reduced to the following (the primes are omitted)
q0 2l R0 u þ ugx þ gux 1 ggt ¼ 0; lR0 x l q0 1 ðu þ uux Þ þ gut Px ¼ 0; 3 lR0 t P þ ,1 Pt þ gP þ 3n,1 gt P ¼ cgttt cggttt ð3n þ 4Þcgt gtt ðb1 þ b2 Þgtt ð2b2 b1 Þggtt
gt
ð2:11Þ
3n 1 3 b1 þ b2 g2t kgt 3g þ 3g2 ; 2 2
where the parameters are determined by formulae
k¼
c¼
4mql c0 þ 3n,1 ; 3P0 l
b1 ¼
,ql R20 c30
,c 0
3
P0 l
;
,1 ¼
l
4m,ql c20 3P0 l
2
;
b2 ¼
ql c20 R20 P0 l
2
; ð2:12Þ
:
Numerical and analytical task by means of the system of Eq. (2.11) is a difficult problem. 3. Basic nonlinear evolution equation for describing nonlinear waves in liquid with bubbles We cannot find the exact solutions of the system of nonlinear differential Eq. (2.11). To account for the slow variation of the waveform, we introduce a scale transformation of independent variables [17]
n ¼ em ðx tÞ; s ¼ emþ1 t; m > 0; @ @ @ @ @ em : ¼ em ; ¼ emþ1 @x @n @t @s @n
e 1;
ð3:1Þ
Substituting (3.1) into (2.11) and dividing on em in first two equations we have the following system of equations
q0 2l R0 2l R0 u þ gun þ ugn e 1 ggs þ 1 ggn ¼ 0; lR0 n l l q0 q0 q0 1 u þ e u þ egus gun uu P ¼ 0; lR0 s lR0 n l R0 n 3 n P þ emþ1 ,1 Ps em ,1 Pn þ gP þ emþ1 3n,1 gs P em 3n,1 gn P ¼ e3mþ3 cgsss þ e3mþ2 3cgssn e3mþ1 3cgsnn þ e3m cgnnn e3mþ3 cggsss þ e3mþ2 3cggssn e3mþ1 3cggsnn þ e3m cggnnn e3mþ3 ð3n þ 4Þcgs gss þ e3mþ2 ð3n þ 4Þcgn gss
egs gn
ð3:2Þ ð3:3Þ
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
417
þ e3mþ2 2ð3n þ 4Þcgs gsn e3mþ1 2ð3n þ 4Þcgn gsn e3mþ1 ð3n þ 4Þcgs gnn þ e3m ð3n þ 4Þcgn gnn e2mþ2 ðb1 þ b2 Þgss 2mþ1
þe
2m
2mþ2
2ðb1 þ b2 Þgsn e ðb1 þ b2 Þgnn e
ð3:4Þ
ð2b2 b1 Þggss
þ e2mþ1 2ð2b2 b1 Þggsn e2m ð2b2 b1 Þggnn 3n 1 3 3n 1 3 b1 þ b2 g2s þ e2mþ1 2 b1 þ b2 gs gn e2mþ2 2 2 2 2 3n 1 3 e2m b1 þ b2 g2n emþ1 kgs þ em kgn 3g þ 3g2 : 2 2 Now we assume that the variables u, g and P can be represented asymptotically as series in powers of e about an equilibrium state
u ¼ eu1 þ e2 u2 þ . . . ;
g ¼ eg1 þ e2 g2 þ . . . ; P ¼ eP1 þ e2 P2 þ . . .
ð3:5Þ
Substituting (3.5) in (3.2)–(3.4) and equating expressions at e, we have the equations
g1n
q0 q 1 u ¼ 0; 0 u1n þ P1n ¼ 0; P1 ¼ 3g1 : 3 lR0 1n lR0
Solving this system of equation, we have
g1 ðn; sÞ ¼
q0 u ðn; sÞ þ wðsÞ; P1 ðn; sÞ ¼ 3g1 ðn; sÞ; l R0 1
ð3:6Þ
where w(s) is an arbitrary function. We suppose below that w(s) = 0. Substituting (3.5) in (3.2)–(3.4) and equating factors at e2 to zero, we have the system of equations
q0 2l R0 u þ g u þ u g þ 1 g1 g1n ¼ 0; lR0 2n 1 1n 1 1n l q0 q0 q0 1 u þ u g u u u P ¼ 0; lR0 1s lR0 2n 1 1n lR0 1 1n 3 2n P2 em1 ,1 P n þ g1 P 1 em 3n,1 g1n P1 ¼ e3m1 cg1nnn þ e3m cg1 g1nnn þ e3m ð3n þ 4Þcg1n g1nn e2m1 ðb1 þ b2 Þg1nn
g1s g2n
e2m ð2b2 b1 Þg1 g1nn e2m
ð3:7Þ
3n 1 3 b1 þ b2 g21n þ em1 kg1n 3g2 þ 3g21 : 2 2
Taking Eqs. (3.6) and (3.7) into account we have the equation for pressure P1 in the form
b1 þ b2 2b b1 3n 2 5 k ,1 b1 þ b2 P 1n P1nn ¼ em1 P1nnn e2m 2 P1 P 1nnn e2m P1nn 18 18 6 2 6 18 n,1 c c ð3n þ 5Þc ð3n þ 4Þc 2 þ em ðP 1 P1n Þn þ e3m1 P1nnnn e3m P1 P1nnnn e3m P1n P1nnn e3m P1nn ; 2 6 18 18 18 3lR0 3l1 R0 2 a¼ þ : 3 q0 l P1s þ aP1 P1n þ e2m1
ð3:8Þ
Assuming m = 1 we obtain the Burgers equation from Eq. (3.8) when e ? 0
P1s þ aP1 P1n ¼
k ,1 P1nn : 6 2
ð3:9Þ
Many solutions of the Burgers equation are well known [3,18,19]. By means of the Cole–Hopf transformation [20,21] the Burgers equation can be transformed to the linear heat equation. The Cauchy problem for this equation can be solved [3]. In the case m ¼ 12 ; 6k ,21 ’ e2 we have the famous Korteweg–de Vries equation [2] at e ? 0
P1s þ aP1 P1n þ
b1 þ b2 P 1nnn ¼ 0: 6
ð3:10Þ
This equation has the soliton solutions [22]. The Cauchy problem for the Korteweg–de Vries equation can be solved by the inverse scattering transform [23]. There are many methods to find exact solutions of the Korteweg–de Vries equation [24,25]. Unfortunately, some of these approaches lead to the erroneous solutions [26–28]. Assuming m ¼ 12 ; 6k ,21 ’ e we have the Korteweg–de Vries–Burgers equation at e ? 0
P1s þ aP1 P1n þ
b1 þ b2 P 1nnn ¼ 6 1 3
In the case m ¼ 13 ; b1 þ b2 ’ 6e ;
k
c
6
k ,1 P1nn : 6 2
ð3:11Þ
,21 ’ e , we have the famous Kuramoto–Sivashinsky equation [29,30] when e ? 0
P1s þ aP1 P1n þ P1nnn ¼ P1nn þ P1nnnn : 6
2 3
ð3:12Þ
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N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
The cauchy problem for Eq. (3.12) cannot be solved by the inverse scattering transform but this equation has some exact solutions. The Kuramoto–Sivashinsky equation was studied in many papers (see, for the example [31–48]). From Eq. (3.8) one can find some other nonlinear evolution equations for describing waves in a mixture of liquid and gas bubbles. 4. Fourth order equation for describing wave processes in liquid with gas bubbles Assuming m ¼ 13 ; k ¼ 6e2=3 k0 ; k0 1; ,1 ¼ 2e2=3 ,01 ; ,01 1; b1 ¼ 6e1=3 b01 ; b01 1; b2 ¼ 6e1=3 b02 ; b02 1; d > 0 in Eq. (3.8) and using transformations P 1 ¼ 1e P01 ; a ¼ ea0 we have nonlinear evolution equation of the fourth order in the form
P01s þ a0 P01 P01n þ ðb01 þ b02 ÞP01nnn
2b02 b01 0 0 3n 2 0 5 0 b1 þ b2 P01n P01nn P 1 P1nnn 3 3 3
c
c
¼ ðk0 ,01 ÞP01nn þ n,01 ðP01n P01 Þn þ P1nnnn P0 P0 6 18 1 1nnnn
ð3n þ 5Þc 0 0 ð3n þ 4Þc 02 P1n P1nnn P1nn : 18 18
ð4:1Þ
Coefficient c/6 at the fourth derivative in (4.1) can be presented in the form
C¼
c 6
¼
4ql c30 R40 9vg T 0 Nuðn 1Þl
3
ð4:2Þ
:
We can see that, the value C in (4.2) depends on the initial radius of a bubble, sound velocity, coefficient of the heat transfer and the Nusselt mumber. The heat capacity of liquid is much more than the the heat capacity of the gas in bubble. As consequence the temperature of liquid is not changed but the gas in bubble can be heated and cooled. In the case of the isothermal processes we have Nu ? 1 and Eq. (4.1) becomes the third order equation. Let us look for the exact solutions of Eq. (4.1). Using the variables
c
n0 ; n¼ 0 6 b1 þ b02
s¼
c3
0
4 s ; 216 b01 þ b02
P01
3 b01 þ b02 ¼ v: 2b02 b01
we obtain the equation (the primes are omitted)
v s þ a1 vv n þ v nnn ðvv nn Þn bv n v nn ¼ rv nn þ r1 ðvv n Þn þ v nnnn c2 ðvv nnn Þn c3 ðv n v nn Þn ;
ð4:3Þ
where parameters of Eq. (4.3) can be written in the form
a1 ¼
a0 c2
2 ; 12 b01 þ b02 2b02 b01 n, c ; 2 2b02 b01 b01 þ b02 0 1
r1 ¼
b¼
c2 ¼
ð3n 1Þb01 þ 3b02 ; 2b02 b01
b01 þ b02 2b02 b01
;
c k0 ,01 r¼ 0 ; 0 2
6 b þb 1 2 ð3n þ 4Þ b01 þ b02 c3 ¼ : 2b02 b01
Using traveling wave ansatz v(n, s) = y(z), z = n C0s and integrating the equation on z, we have
C1 C0 y þ
a1 2
b y2 þ yzz yyzz y2z ryz r1 yyz yzzz þ c2 yyzzz þ c3 yz yzz ¼ 0: 2
ð4:4Þ
To find the solution of (4.4) we use the simplest equation method [49–51]
yðzÞ ¼ a0 þ w;
w wðzÞ;
ð4:5Þ
where w(z) is the solution of the linear equation
wzz ¼ Awz þ Bw:
ð4:6Þ
Substituting the transformation (4.5) and taking into account Eq. (4.6) we obtain the following solution of the Eq. (4.4)
yðzÞ ¼ a0 þ C 3 ekz þ C 2 ekz ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bðc3 þ c2 Þ þ b2 ðc3 þ c2 Þðc3 3c2 Þ þ 8c3 ðc3 þ c2 Þðb þ 2r1 c3 Þ ; k¼ 4c3 ðc3 þ c2 Þ where C2 and C3 are arbitrary constants, parameters a0, A, B, a1, C0 and C1 are determined by formulae
ð4:7Þ
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
a0 ¼ A¼
2bðc2 þ c3 1Þ 4rc3 ðc3 þ c2 Þ 4r1 c3 b2 2c3 b þ 4r1 c23 c2 b2
b ; 2c3
B¼
419
;
c2 b2 þ 2c3 b þ 4r1 c23 ; 4ðc3 þ c2 Þc23
a1 ¼
4c23 b þ c22 b3 4c2 b2 c3 þ 8r1 c33 4r1 c23 bc2 ; 4c33 ðc3 þ c2 Þ
C0 ¼
1 4r1 c33 4r1 c23 2r1 c23 b þ 2r1 bc2 c3 þ c2 b2 4c33 ðc3 þ c2 Þ
þ2rc23 bc2 4c33 r 4c23 c2 r þ 4c23 b þ 2bc2 c3 2b2 c3 2c2 b2 c3 þ 2c3 rbc22 2c3 b þ b3 c2 c22 b2 ; ðb 2c3 Þ b2 2bc2 þ 2b 2c3 b þ 4c2 rc3 þ 4r1 c3 þ 4rc23 C1 ¼ : 8c33 ðc3 þ c2 Þ For finding the periodic solutions of Eq. (4.4) we use the expression in the form
yðzÞ ¼ a0 þ a1 w þ a2 w2 þ a3 w3 ;
w wðzÞ
ð4:8Þ
where w(z) also satisfies linear ordinary differential equation (4.6). Substituting Eq. (4.8) into Eq. (4.4) and equating the different expressions at w(z) to zero, we have the system of the algebraic equations. Solving this system of equations we obtain the following values of parameters
a2 ¼ a1 ¼ 0;
1 3
c2 ¼ 1; c3 ¼ ; B ¼
a0 ¼ 1;
2
3
a1 6ðA 1Þ
2
;
3
2A þ A a1 þ A 2A þ A a1 þ A ; r1 ¼ ; A1 A1 10A 4 a1 ; C 0 ¼ a1 ; C 1 ¼ ; b¼ 3 2
ð4:9Þ
r¼
and solution of Eq. (4.4) in the form
pffiffiffi pffiffiffi 3 A z p A z p yðzÞ ¼ 1 þ a3 C 3 exp z þ ; þ C 2 exp z 2 6ðA 1Þ 2 6ðA 1Þ 3
2
p ¼ 3ðA 1Þð3A 3A 2a1 Þ;
A – 0;
A – 1;
ð4:10Þ
a3 – 0:
In the case of
pffiffiffi 3A þ 3A2 þ p 6ðA 1Þ
< 0;
3A þ 3A2
pffiffiffi p <0
6ðA 1Þ
solution (4.10) is finite function. Consider case A = 0. In this case the parameters of equation and coefficients of expansion take the form
a1 ¼ a2 ¼ 0;
a0 ¼ 1;
1 3
c2 ¼ 1; c3 ¼ ; B ¼ 4 3
r ¼ a1 ; r1 ¼ a1 ; b ¼ ; C 0 ¼ a1 ; C 1 ¼
a1 6
a1 2
; :
At a1 > 0 we have the following solution of Eq. (4.4)
pffiffiffiffiffi !3
pffiffiffiffiffi yðzÞ ¼ 1 þ a3 C 3 e
6a1 z 6
þ C 2 e
6a1 z 6
;
a3 – 0
ð4:11Þ
In the case of a1 < 0 the solution of Eq. (4.4) can be written as
yðzÞ ¼ 1 þ a3
(pffiffiffiffiffiffiffiffiffiffiffi ) (pffiffiffiffiffiffiffiffiffiffiffi )!3 6ja1 jz 6ja1 jz þ C 2 cos C 3 sin ; 6 6
a3 – 0:
ð4:12Þ
Solutions (4.11), (4.12) of Eq. (4.4) have two arbitrary constants C2 and C3. Solutions (4.10), (4.12) have physical sense. The first of them is limited at the values of parameters at conditions of the above mentioned. The solution (4.12) is the periodical solution. The velocity of the wave described (4.10)–(4.12), is equal to parameter characterizing nonlinear transfer. Dependence of a solution (4.12) of Eq. (4.4) with respect to z at values of parameters a1 = 1, C2 = 0.04, C3 = 0.05 is given on Fig. 1.
420
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
y 1.2
1.1
1
0.9
0.8
-8
-4
0
4
8
z
Fig. 1. Periodic solution (4.12) of Eq. (4.4).
5. Conclusion We have studied the propagation of nonlinear waves in a mixture of liquid and gas bubbles taking into consideration the influence of heat transfer and viscosity. We have applied different scales of time and coordinate and the method of perturbation to obtain nonlinear evolution equations for describing pressure waves. As a result we have found the nonlinear evolution Eq. (3.8) of the fourth order for describing the pressure waves in a liquid with bubbles. This evolution equation allow us to take into account the influence of the viscosity and the heat transfer on the pressure waves. It is likely this nonlinear differential equation is new and generalizes many well – known nonlinear equations as the Burgers equation, the Korteweg– de Vries equation, the Korteweg–de Vries–Burgers equation and the Kuramoto–Sivashinsky equation. Generally this equation is not integrable but this equation has some exact solutions that have been found in this paper. References [1] L. Van Wijngaarden, On the equations of motion for mixtures of liquid and gas bubbles, J. Fluid Mech. 33 (1968) 465–474. [2] D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new tipe of long stationary waves, Phil. Mag. 39 (1895) 422–443. [3] V. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974. [4] P.G. Drazin, R.S. Johnson, Soliton: An Introduction, Cambridge university press, 1989. [5] V.E. Nakoryakov, B.G. Pokusaev, I.R. Shraiber, Wave Dynamics of Gas– and Vapor–Liquid Media, Energoatomizdat, Moscow, 1990 (in Russian). [6] H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Rev. 43 (1915) 163–170. [7] J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948) 171–189. [8] P.M. Jordan, C. Feuillade, On the propagation of transient acoustic waves in isothermal bubbly liquids, Phys. Lett. A 350 (2006) 56–62. [9] G.G. Oganyan, Increase in the compression wave amplitude in a liquid with dissolved – gas bubbles, Fluid Dynamics 40 (2005) 95–102. [10] R.I. Nigmatulin, N.S. Khabeev, Heat transfer of gas bubbles in liquid, Fluid Dynamics 9 (1974) 759–769. [11] R.R. Aidagulov, N.S. Khabeev, V.Sh. Shagapov, Shock wave structure in a liquid with gas bubbles with allowance for unsteady interphase heat transfer, J. Appl. Mech. Tech. Phys. 18 (1977) 334. [12] V.E. Nakoryakov, V.V. Sobolev, I.R. Shreiber, Long wave perturbation in a gas–liquid medium, Fluid Dynamics 7 (1972) 763–772. [13] R.I. Nigmatulin, Dynamics of Multiphase Media, Part 2, Hemisphere, New York, 1991. [14] Rayleigh, Lord, On the pressure developed in a liquid during the collapse of a spherical void, Philos. Mag. 34 (1917) 94–98. [15] M.S. Plesset, A. Prosperetti, Bubble dynamics and cavitation, Ann. Rev. Fluid Mech. 9 (1977) 145–185. [16] L. Van Wijngaarden, One-dimensional flow of liquid containing small gas flow, Annu. Rev. Fluid Mech. 4 (1972) 369–396. [17] N.A. Kudryashov, I.L. Chernyavskii, Nonlinear waves in fluid flow through viscoelastic tube, Fluid Dynamics 41 (2006) 49–62. [18] N.A. Kudryashov, D.I. Sinechshikov, Exact solutions of equations for the Burgers hierarchy, Appl. Math. Comput. 215 (2009) 1293–1300. [19] N.A. Kudryashov, Self – similar of the Burgers hierarchy, Appl. Math. Comput. 215 (2009) 1990–1993. [20] E. Hopf, The partial differential equation ut + uux = uxx, Commun. Pure Appl. Math. 3 (1950) 201–230. [21] J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1950) 225–236. [22] N.J. Zabuski, M.D. Kruskal, Integration of ‘‘solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965) 240–242. [23] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg–de Vries equation, J. Math. Phys. 19 (1967) 1095–1097. [24] R. Hirota, Exact solution of the Korteweg–de Vries for multiple collisions of solutions, Phys. Rev. Lett. 27 (1971) 1192–1194. [25] N.A. Kudryashov, Analytical Theory of Nonlinear Differential Equations, Institute of computer investigations, Moskow – Igevsk, 2004 (in Russian). [26] N.A. Kudryashov, N.B. Loguinova, Be careful with exp-function method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1881–1890. [27] N.A. Kudryashov, On ‘‘new travelling wave solutions” of the KdV and the KdV–Burgers equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1891–1900. [28] N.A. Kudryashov, Seven common errors in finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3507–3523.
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration Nikolai A. Kudryashov *, Dmitry I. Sinelshchikov Department of Applied Mathematics, National Research Nuclear University MEPHI, 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation
a r t i c l e
i n f o
Keywords: Liquid with gas bubbles Nonlinear evolution equation Nonlinear ordinary differential equation Exact solution Simplest equation method
a b s t r a c t Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction It is well known that a mixture of liquid and gas bubbles can be considered as an example of a nonlinear medium. Van Wijngaarden [1] was first who derived that the famous Korteweg–de Vries equation [2–4] can be used for describing nonlinear waves in liquid with gas bubbles. Taking into account the viscosity of liquid, Nakoryakov et al. [5] obtained the Burgers equation [6,7] and the Korteweg–de Vries–Burgers equation to describe the pressure waves in bubbly liquids. The dynamic propagation of acoustic waves in a half – space filled with a viscous, bubbly liquid under Van Wijngaarden linear theory was considered in the recent work [8]. Oganyan in [9] investigated a quasi – isothermal wave propagation in a gas–liquid mixture, in which he take into consideration the thermodynamic behavior of the near – isothermal gas in the bubbles. Many authors studied nonlinear processes in a liquid with gas bubbles using the numerical methods. For an example Nigmatulin and Khabeev [10,11] considered the heat transfer between an gas bubble and a liquid and studied the structure of shock waves in a liquid with gas bubbles with consideration for the heat transfer. The purpose of this work is to obtain nonlinear evolution equations for describing the pressure waves of in a liquid with gas bubbles taking into account the viscosity of liquid and the heat transfer on boundary a bubble and a liquid. 2. System of equations for describing waves in a liquid with gas bubbles with consideration for heat transfer and viscosity Suppose that a mixture of a liquid and gas bubbles is homogeneous medium [12,13]. In this case for description of this mixture we use the averaged temperature, velocity, density and pressure. We also assume that, the gas bubbles has the same size and the amount of bubbles in the mass unit is constant. We take the processes of the heat transfer and viscosity on the boundary of bubble and liquid into account. We do not consider the processes of formation, destruction, and conglutination for the gas bubbles.
* Corresponding author. E-mail addresses: [email protected], [email protected] (N.A. Kudryashov). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.033
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
415
We have that the volume and the mass of gas in the unit of the mass mixture can be written as
V¼
4 3 pR N; 3
X ¼ V qg ;
where R = R(x, t) is bubble radius, N is number of bubbles in mass unit, qg = qg(x, t) is the gas density. Here and later we believe that the subscript g corresponds to the gas phase and subscript l corresponds to the liquid phase. We consider the long wavelength perturbations in a mixture of the liquid and the gas bubbles assuming that characteristic length of waves of perturbation more than distance between bubbles. We also assume that, distance between bubbles much more than the averaged radius of a bubble. We describe dynamics of a bubble using the Rayleigh–Lamb equation. We also take the equation of energy for a bubble and the state equation for the gas in a bubble into account. The system of equation for the description of the gas bubble takes the form [12–16]
4m Rt ¼ Pg P; 3R 3vg Nuðn 1Þ 3nP g Pg;t þ ðT g T l Þ ¼ 0; Rt þ R 2R2 3 T 0 Pg R ; Tg ¼ Pg;0 R0 3 2
ql RRtt þ R2t þ
ð2:1Þ ð2:2Þ ð2:3Þ
where P(x, t) is a pressure of a gas–liquid mixture, Pg is a gas pressure in a bubble, Tg and Tl are temperatures of liquid and gas accordingly, vg is a coefficient of the gas thermal conduction, Nu is the Nusselt number, n is a polytropic exponent, m is the viscosity of a liquid. The expression for the density of a mixture can be presented in the form [5]
1
q
¼
1X
ql
þV )q¼
ql 1 X þ V ql
ð2:4Þ
:
Considering the small deviation of the bubble radius in comparison with the averaged radius of bubble, we have
Rðx; tÞ ¼ R0 þ gðx; tÞ;
R0 ¼ const; kgk R0 ;
Rðx; 0Þ ¼ R0 :
ð2:5Þ
Assume that the liquid temperature is constant and equal to the initial value
T l ¼ Tjt¼0 ¼ T 0 ;
T 0 ¼ const:
At the initial moment, we also have
t ¼ 0 : P ¼ Pg ¼ P0 ;
P0 ¼ const; V ¼ V 0 ¼
4 3 pR N: 3 0
Substituting Pg and Tg from Eqs. (2.1) and (2.3) into Eq. (2.2) and taking relation (2.5) into account we have the pressure dependence of a mixture on the radius perturbation in the form
g 3n, Pþ P gt þ ,P t þ ,ql ðR0 þ gÞgttt þ ð3n þ 4Þ,ql gt gtt þ R0 R0 ql 3R20 þ 4m, ql 6R20 4m, ql 8m,ð3n 1Þ þ 9R20 4mql 2P 3P gtt þ ggtt þ g2t þ gt þ 0 g þ 20 g2 ¼ 0; þ 2 2 3R0 3R R 0 0 R0 3R0 6R0
P P0 þ
,¼
2R20 P0 : 3vNuðn 1ÞT 0
ð2:6Þ
From Eq. (2.4) we also have the dependence q on g by means of formula (2.5)
q ¼ q0 lg þ l1 g2 ; q0 ¼
l¼
3q
2 l V0
R0 ð1 X þ V 0 ql Þ
2
;
ql 1 X þ V 0 ql
l1 ¼
;
2 l V 0 ð2 l V 0 R20 ð1 X þ
6q
q
1 þ XÞ 3
ql V 0 Þ
ð2:7Þ :
We use the system of equations for description of the motion of a gas–liquid mixture flow in the form
@ q @ðquÞ ¼ 0; þ @x @t
@u @u @P þ þu ¼ 0; @t @x @x
q
ð2:8Þ
where u = u(x, t) is a velocity of a flow of a gas–liquid mixture. Eq. (2.8) together with Eqs. (2.6) and (2.7) can be applied for describing nonlinear waves in a gas–liquid medium. This system of equations can be written as
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gt þ ugx þ gux
q0 2l u 1 gg ¼ 0; l x l t
q0 1 ðu þ uux Þ þ gut P x ¼ 0; l t l g
ð2:9Þ
3n, P gt þ ,P t ¼ ,ql ðR0 þ gÞgttt ð3n þ 4Þ,ql gt gtt R0 R0 ql 3R20 þ 4m, ql 6R20 4m, q ð8m,ð3n 1Þ þ 9R20 Þ 2 4mql 2P 3P gtt ggtt l gt g 0 g þ 20 g2 þ P0 : 2 3R0 t R0 3R0 6R20 R0 3R0
Pþ
Pþ
Consider the linear case of the system of Eq. (2.9). Assuming, that pressure in a mixture is proportional to perturbation radius, we obtain the linear wave equation for the radius perturbations from Eq. (2.9)
gtt ¼
c20
sffiffiffiffiffiffiffiffiffi 3P0 : c0 ¼ lR0
gxx ;
ð2:10Þ
Let us introduce the following dimensionless variables
t¼
l 0 t; c0
0
x ¼ lx ;
u ¼ c0 u0 ;
g ¼ R0 g 0 ; P ¼ P 0 P 0 þ P 0 ;
where l is the characteristic length of wave. Using the dimensionless variables the system of Eq. (2.9) can be reduced to the following (the primes are omitted)
q0 2l R0 u þ ugx þ gux 1 ggt ¼ 0; lR0 x l q0 1 ðu þ uux Þ þ gut Px ¼ 0; 3 lR0 t P þ ,1 Pt þ gP þ 3n,1 gt P ¼ cgttt cggttt ð3n þ 4Þcgt gtt ðb1 þ b2 Þgtt ð2b2 b1 Þggtt
gt
ð2:11Þ
3n 1 3 b1 þ b2 g2t kgt 3g þ 3g2 ; 2 2
where the parameters are determined by formulae
k¼
c¼
4mql c0 þ 3n,1 ; 3P0 l
b1 ¼
,ql R20 c30
,c 0
3
P0 l
;
,1 ¼
l
4m,ql c20 3P0 l
2
;
b2 ¼
ql c20 R20 P0 l
2
; ð2:12Þ
:
Numerical and analytical task by means of the system of Eq. (2.11) is a difficult problem. 3. Basic nonlinear evolution equation for describing nonlinear waves in liquid with bubbles We cannot find the exact solutions of the system of nonlinear differential Eq. (2.11). To account for the slow variation of the waveform, we introduce a scale transformation of independent variables [17]
n ¼ em ðx tÞ; s ¼ emþ1 t; m > 0; @ @ @ @ @ em : ¼ em ; ¼ emþ1 @x @n @t @s @n
e 1;
ð3:1Þ
Substituting (3.1) into (2.11) and dividing on em in first two equations we have the following system of equations
q0 2l R0 2l R0 u þ gun þ ugn e 1 ggs þ 1 ggn ¼ 0; lR0 n l l q0 q0 q0 1 u þ e u þ egus gun uu P ¼ 0; lR0 s lR0 n l R0 n 3 n P þ emþ1 ,1 Ps em ,1 Pn þ gP þ emþ1 3n,1 gs P em 3n,1 gn P ¼ e3mþ3 cgsss þ e3mþ2 3cgssn e3mþ1 3cgsnn þ e3m cgnnn e3mþ3 cggsss þ e3mþ2 3cggssn e3mþ1 3cggsnn þ e3m cggnnn e3mþ3 ð3n þ 4Þcgs gss þ e3mþ2 ð3n þ 4Þcgn gss
egs gn
ð3:2Þ ð3:3Þ
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
417
þ e3mþ2 2ð3n þ 4Þcgs gsn e3mþ1 2ð3n þ 4Þcgn gsn e3mþ1 ð3n þ 4Þcgs gnn þ e3m ð3n þ 4Þcgn gnn e2mþ2 ðb1 þ b2 Þgss 2mþ1
þe
2m
2mþ2
2ðb1 þ b2 Þgsn e ðb1 þ b2 Þgnn e
ð3:4Þ
ð2b2 b1 Þggss
þ e2mþ1 2ð2b2 b1 Þggsn e2m ð2b2 b1 Þggnn 3n 1 3 3n 1 3 b1 þ b2 g2s þ e2mþ1 2 b1 þ b2 gs gn e2mþ2 2 2 2 2 3n 1 3 e2m b1 þ b2 g2n emþ1 kgs þ em kgn 3g þ 3g2 : 2 2 Now we assume that the variables u, g and P can be represented asymptotically as series in powers of e about an equilibrium state
u ¼ eu1 þ e2 u2 þ . . . ;
g ¼ eg1 þ e2 g2 þ . . . ; P ¼ eP1 þ e2 P2 þ . . .
ð3:5Þ
Substituting (3.5) in (3.2)–(3.4) and equating expressions at e, we have the equations
g1n
q0 q 1 u ¼ 0; 0 u1n þ P1n ¼ 0; P1 ¼ 3g1 : 3 lR0 1n lR0
Solving this system of equation, we have
g1 ðn; sÞ ¼
q0 u ðn; sÞ þ wðsÞ; P1 ðn; sÞ ¼ 3g1 ðn; sÞ; l R0 1
ð3:6Þ
where w(s) is an arbitrary function. We suppose below that w(s) = 0. Substituting (3.5) in (3.2)–(3.4) and equating factors at e2 to zero, we have the system of equations
q0 2l R0 u þ g u þ u g þ 1 g1 g1n ¼ 0; lR0 2n 1 1n 1 1n l q0 q0 q0 1 u þ u g u u u P ¼ 0; lR0 1s lR0 2n 1 1n lR0 1 1n 3 2n P2 em1 ,1 P n þ g1 P 1 em 3n,1 g1n P1 ¼ e3m1 cg1nnn þ e3m cg1 g1nnn þ e3m ð3n þ 4Þcg1n g1nn e2m1 ðb1 þ b2 Þg1nn
g1s g2n
e2m ð2b2 b1 Þg1 g1nn e2m
ð3:7Þ
3n 1 3 b1 þ b2 g21n þ em1 kg1n 3g2 þ 3g21 : 2 2
Taking Eqs. (3.6) and (3.7) into account we have the equation for pressure P1 in the form
b1 þ b2 2b b1 3n 2 5 k ,1 b1 þ b2 P 1n P1nn ¼ em1 P1nnn e2m 2 P1 P 1nnn e2m P1nn 18 18 6 2 6 18 n,1 c c ð3n þ 5Þc ð3n þ 4Þc 2 þ em ðP 1 P1n Þn þ e3m1 P1nnnn e3m P1 P1nnnn e3m P1n P1nnn e3m P1nn ; 2 6 18 18 18 3lR0 3l1 R0 2 a¼ þ : 3 q0 l P1s þ aP1 P1n þ e2m1
ð3:8Þ
Assuming m = 1 we obtain the Burgers equation from Eq. (3.8) when e ? 0
P1s þ aP1 P1n ¼
k ,1 P1nn : 6 2
ð3:9Þ
Many solutions of the Burgers equation are well known [3,18,19]. By means of the Cole–Hopf transformation [20,21] the Burgers equation can be transformed to the linear heat equation. The Cauchy problem for this equation can be solved [3]. In the case m ¼ 12 ; 6k ,21 ’ e2 we have the famous Korteweg–de Vries equation [2] at e ? 0
P1s þ aP1 P1n þ
b1 þ b2 P 1nnn ¼ 0: 6
ð3:10Þ
This equation has the soliton solutions [22]. The Cauchy problem for the Korteweg–de Vries equation can be solved by the inverse scattering transform [23]. There are many methods to find exact solutions of the Korteweg–de Vries equation [24,25]. Unfortunately, some of these approaches lead to the erroneous solutions [26–28]. Assuming m ¼ 12 ; 6k ,21 ’ e we have the Korteweg–de Vries–Burgers equation at e ? 0
P1s þ aP1 P1n þ
b1 þ b2 P 1nnn ¼ 6 1 3
In the case m ¼ 13 ; b1 þ b2 ’ 6e ;
k
c
6
k ,1 P1nn : 6 2
ð3:11Þ
,21 ’ e , we have the famous Kuramoto–Sivashinsky equation [29,30] when e ? 0
P1s þ aP1 P1n þ P1nnn ¼ P1nn þ P1nnnn : 6
2 3
ð3:12Þ
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The cauchy problem for Eq. (3.12) cannot be solved by the inverse scattering transform but this equation has some exact solutions. The Kuramoto–Sivashinsky equation was studied in many papers (see, for the example [31–48]). From Eq. (3.8) one can find some other nonlinear evolution equations for describing waves in a mixture of liquid and gas bubbles. 4. Fourth order equation for describing wave processes in liquid with gas bubbles Assuming m ¼ 13 ; k ¼ 6e2=3 k0 ; k0 1; ,1 ¼ 2e2=3 ,01 ; ,01 1; b1 ¼ 6e1=3 b01 ; b01 1; b2 ¼ 6e1=3 b02 ; b02 1; d > 0 in Eq. (3.8) and using transformations P 1 ¼ 1e P01 ; a ¼ ea0 we have nonlinear evolution equation of the fourth order in the form
P01s þ a0 P01 P01n þ ðb01 þ b02 ÞP01nnn
2b02 b01 0 0 3n 2 0 5 0 b1 þ b2 P01n P01nn P 1 P1nnn 3 3 3
c
c
¼ ðk0 ,01 ÞP01nn þ n,01 ðP01n P01 Þn þ P1nnnn P0 P0 6 18 1 1nnnn
ð3n þ 5Þc 0 0 ð3n þ 4Þc 02 P1n P1nnn P1nn : 18 18
ð4:1Þ
Coefficient c/6 at the fourth derivative in (4.1) can be presented in the form
C¼
c 6
¼
4ql c30 R40 9vg T 0 Nuðn 1Þl
3
ð4:2Þ
:
We can see that, the value C in (4.2) depends on the initial radius of a bubble, sound velocity, coefficient of the heat transfer and the Nusselt mumber. The heat capacity of liquid is much more than the the heat capacity of the gas in bubble. As consequence the temperature of liquid is not changed but the gas in bubble can be heated and cooled. In the case of the isothermal processes we have Nu ? 1 and Eq. (4.1) becomes the third order equation. Let us look for the exact solutions of Eq. (4.1). Using the variables
c
n0 ; n¼ 0 6 b1 þ b02
s¼
c3
0
4 s ; 216 b01 þ b02
P01
3 b01 þ b02 ¼ v: 2b02 b01
we obtain the equation (the primes are omitted)
v s þ a1 vv n þ v nnn ðvv nn Þn bv n v nn ¼ rv nn þ r1 ðvv n Þn þ v nnnn c2 ðvv nnn Þn c3 ðv n v nn Þn ;
ð4:3Þ
where parameters of Eq. (4.3) can be written in the form
a1 ¼
a0 c2
2 ; 12 b01 þ b02 2b02 b01 n, c ; 2 2b02 b01 b01 þ b02 0 1
r1 ¼
b¼
c2 ¼
ð3n 1Þb01 þ 3b02 ; 2b02 b01
b01 þ b02 2b02 b01
;
c k0 ,01 r¼ 0 ; 0 2
6 b þb 1 2 ð3n þ 4Þ b01 þ b02 c3 ¼ : 2b02 b01
Using traveling wave ansatz v(n, s) = y(z), z = n C0s and integrating the equation on z, we have
C1 C0 y þ
a1 2
b y2 þ yzz yyzz y2z ryz r1 yyz yzzz þ c2 yyzzz þ c3 yz yzz ¼ 0: 2
ð4:4Þ
To find the solution of (4.4) we use the simplest equation method [49–51]
yðzÞ ¼ a0 þ w;
w wðzÞ;
ð4:5Þ
where w(z) is the solution of the linear equation
wzz ¼ Awz þ Bw:
ð4:6Þ
Substituting the transformation (4.5) and taking into account Eq. (4.6) we obtain the following solution of the Eq. (4.4)
yðzÞ ¼ a0 þ C 3 ekz þ C 2 ekz ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bðc3 þ c2 Þ þ b2 ðc3 þ c2 Þðc3 3c2 Þ þ 8c3 ðc3 þ c2 Þðb þ 2r1 c3 Þ ; k¼ 4c3 ðc3 þ c2 Þ where C2 and C3 are arbitrary constants, parameters a0, A, B, a1, C0 and C1 are determined by formulae
ð4:7Þ
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
a0 ¼ A¼
2bðc2 þ c3 1Þ 4rc3 ðc3 þ c2 Þ 4r1 c3 b2 2c3 b þ 4r1 c23 c2 b2
b ; 2c3
B¼
419
;
c2 b2 þ 2c3 b þ 4r1 c23 ; 4ðc3 þ c2 Þc23
a1 ¼
4c23 b þ c22 b3 4c2 b2 c3 þ 8r1 c33 4r1 c23 bc2 ; 4c33 ðc3 þ c2 Þ
C0 ¼
1 4r1 c33 4r1 c23 2r1 c23 b þ 2r1 bc2 c3 þ c2 b2 4c33 ðc3 þ c2 Þ
þ2rc23 bc2 4c33 r 4c23 c2 r þ 4c23 b þ 2bc2 c3 2b2 c3 2c2 b2 c3 þ 2c3 rbc22 2c3 b þ b3 c2 c22 b2 ; ðb 2c3 Þ b2 2bc2 þ 2b 2c3 b þ 4c2 rc3 þ 4r1 c3 þ 4rc23 C1 ¼ : 8c33 ðc3 þ c2 Þ For finding the periodic solutions of Eq. (4.4) we use the expression in the form
yðzÞ ¼ a0 þ a1 w þ a2 w2 þ a3 w3 ;
w wðzÞ
ð4:8Þ
where w(z) also satisfies linear ordinary differential equation (4.6). Substituting Eq. (4.8) into Eq. (4.4) and equating the different expressions at w(z) to zero, we have the system of the algebraic equations. Solving this system of equations we obtain the following values of parameters
a2 ¼ a1 ¼ 0;
1 3
c2 ¼ 1; c3 ¼ ; B ¼
a0 ¼ 1;
2
3
a1 6ðA 1Þ
2
;
3
2A þ A a1 þ A 2A þ A a1 þ A ; r1 ¼ ; A1 A1 10A 4 a1 ; C 0 ¼ a1 ; C 1 ¼ ; b¼ 3 2
ð4:9Þ
r¼
and solution of Eq. (4.4) in the form
pffiffiffi pffiffiffi 3 A z p A z p yðzÞ ¼ 1 þ a3 C 3 exp z þ ; þ C 2 exp z 2 6ðA 1Þ 2 6ðA 1Þ 3
2
p ¼ 3ðA 1Þð3A 3A 2a1 Þ;
A – 0;
A – 1;
ð4:10Þ
a3 – 0:
In the case of
pffiffiffi 3A þ 3A2 þ p 6ðA 1Þ
< 0;
3A þ 3A2
pffiffiffi p <0
6ðA 1Þ
solution (4.10) is finite function. Consider case A = 0. In this case the parameters of equation and coefficients of expansion take the form
a1 ¼ a2 ¼ 0;
a0 ¼ 1;
1 3
c2 ¼ 1; c3 ¼ ; B ¼ 4 3
r ¼ a1 ; r1 ¼ a1 ; b ¼ ; C 0 ¼ a1 ; C 1 ¼
a1 6
a1 2
; :
At a1 > 0 we have the following solution of Eq. (4.4)
pffiffiffiffiffi !3
pffiffiffiffiffi yðzÞ ¼ 1 þ a3 C 3 e
6a1 z 6
þ C 2 e
6a1 z 6
;
a3 – 0
ð4:11Þ
In the case of a1 < 0 the solution of Eq. (4.4) can be written as
yðzÞ ¼ 1 þ a3
(pffiffiffiffiffiffiffiffiffiffiffi ) (pffiffiffiffiffiffiffiffiffiffiffi )!3 6ja1 jz 6ja1 jz þ C 2 cos C 3 sin ; 6 6
a3 – 0:
ð4:12Þ
Solutions (4.11), (4.12) of Eq. (4.4) have two arbitrary constants C2 and C3. Solutions (4.10), (4.12) have physical sense. The first of them is limited at the values of parameters at conditions of the above mentioned. The solution (4.12) is the periodical solution. The velocity of the wave described (4.10)–(4.12), is equal to parameter characterizing nonlinear transfer. Dependence of a solution (4.12) of Eq. (4.4) with respect to z at values of parameters a1 = 1, C2 = 0.04, C3 = 0.05 is given on Fig. 1.
420
N.A. Kudryashov, D.I. Sinelshchikov / Applied Mathematics and Computation 217 (2010) 414–421
y 1.2
1.1
1
0.9
0.8
-8
-4
0
4
8
z
Fig. 1. Periodic solution (4.12) of Eq. (4.4).
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