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Research Papers: Flows in Complex Systems

# Combined Effect of Viscosity, Surface Tension and Compressibility on Rayleigh-Taylor Bubble Growth Between Two Fluids

[+] Author and Article Information
Sourav Roy

Department of Instrumentation Science,
Kolkata-700032, India
e-mail: phy.sou82@gmail.com

L. K. Mandal

Department of Physics,
Budge Budge Institute of Technology,
Kolkata-137, India
e-mail: labakanta@gmail.com

Manoranjan Khan

Department of Instrumentation Science,
Kolkata-700032, India
e-mail: mkhan_ju@yahoo.com

M. R. Gupta

Department of Instrumentation Science,
Kolkata-700032, India

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 21, 2013; final manuscript received May 9, 2014; published online July 9, 2014. Assoc. Editor: Stuart Dalziel.

J. Fluids Eng 136(9), 091101 (Jul 09, 2014) (6 pages) Paper No: FE-13-1033; doi: 10.1115/1.4027655 History: Received January 21, 2013; Revised May 09, 2014

## Abstract

The combined effect of viscosity, surface tension, and the compressibility on the nonlinear growth rate of Rayleigh-Taylor (RT) instability has been investigated. For the incompressible case, it is seen that both viscosity and surface tension have a retarding effect on RT bubble growth for the interface perturbation wave number having a value less than three times of a critical value ($kc=(ρh-ρl)g/T$, $T$ is the surface tension). For the value of wave number greater than three times of the critical value, the RT induced unstable interface is stabilized through damped nonlinear oscillation. In the absence of surface tension and viscosity, the compressibility has both a stabilizing and destabilizing effect on RTI bubble growth. The presence of surface tension and viscosity reduces the growth rate. Above a certain wave number, the perturbed interface exhibits damped oscillation. The damping factor increases with increasing kinematic viscosity of the heavier fluid and the saturation value of the damped oscillation depends on the surface tension of the perturbed fluid interface and interface perturbation wave number. An approximate expression for asymptotic bubble velocity considering only the lighter fluid as a compressible one is presented here. The numerical results describing the dynamics of the bubble are represented in diagrams.

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## Figures

Fig. 1

Solution of the set of Eqs. (30) neglecting the last term of the third equation for the bubble considering initial values ξ1 = 0.1,ξ2 = -0.1,ξ3 = 0.1 and r = 1.545, s = 1. The solid line represents the solution for χ = 0, k2/kc2 = 0; the dashed line represents that for χ = 0, k2/kc2 = 2, and the dotted line represents that for χ = 0.3, k2/kc2 = 2.

Fig. 2

Solution of the set of Eqs. (30) neglecting the last term of the third equation for bubble considering initial values ξ1 = 0.1,ξ2 = -0.1,ξ3 = 0.1 and r = 1.545, s = 1. The solid line represents the solution for χ = 0, k2/kc2 = 0; the dashed line represents that for χ = 0.3, k2/kc2 = 5; the dotted line represents that for χ = 0.7, k2/kc2 = 5; and the dashed–dotted line represents that for χ = 0.7, k2/kc2 = 5.

Fig. 3

Solution of the set of Eqs. (30) for bubble considering initial values ξ1 = 0.1,ξ2 = -0.1,ξ3 = 0.1,c =  100 and r = 1.545, s = 1. The solid line represents the solution for ρh1/ρh0* = 0, ρl1/ρl0* = 0, χ = 0, k2/kc2 = 0; the dashed line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0, k2/kc2 = 0; the dotted line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0, k2/kc2 = 1; the dashed–dotted line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0.3, k2/kc2 = 0 and the dashed–dotted–dotted line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0.3, k2/kc2 = 1.

Fig. 4

Solution of the set of Eqs. (30) for the bubble considering initial values ξ1 = 0.1,ξ2 = -0.1,ξ3 = 0.1,c = 100 and r = 1.545, s = 1. The solid line represents the solution for ρh1/ρh0* = 0, ρl1/ρl0* = 0, χ = 0, k2/kc2 = 0; the dashed line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0.3, k2/kc2 = 6; the dotted line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0.7, k2/kc2 = 6; and the dashed–dotted line represents that for ρh1/ρh0* = -0.002, ρl1/ρl0* = -0.002, χ = 0.7, k2/kc2 = 7.

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