Mathematical Modelling of the Dynamics of a Viscous Fluid with Gas BubblesJaden R. Dasiuk and Alexei F. Cheviakov

Department of Mathematics and Statistics, University of Saskatchewan

Motivation & Application Areas

Bubble flow: a two-phase flow; small bubbles are dispersed or suspended in liquid continuum.

I General interest: bubbles change flow dynamics by increasing or decreasing local turbulence.

I A specific application: bubble regime of laminar magma flow in a volcanic conduit.

Retrieved July 27, 2015 fromhttp://www.pitt.edu/ cejones/GeoImages/

Vesicular basalt; retrieved July 27, 2015 fromhttp://www.sciencebuzz.org/buzz-tags/magmatic-differentiation

A schematic of magma flow regimes in a volcanic conduit [2].

Dynamics of a Single Gas Bubble

R(t)

Gas: P2(t),T2(t)

Fluid: P1(t),T1(t)

R(t) is the radius of the bubble; P = P1(t), P2(t) denote thepressure of fluid and gas; T1(t), T2(t), the temperature of fluidand gas; P0, T0 and R0 are the initial values of gas pressure,temperature, and the gas bubble radius; ρ1 is the mass densityof the liquid; ν is the fluid kinematic viscosity coefficient; γ isthe adiabatic constant; χ is the gas thermal conductioncoefficient; Nu is the dimensionless Nusselt number (relativethickness of the thermal layer).

Gas bubble dynamics equations:

I The Rayleigh equation: single bubble radius dynamics [3]

P2 = P + ρ1

(RRtt +

3

2Rt

2 +4ν

3RRt

). (1a)

I Pressure equation, from conservation of energy and assumption that Nusselt number is constant:

(P2)t +3γP2R

Rt +3χNu (γ − 1)

2R2(T2 − T1) = 0, Nu = const. (1b)

I Ideal gas relationship:

T2 =T0P2P0

(R

R0

)3

. (1c)

I Derivatives are denote by subscripts: ∂P2/∂t = (P2)t, etc.

Dynamics of a Fluid with Gas Bubbles 

x

Main assumptions:

I Quantities depend on space and time (R = R(x, t), P = P (x, t), etc.).

I Fluid temperature is constant: T1 = T0.

I The bubble radius is a small deviation from its average:

R(x, t) = R0 + η(x, t). (2)

From these assumptions and (1), the following PDE is obtained (cf. [1]):

P − P0 +1

R0ηP +

3γκ

R0Pηt + κPt +

ρ1(3R02 + 4νκ)

3R0ηtt +

ρl(6R02 − 4νκ)

3R02

ηηtt

+ρ1(8νκ(3γ − 1) + 9R0

2)

6R02

ηt2 +

4νρ13R0

ηt +2P0R0

η − 3P0

R02η2

+ρ1κR0ηttt −3κγ

R02ηηtP + ρ1κηηttt + ρlκ(3γ + 4)ηtηtt = 0,

(3)

where

κ = const =2R0

2P03χNu (γ − 1)T0

.

Density of the fluid mixture:

ρ =ρ1

1−X + V̂ ρ1, (4)

where V̂ is the relative volume of gas (gas volume per unit mass of mixture) and X is the relative masscontent of gas (mass of gas per unit mass of mixture).

Fundamental equations of fluid dynamics:

The standard Euler equations that describe fluid dynamics are also used:

ρt + (ρu)x = 0, (5)

ρ(ut + uux) + Px = 0. (6)

A Dimensionless Model for the Fluid with Gas Bubbles

We recast the model (2) – (6) into the dimensionless form, using the substitution

t =`

c0t′, x = `x′, u = c0u

′, η = R0η′, P = P0P

′ + P0, (7)

where ` is the characteristic wavelength, and c0 =√

3P0µR0

is the characteristic wave speed. The primes on

the new variables will be dropped for convenience.

Making the substitutions, we arrive at the following dimensionless model for a fluid with gasbubbles:

ηt −ρ0µR0

ux + uηx + ηux −2µ1R0

µηηt −

2R0µ1µ

uηηx −R0µ1µ

uxη2 = 0,

− ρ0µR0

(ut + uux) + ηut −1

3Px + ηuux −

R0µ1µ

η2ut −R0µ1µ

η2uux = 0,

P + κ1Pt + ηP + 3γκ1ηtP + (β1 + β2)ηtt + (2β2 − β1)ηηtt +((3γ − 1)β1 +

32β2

)ηt2

+ληt + 3η − 3η2 + ωηttt + ωηηttt + ω(3γ + 4)ηtηtt − 3κ1γηηtP − 3κ1γηηt = 0,

(8a)

(8b)

(8c)

where

β1 =4νκρlc0

2

3P0l2, β2 =

ρlc02R0

2

P0l2, κ1 =

κc0l, ω = κ1β2, λ =

β1κ1

+ 3γκ1,

ρ0 =ρ1

1−X0 + V0ρ1, µ =

3ρ12V0

R0(1−X0 + V0ρ1)2, µ1 =

6ρ12V0(2ρ1V0 − 1 +X0)

R02(1−X0 + V0ρ1)3

,

and V0, X0 are the average relative volume and mass content of gas.

An Asymptotic Approximation

Rescale the independent variables:

ξ = εm(x− t), τ = εm+1t, 0 < ε� 1, m > 0. (9)

I ξ: a large-scale moving spatial variable (ξ ∼ 1 when x ∼ ε−m� 1)

I τ : ‘slow time’.

Assume a standard asymptotic expansion of flow parameters near the equilibrium:

u = εu1 + ε2u2, η = εη1 + ε2η2, P = εP1 + ε2P2. (10)

I (9), (10) are substituted in (8).

I Various m can be chosen.

I Coefficients at different powers of ε must vanish independently.

I Obtain a single PDE on P1(ξ, τ ): a scaled, dimensionless pressure perturbation.

Case A: m = 1

In this case, we arrive at the Burgers’ equation

P1τ + A1P1P1ξ +B1P1ξξ = 0, (11)

where

A1 =1

3

(µR0

ρ0− µ1R0

µ+ 2

), B1 =

(κ12− κ1γ

2− 1

6

β1κ1

).

A change of variables

ξ =

(B1

2

A1

)13

x, τ = −(B1

A12

)13t, P1(ξ, τ ) = −

(B1

A12

)13u(x, t) (12)

maps (11) into the standard form

ut + uux = uxx. (13)

Case B: m = 12

In this case, we arrive at the PDEs

P1τ + A2P1P1ξ +B2P1ξξξ = 0, C2P1ξ = 0, (14)

where

A2 =1

3

(µR0

ρ0− µ1R0

µ+ 2

), B2 =

1

6(β1 + β2), C2 =

(3γκ1

2 − 3κ12 + β1

). (15)

When C2 = 0, a change of variables

ξ =

(6B2

2

A2

)15

x, τ =

(216B2

A23

)15t, P1(ξ, τ ) =

(216B2

A23

)15u(x, t) (16)

maps (14) into a canonical Korteweg - de Vries (KdV) equation

ut + 6uux + uxxx = 0. (17)

Other Cases and Further Work

Physical motivations were not considered in the choosing of m. Cases m = 1 and m = 12 work well with

the Taylor expansion (10). Some other cases of m that were tried yielded results that were similar(differed by a term) to the cases m = 1 and m = 1

2, while other cases were degenerate.

In particular, in both Cases (A) and (B), a determining equation was found that gave the relationshipbetween P1 and η1 as:

P1 + 3η1 = 0 (18)

I Physically, η1 is the dimensionless scaled radius perturbation of the gas bubbles. The meaning of therelationship (18) is that higher pressure leads to lower bubble radius.

I It remains to systematically study the case of general m and its compatibility with more generalasymptotic expansions (10), which will possibly lead to more general nonlinear PDE models.

Traveling Wave Solutions of the Burgers Equation

One can obtain particular solutions of (13) and (17) using the traveling wave ansatz: u(x, t) = g(z)where z = x− ct. This approach is based on symmetries of PDEs, and is useful in many cases, since aPDE can be reduced to a much simpler ODE. For example, the Burgers equation reduces to the followingODE:

− cg′(z) + g(z)g′(z)− g′′(z) = 0. (19)

The latter can be solved by integrating twice:

u(x− ct) = g(z) = α+β e12 (α−β)z

1+e12 (α−β)z

, (20)

where α and β are arbitrary constants, and c = 12(α + β). Assuming α > β, one observes that α is the

top of the waveform and β is its bottom.

x

u

-

,u(x ! ct0)

u(x ! ct1); t1 > t0

u(x ! ct2); t2 > t1

Traveling wave solution for the Burgers equation (13).

One-Soliton Solution of the KdV equation

We use travelling wave ansatz for the KdV, reducing it to an ODE. The latter admits the well-knownsolitary wave-type exact solutions

u(x− ct) = g(z) =c

2 cosh2(−√c2 z), (21)

where c, the wave’s speed, is an arbitrary constant. (21) is known as a single-soliton travelling wavesolution for the KdV equation.

x

u

u(x − ct0)

u(x − ct1), t1 > t0

u(x − ct2), t2 > t1

Travelling wave soliton solution of the KdV equation.

Multi-Soliton Solutions of the KdV equation

Multi-soliton solutions exist for (17); a two-soliton solution, for example, is given by

u(x, t) = −2(cosh

(k1(4 k1

2t− x)))2

k12k2

2 −(cosh

(k1(4 k1

2t− x)))2

k24 +

(cosh

(k2(4 k2

2t− x)))2

k14 −

(cosh

(k2(4 k2

2t− x)))2

k12k2

2 − k14 + k12k2

2(cosh

(k1(4 k1

2t− x))k2 cosh

(k2(4 k2

2t− x))− sinh

(k2(4 k2

2t− x))k1 sinh

(k1(4 k1

2t− x)))2 , (22)

where k1 and k2 are constants.

I KdV is an integrableequation.

I Nonlinear solitarywaves interact likeparticles.

I Higher waves havehigher speeds.Increasing c in (21)will increase thewave’s speed andheight.

0

1

2

0

1

2

0

1

2

9

-10 0 10 20 30 40 500

1

2

t = 1

t = 0

t = 5

t = 10

Evolution of a two soliton solution when k1 =12 and k2 = 1.

Conclusions & Future Work

Boat following a solitary waveRetrieved Aug 4, 2015 from http://www.maplesoft.com/

I Solitons on water surface arestraightforward to reproduce in shallowconstant-depth channels.

I Stable solitary wave-type arise for variousnonlinear models, including fluid dynamics,plasma physics, and nonlinear optics, andare observed in laboratory and numericalexperiments.

I It has been shown that the classicalBurgers’ and KdV equations arise in thecontext of bubble flow dynamics.

I Ongoing work: consider more generalasymptotic expansions of the form (9), (10)to model a wider range of physicalsituations for the bubble flows.

References

[1] N.A. Kudryashov and D.I. Sinelshchikov.Nonlinear waves in bubbly liquids with considersation for viscosity and heat transfer.Physics Letters A, 374:2011–2016, 2010.

[2] O. Melnik, A.A. Barmin, and R.S.J. Sparks.Dynamics of magma flow inside volcanic conduits with bubble overpressure buildup and gas loss throughpermeable magma.Journal of volcanology and geothermal research, 143:53–68, 2005.

[3] V.E. Nakoryakov, B.G. Pokusaev, and I.R. Shreiber.Wave propagation in gas-liquid media.CRC Press, 1993.

Department of Mathematics and Statistics - University of Saskatchewan - Saskatoon, Saskatchewan