Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer

Abstract

Nonlinear waves are studied in a mixture of liquid and gas bubbles. Influence
of viscosity and heat transfer is taken into consideration on propagation of
the pressure waves. Nonlinear evolution equations of the second and the third
order for describing nonlinear waves in gas-liquid mixtures are derived. Exact
solutions of these nonlinear evolution equations are found. Properties of
nonlinear waves in a liquid with gas bubbles are discussed.

Full text

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arXiv:1112.5436v1 [nlin.PS] 22 Dec 2011
Nonlinear waves in bubbly liquids with
consideration for viscosity and heat transfer
Nikolay A. Kudryashov, Dmitry I. Sinelshchikov
Department of Applied Mathematics, National Research
Nuclear University MEPHI, 31 Kashirskoe Shosse, 115409
Moscow, Russian Federation
Abstract
Nonlinear waves are studied in a mixture of liquid and gas bubbles.
Influence of viscosity and heat transfer is taken into consideration
on propagation of the pressure waves. Nonlinear evolution equations
of the second and the third order for describing nonlinear waves in
gas-liquid mixtures are derived. Exact solutions of these nonlinear
evolution equations are found.Properties of nonlinear waves in a
liquid with gas bubbles are discussed.
1 Introduction
A mixture of liquid and gas bubbles of the same size may be considered as an
example of a classic nonlinear medium. In practice analysis of propagation of
the pressure waves in a liquid with gas bubbles is important problem. Similar
two - phase medium describes many processes in nature and engineering
applications. In particular, such mathematical models are useful for studying
dynamics of contrast agents in the blood flow at ultrasonic researches [1,2].
The literature on this subject deals with theoretical and experimental studies
of the various aspects for propagation of the pressure waves in bubbly liquids.
The first analysis of a problem bubble dynamics was made by Rayleigh [3],
who had solved the problem of the collapse of an empty cavity in a large mass
of liquid. He also considered the problem of a gas - filled cavity under the
assumption that the gas undergoes isothermal compression [4]. Based on
the works of Rayleigh [3] and Foldy [5] van Wijngarden [6] showed that in
the case of one spatial dimension, the propagation of linear acoustic waves
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in isothermal bubbly liquids, wherein the bubbles are of uniform radius, is
described by the linear partial differential equation of the fourth order [7].
The dynamic propagation of acoustic waves in a half - space filled with
a viscous, bubbly liquid under van Wijngaarden linear theory was consid-
ered in the recent work [7]. However we know that there are solitary and
periodic waves in a mixture of a liquid and gas bubbles and these waves can
be described by nonlinear partial differential equations. As for examples of
nonlinear differential equations to describe the pressure waves in bubbly liq-
uids we can point out the Burgers equation [8–10], the Korteweg – de Vries
equation [11–14], the Burgers – Korteweg – de Vries equation [14] and so on.
Many authors applied the numerical methods to study properties of the
nonlinear pressure waves in a mixture of liquid and gas bubbles. Nigmatullin
and Khabeev [15] studied the heat transfer between a gas bubble and a
liquid by means of the numerical approach. Later Aidagulov and et al. [16]
investigated the structure of shock waves in a liquid with gas bubbles with
consideration for the heat transfer between gas and liquid. Oganyan in [17,18]
tried to take into account the heat transfer between a gas bubble and a liquid
to obtain the nonlinear evolution equations for the description of the pressure
waves in a gas-liquid mixture. However, we have some different characteristic
times of nonlinear waves in these processes and it turned out that the solution
of this task is difficult problem.
The purpose of this work is to obtain more common nonlinear partial
differential equations for describing the pressure waves in a mixture liquid
and gas bubbles taking into consideration the viscosity of liquid and the
heat transfer. We also look for exact solutions of these nonlinear differential
equations to study the properties of nonlinear waves in a liquid with gas
bubbles.
This Letter is organized as follows. System of equations for description
of nonlinear waves in a mixture of liquid and gas bubbles taking into con-
sideration for the heat transfer and the viscosity of liquid is given in section
2. In section 3 we obtain the basic nonlinear evolution equation to describe
the pressure waves in liquid with gas bubbles. In sections 4 and 5 we present
nonlinear evolution equations of the second and the third order and search
for exact solutions of these nonlinear differential equations.
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2 System of equations for description of mo-
tion of liquid with gas bubbles with consid-
eration for heat exchange and viscosity
Suppose that a mixture of a liquid and gas bubbles is homogeneous medium
[19]. In this case for description of this mixture we use the averaged temper-
ature, 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
N. We take the processes of the heat transfer and viscosity on the bound-
ary of bubble and liquid into account. We do not consider the processes of
formation, destruction, and conglutination for the gas bubbles.
We have that the volume and the mass of gas in the unit of the mass
mixture can be written as
V =4
3πR3N,X = V ρg,
where R = R(x,t) is bubble radius, ρg = ρg(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 pertur-
bation 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 [19,20]
ρl
?
RRtt+3
2R2
t+4ν
3RRt
?
= Pg− P, (1)
Pg,t+3nPg
R
Rt+3χgNu(n − 1)
2R2
(Tg− Tl) = 0,(2)
Tg=T0Pg
Pg,0
?R
R0
?3
,(3)
where P(x,t) is a pressure of a gas-liquid mixture, Pg is a gas pressure in
a bubble, Tgand Tlare temperatures of liquid and gas accordingly, χgis a
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coefficient of the gas thermal conduction, Nu is the Nusselt number, n is a
polytropic exponent, ν is the viscosity of a liquid.
The expression for the density of a mixture can be presented in the form
[19]
1
ρ=1 − X
ρl
+ V ⇒ ρ =
ρl
1 − X + V ρl. (4)
Considering the small deviation of the bubble radius in comparison with
the averaged radius of bubble, we have
R(x,t) = R0+ η(x,t),R0= const, ||η|| << R0,R(x,0) = R0. (5)
Assume that the liquid temperature is constant and equal to the initial
value
Tl= T |t=0= T0,
At the initial moment, we also have
T0= const.
t = 0 :P = Pg= P0,P0= const,V = V0=4
3πR3
0N.
Substituting Pgand Tgfrom Eqs.(1) and (3) into Eq.(2) and taking rela-
tion (5) into account we have the pressure dependence of a mixture on the
radius perturbation in the form
P − P0+
η
R0P +3nκ
+ρl(8νκ(3n − 1) + 9R2
6R2
R0
Pηt+ κPt+ρl(3R2
0+ 4νκ)
3R0
3R0ηt+2P0
ηtt+ρl(6R2
0− 4νκ)
3R2
0
ηηtt+
0)
0
η2
t+4νρl
R0η +3P0
R2
0
η2= 0,
κ =
2R2
0P0
3χNu(n − 1)T0.
(6)
From Eq.(4) we also have the dependence ρ on η using formula (5)
ρ = ρ0− µη + µ1η2,ρ0=
ρl
1 − X + V0ρl,
µ =
3ρ2
lV0
R0(1 − X + V0ρl)2,µ1=6ρ2
lV0(2ρlV0− 1 + X)
R2
0(1 − X + ρlV0)3.
(7)
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We use the system of equations for description of the motion of a gas-
liquid mixture flow in the form
∂ρ
∂t+∂(ρu)
∂x
= 0,ρ
?∂u
∂t+ u∂u
∂x
?
+∂P
∂x= 0, (8)
where u = u(x,t) is a velocity of a flow of a gas-liquid mixture.
Eq.(8) together with Eqs.(6) and (7) can be applied for description of
nonlinear waves in a gas-liquid medium.
Consider the linear case of the system of equations (6), (7)and (8). As-
suming, that pressure in a mixture is proportional to perturbation radius,
we obtain the linear wave equation for the radius perturbations
ηtt= c2
0ηxx,c0=
?
3P0
µR0. (9)
Let us introduce the following dimensionless variables
t =
l
c0t′,x = lx′,u = c0u′,η = R0η
′,P = P0P′+ P0,
where l is the characteristic length of wave.
Using the dimensionless variables the system of equations (6), (7) and (8)
can be reduced to the following (the primes of the variables are omitted)
ηt−
ρ0
µR0ux+ uηx+ ηux−2µ1R0
µ
ηηt= 0,
−ρ0
µR0(ut+ uux) + ηut−1
3Px= 0,
P + κ1Pt+ ηP + 3nκ1ηtP = −(β1+ β2)ηtt− (2β2− β1)ηηtt−
−
2
?3n − 1
β1+3
2β2
?
η2
t− ληt− 3η + 3η2,
(10)
where the parameters are determined by formulae
λ =4νρlc0
3P0l
+ 3nκ1,β1=4ν κ ρlc2
3P0l2,
0c3
0
P0l3
0
β2=ρlc2
0R2
P0l2,
0
γ =κρlR2
,
κ1=κc0
l
.
(11)
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