Cavitation of spherical bubbles with =1=surface tension ...

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IntroductionRayleigh-Plesset with no surface tension

Rayleigh-Plesset with surface tension via AbelRayleigh-Plesset equation with viscosity

ConclusionOld Nonlinear Waves Lab

New Nonlinear Waves Lab

.

......

Cavitation of spherical bubbles with surface tensionand viscosity: theory and experiments

Stefan C. [email protected]

Center of Nonlinear Waves, Department of MathematicsEmbry-Riddle Aeronautical University

Daytona Beach, FL. 32114

A topical conference on elementary particles astrophysics, andcosmology

U. of Miami, 2015 FRW–RP

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.. Outline...1 Introduction

Rayleigh-Plesset equationConnection with cosmology

...2 Rayleigh-Plesset with no surface tensionHypergeometric solutions

...3 Rayleigh-Plesset with surface tension via AbelODE → Abel’s equationCapillarity included

...4 Rayleigh-Plesset equation with viscosityGeneral approach - Lemke/Kamke

...5 Conclusion

...6 Old Nonlinear Waves Lab

...7 New Nonlinear Waves LabU. of Miami, 2015 FRW–RP

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.. Summary

to obtain exact solutions nonlinear ODEs via Abel eq.we apply the derivation to Rayleigh-Plesset (RP) eq. via reductionsto 1st kind Abel eq.we present the connection with barotropic FRW eqs.we present graphs of hypergeometric solutions to RP eq. and ℘solutions in presence of surface tensionwhen viscozity is present we present numerical results


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...5 Conclusion



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.. Introduction of RP eq.

RP for a 3D vacuous bubble in watereffects of surface tension are neglected → radius and time of theevolution as parametric closed-form solutions in terms ofhypergeometric functions.including capillarity, via Abel → parametric rational Weierstrassperiodic solutionswith viscosity → numerical solutions only


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.Rayleigh-Plesset eq...

......ρw

(RR +

32

R2)= p − P∞ − 2

R

(σ + 2µw R

)(1)

ρw is the density of the water, R(t) is the radius of bubblep and P∞ are respectively the pressures inside the bubble and atlarge distanceσ is the surface tension of the bubble, µw the dynamic viscosity ofwater1917 with only the pressure difference in the right hand side wasfirst derived by Rayleigh, [7]1949 that Plesset developed the full form of the equation andapplied it to the problem of traveling cavitation bubbles, [8]


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...5 Conclusion



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.. Barotropic FRW eqs. with cosmological constant

Einstein’s field eqs. for spatialy curved FRW cosmology with perfectfluid matter

aa = −4πG

3 (ρ+ 3p) + Λ3

H2 =(

aa

)2= 8πGρ

3 − κa2 + Λ

3(2)

energy-momentum cons. , a(t) scale factor of the univ. in comovingtime, H- Hubble exp. paramρ,p - energy density/ pressure, G gravit. const. Λ cosmologicalconst.using barotropic eq. of state p = (γ − 1)ρ


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.. Comparison

.FRW..

......aa + γa2 + γκ =

Λ

3(γ + 1)a2, (3)

.RP eq...

......ρw

(RR +

32

R2)= p − P∞ − 2

R

(σ + 2µw R

)(4)


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Hypergeometric solutions






...5 Conclusion



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.. No surface tension (no capillarity)

2RR + 3R2 = −2P∞ρw

(5)

R2R is int. fact.

R2 =23

P∞ρw

[(R0

R

)3

− 1

]. (6)

susbst. (6) into (5) we obtain Emden-Fowler eq. R = AtnRm with

part. sol. Rp(t) =5

√256

P∞R30

ρwt

25

for general sol. we use Kudryashov’s [9] subst. R = Sϵ,dt = Rδdτ ,ϵ, δ are constants that depend on the dimension of the bubble


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.. Ansatz

S2τ =

23

P∞ρw

1ϵ2 (R

30 S−3ϵ − 1)S2+2ϵδ−2ϵ . (7)

set ϵ = 13 , δ = 4

Sτ =

√6P∞ρw

S√

R30 S − S2 . (8)

with rational solution

S(τ) =R3

0Bτ2 + 1

, (9)

where B = 32

P∞ρw

R60 .


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.. Parametric solutions

R(τ) =R0

(Bτ2 + 1)13

,

t(τ) = R40

∫ τ

0

dξ

(Bξ2 + 1)43

.

(10)

t(R) = R0

√ρw

6P∞

(R0

R

) 32

[3(R0

R

)−1− 2F1

(13,12;32;

√1 −

( RR0

)3)](11)

see Fig. 1.


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Figure: Radius of the bubble from equation (11) without surface tension.


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ODE → Abel’s equationCapillarity included






...5 Conclusion



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.. Abel.ODEs..

...... R + f2(R)R + f3(R) + f1(R)R2 + f0(R)R3 = 0 (12)

letting R = η(R(t)), we obtain a 2nd kind Abel’s equation

ηη + f3(R) + f2(R)η + f1(R)η2 + f0(R)η3 = 0 . (13)

2nd kind to a 1st kind via η(R(t)) = 1y(R(t))

dydR

= f0(R) + f1(R)y + f2(R)y2 + f3(R)y3 (14)

it is still not known how to integrate it for general fi(R), for specialcases, see Kamke [2] (canonical form) and Lemke [4] (normal form).


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.. Kamke’s transformation

see [2] : y(R) = ω(R)η(ξ)− f23f3

where ω(R) = exp∫ (

f1 −f 22

3f3

)dR

and ξ(R) =∫ω2f3dR

dηdξ

= η3 +Φ(R), (15)

where the invariant is

Φ(R) = f0f 23 +

13

(df2dR

f3 − f2df3dR

− f1f2f3 +29

f 32

)(16)

so when Φ ≡ const . then (26) is separable


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...5 Conclusion



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.. Abel’s eq. for surface tension

R + f1(R)R2 + f3(R) = 0, (17)

where f1(R) = 32R , f2 = f0 = 0, f3 = K1

R + K2R2

and K1 = P∞−Pρw

and K2 = 2σρw

thus, (12) becomes.Bernoulli..

......dydR

= f1(R)y + f3(R)y3 (18)


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.. Weierstrass

R2 =2K1(R3

0 − R3) + 3K2(R20 − R2)

3R3 . (19)

R = Sϵ,dt = Rδdτ

S2τ =

S2+2ϵδ

ϵ2 (a0S−2ϵ + a1S−3ϵ + a3S−5ϵ) , (20)

with ϵ = −1, δ = 2, and a3 = R20

(K2 +

2K13 R0

), a1 = −K2, and

a0 = −2K13 we obtain the Weierstrass elliptic equation

S2τ = a0 + a1S + a3S3 . (21)


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.. Weierstrass - solutions

standard form using scale transformation S(τ) = 4a3℘(τ ;g2,g3)

℘τ2 = 4℘3 − g2℘− g3 (22)

and germs

g2 = −a1a3

4=

K2R20

4(K2 +

2K1R0

3)

g3 = −a0a2

316

=K1R4

024

(K2 +

2K1R0

3)2

(23)


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.. Weierstrass - solutions

S(τ) =6ρw

R20(P∞R0 + 3σ)

℘(τ ;g2,g3), (24)

R(τ) =R2

0(P∞R0 + 3σ)6ρw

1

℘

(τ ;

R20σ

3ρ2w(P∞R0 + 3σ),

R40P∞

54ρ3w(P∞R0 + 3σ)2

)t(τ) =

R40(P∞R0 + 3σ)2

36ρ2w

∫ τ

0

dξ

℘

(ξ;

R20σ

3ρ2w(P∞R0 + 3σ),

R40P∞

54ρ3w(P∞R0 + 3σ)2

)2

(25)see Fig. 2.


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.. Weierstrass - solutions..

Figure: Parametric solutions for parametric time (left), parametric radius (center), andradius vs time (right) from eq. (25) when surface tension is present.


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General approach - Lemke/Kamke






...5 Conclusion



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.. Lemke’s transformation f0 = 0

progress of integration of Abel’s eq. (12) is based on Lemke’stransformations, see [4] : y(R) = η(ξ)u(R) whereu(R) = exp

∫f1(R)dR and ξ =

∫uf2dR

dηdξ

= η2 + g(ξ)η3, (26)

where g(ξ) = u f3f2

is the Appell invariant

letting η = −1ρ

dρdξ , we obtain the 2nd order non-autonomous system

ρ2 d2ξ

dρ2 + g(ξ) = 0 (27)

to integrate the above (for special g’s) see Poincare [5], Ince [1]


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.. Abel’s eq. with viscosity

R + f2(R)R + f1(R)R2 + f3(R) = 0, (28)

where f2(R) = K3R2 , K3 = 4µw

ρw

thus, (26) becomesdηdξ

= η2 + ξ3(b3 + b5ξ2)η3, (29)

b3 =K2

8K 43=

σρ3w

210µ4w,

b5 =K1

32K 63=

P∞ρ5w

217µ6w

(30)


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.. Abel’s eq. with viscosity solution

the invariant is

Φ(R) =2µw [64µ2

w − 9ρwR(6σ + 5P∞R)]

27ρ3wR6

(31)

thus, we resort to numerics only, see Fig. 3.

Figure: Numerical solution for RP eq. with surface tension


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.. Conclusion.proposed FRW..

......aa + γa2 + γκ =

Λ

3(γ + 1)a2 − σ

1a− µ

aa, (32)

.proposed RP..

......RR +

32

R2 − p − P∞ρw

=5Λ6

R2 − 2σρw

1R

− 4µw

ρw

RR

(33)

γ = 32 = 3

2γ − 1 → γ = 53 > 4

3 for radiation and κ = 2P∞3ρw

Λ in RP? σ surface tension and µ viscosity in FRW?U. of Miami, 2015 FRW–RP

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.. Small water tank in Engr. building

16′ × 4′ × 4′

3′ water max192ft3 ≈ 5.5l tones1.5 yrs to buildattracted > 120k USDshallow and deep water for UWV research, supercavitation


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.. Tank

Figure: 16’ long wave tank


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.. Longer water tank in COAS

32′ × 4′ × 4′, 3′ water max, 384ft3 ≈ 11 tones3D numerical simulations of waves, small scale earthquake andtsunami formationEco-dolphin project involving a fleet of 10 AUV (autonomousunderwater vehicle)Dissipative 2D vortex solitons of the complex cubic-quinticGinzburg-Landau equation with applications in nonlinear optics,microbial growth, transport and fate of micro-organisms inunsaturated porous media where the flow is governed by Richards’equationHydrodynamics of tornadoes, mixing of two fluids in diversegeometriesFlow in an artificial heart system built in the lab, optimum design ofa ship U. of Miami, 2015 FRW–RP

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.. COAS

Figure: new COAS buildingU. of Miami, 2015 FRW–RP

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.. Lab

Figure: 32’ long wave tankU. of Miami, 2015 FRW–RP

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.. Experiments : Microcavitation

use microcavitation to control the depth of an immersed submarinecontrol the formation of bubbles via compressed airit can be used to create 0 g so that a submarine can immerse veryfastmanipulate buoyancyarXiv [10]


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Appendix References

.. References I

E. Ince,Ordinary differential equationsDover, N.Y. (1956)

E. KamkeDifferentialgleichungen: Losungsmethoden und LosungenChelsea, N.Y. (1959)

E. Whittaker, G. WatsonModern analysisCambridge, Univ. Press. (1927)


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Appendix References

.. References II

H. LemkeOn a first order differential equation studied by R. LiouvilleSitzungsberichte der berliner math. Ges., 18. (1920)

H. PoincareSur une theorem de M. FuchsActa Math, 7. (1885)

S.C. Mancas, H. C. RosuCavitation of spherical bubbles: closed-form, parametric, andnumerical solutionsarXiv:1508.01157v2. (2015)


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Appendix References

.. References III

M.S. PlessetThe dynamics of cavitation bubblesASME J. Appl. Mech., 16. (1949)

Lord RayleighVIII. On the pressure developed in a liquid during the collapse of aspherical cavityPhilos. Mag. Ser. 6 , 34. (1917)

N. A. Kudryashov, D. I. SinelshchikovAnalytical solutions for problems of bubble dynamicsPhys. Lett. A , 379 (8). (2015)


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Appendix References

.. References IV

S. Mancas, S. Sajjadi, A. Anderson, D. HoffmanMicro cavitation bubbles on the movement of an experimentalsubmarinearXiv:1407.7711v2. (2015)

H.C. Rosu, S. Mancas, P. ChenBarotropic FRW cosmologies with Chiellini dampingPhysics Letters A , 379 (10-11). (2015)


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Appendix References

.. References V

H.C. Rosu, S. Mancas, P. ChenBarotropic FRW cosmologies with Chiellini damping in comovingtimeModern Physics Letters A , 30 (20). (2015)


Cavitation of spherical bubbles with =1=surface tension ...

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