ARTICULOS

Influence of surface tension on two fluids shearing instability

 

Rahul Banerjee,¹* S. Kanjilal¹

* E-mail:rbanerjee.math@gmail.com

¹ St. Paul's Cathedral Mission College, 33/1, Raja Ram-mohan Roy, Sarani, 700 009 Kolkata, India.

---

Received: 20 March 2014, Accepted: 7 August 2014
Edited by: A. Marti

Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.060006 _ ISSN 1852-4249

---

 

Using extended Layzer's potential flow model, we investigate the effects of surface tension on the growth of the bubble and spike in combined Rayleigh-Taylor and Kelvin-Helmholtz instability. The nonlinear asymptotic solutions are obtained analytically for the velocity and curvature of the bubble and spike tip. We find that the surface tension decreases the velocity but does not affect the curvature, provided surface tension is greater than a critical value. For a certain condition, we observe that surface tension stabilizes the motion. Any perturbation, whatever its magnitude, results stable with nonlinear oscillations. The nonlinear oscillations depend on surface tension and relative velocity shear of the two fluids.

I. Introduction

When two different density fluids are divided by an interface, the interface becomes unstable with exponential growth under the action of a constant acceleration acting in the direction perpendicular to the interface from the heavier to lighter fluid or under the action of relative velocity shear of two fluids. These two types of instabilities are known as Rayleigh-Taylor and Kelvin-Helmholtz instabilities, respectively. Temporal development of the nonlinear structure of the interface consequent to Rayleigh-Taylor or Kelvin-Helmholtz instability is currently a topic of interest both from theoretical and experimental points of view. The nonlinear structure is called a bubble if the lighter fluid penetrates across the unperturbed interface into the heavier fluid and it is called a spike if the opposite takes place. The instabilities arise in connection with a wide range of problems ranging from direct or indirect laser driven experiments in the ablation region at compression front during the process of inertial confinement fusion [1, 2] to mixing of plasmas in space plasma systems, such as boundary of planetary magnetosphere, solar wind and cluster of galaxies [3]. In high energy density physics(HEDP), formation of supernova remnant or formation of astrophysical jets [4-8] are also seen in these types of instabilities. In high energy density plasma experiments using Omega laser [9], Kelvin-Helmholtz instability growth has recently been observed .

There are several methods to describe the nonlinear structure of the interface of two constant density fluids under potential theory and the associated nonlinear dynamics has been studied by many authors [10-13]. Layzer [10] described the formation of the structure using an expansion near the tip of the bubble or the spike up to second order in the transverse coordinates in two dimensional motion and this approach was extended in Ref. [14] for Kelvin-Helmholtz instability. It is well known [15] that the surface tension reduces the linear Rayleigh-Taylor Growth rate. The lowering in the growth rate is seen to increase with increase in the wave number k up to a critical wave number, where T denotes surface tension, and p_h and p are the densities of the heavier and lighter fluids, respectively. The same effect has been described by Mikaelian [16] for Rayleigh-Taylor instability in finite thickness and Sung-Ik Sohn [17] described the effect using the Layzer nonlinear potential model. The nonlinear theory influence of surface tension was elaborately studied by Pullin [18] and Garnier et al. [19] using numerical methods.

The present paper addresses to the problem of the time development of the nonlinear interfacial structure caused by combined Rayleigh-Taylor and Kelvin-Helmholtz instability in presence of surface tension. It is shown that the growth rate of the instabilities is affected by the surface tension. The growth rate of the tip of the bubble or spike are significantly reduced due to the surface tension. We observed an oscillatory stabilization of the interface for large surface tension. This oscillation depends on the relative velocity shear also. Section II deals with the basic hydrodynamical equations together with the geometry involved. Here we assume that the fluids are inviscid and the motion is irrota-tional. The investigation of the nonlinear aspect of the structure of the two fluids interface is facilitated by Bernoull's equation together with the pressure balance equation at the interface. The long time asymptotic behavior of the bubble and spike tip for combined Rayleigh-Taylor and Kelvin-Helmholtz instabilities is derived in section III.A and III.B, respectively. We have also discussed the characteristics of the tip of the bubble and the spike derived analytically and numerically. Finally, we have concluded the results in section IV.

II. Basic mathematical model

We have considered two incompressible fluids separated by an interface located at y = 0 in a two-dimensional x - y plane, where x axis lying normal to the unperturbed fluid interface. The fluid with density p_h is assumed to overlie the fluid with density p_; and gravity is taken along negative y-axis. In the following discussion, we shall denote the properties of the fluid above the interface by the subscript h and below the interface by the subscript l. After perturbation, the nonlinear interface is assumed to take up a parabolic shape, given by

The perturbed interface forms a bubble or spike according to n₀(t) > 0, n₂(t) < 0 or n_o(t) < 0, n₂(t) > 0. Functions n_o(t) and n₁(t) are related to the position of the tip of the bubble from the unperturbed interface, i.e, at time t the position of the bubble tip is (n₁(t),n_o(t)) and n₂(t) is related to the bubble curvature.

In our previous works [14,20-23], we have considered n₁(t) = 0 due to the absence of velocity shear parallel to the unperturbed interface. However, in presence of streaming motion of the fluids, the tip of the bubble moves parallel to unperturbed interface with velocity n₁(t).

According to the extended Layzer model [10,11, 14,20], the velocity potentials describing the motion for the upper (heavier) and lower (lighter) fluids are assumed to be given by

 

where U_h and U_i are streaming velocities of upper and lower fluids, respectively, and k is the perturbed wave number.

The evolution of the interface y = n(x, t) can be determined by the kinematical and dynamical boundary conditions. The kinematical boundary conditions are

and the dynamical boundary condition (first integral of the momentum equation) is of the form

asymptotic stage are obtained for arbitrary Atwood number and velocity shear. Surface tension becomes a stabilizing factor of the instability, provided it is larger than a critical value. In this case, oscillatory behavior of motion described by numerical integration of governing equations. The nature of oscillations depends on both surface tension and relative velocity shear of two fluids. On the other hand, below the critical value, surface tension dominates the growth and growth rate of the instability. This result is expected to improve the understanding of the stabilization factor for the astrophysical instability.

Acknowledgements - This work was supported by the University Grant Commission, Government of India under Ref. No. PSW-43/12-13 (ERO).

1. D K Bradley, D G Barun, S G Glendinning, M J Edwards, J L Milovich, C M Sorce, G W Collins, S W Hann, R H Page, R J Wallace, Very-high-growth-factor planar ablative Rayleigh-Taylor experiments, Phys. Plasmas 14, 056313 (2007).

2. K S Budil, B A Remington, T A Peyser, K O Mikaelian, P L Miller, N C Woolsey, W M Wood-Vasey, A M Rubenchik, Experimental comparison of classical versus ablative Rayleigh-Taylor instability, Phys. Rev.Lett. 76, 4536 (1996).

3. H Hasegawa, M Fujimoto, T-D Phan, H Reme, A Balogh, M W Dunlop, C Hashimoto, R TanDokoro, Transport of solar wind into Earth's magnetosphere through rolled-up Kelvin- Helmholtz vortices, Nature 430, 755 (2004).

4. R P Drake, Hydrodynamic instabilities in astrophysics and in laboratory high-energydensity systems, Plasma Phys. Control. Fusion 47, B419 (2005).

5. J O Kane, H F Robey, B A Remington, R P Drake, J Knauer, D D Ryutov, H Louis, R Teyssier, O Hurricane, D Arnett, R Rosner, A Calder, Interface imprinting by a rippled shock using an intense laser, Phys. Rev. E 63, 055401R (2001).

6. D D Ryutov, B A Remington, Scaling astro-physical phenomena to high-energy-density laboratory experiments, Plasma Phys. Control. Fusion 44, B407 (2002).

7. L Spitzer, Behavior of matter in space, Astro-phys. J. 120, 1 (1954).

8. B A Remington, R P Drake, H Takabe, D Ar-nett, A review of astrophysics experiments on intense lasers, Phys. Plasma 7, 1641 (2000).

9. E C Harding, J F Hansen, O A Hurricane, R P Drake, H F Robey, C C Kuranz, B A Remington, M J Bono, M J Grosskopf, R S Gillespie, Observation of a Kelvin-Helmholtz instability in a high-energy-density plasma on the omega laser, Phys. Rev.Lett. 103, 045005 (2009).

10. D Layzer, On the Instability of Superposed Fluids in a Gravitational Field, Astrophys. J. 122, 1 (1955).

11. V N Goncharov, Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers, Phys. Rev. Lett. 88, 134502 (2002).

12. Sung-Ik Sohn, Simple potential-flow model of Rayleigh-Taylor and Richtmyer-Meshkov instabilities for all density ratios, Phys. Rev. E 67, 026301 (2003).

13. Q Zhang, Analytical solutions of Layzer-type approach to unstable interfacial fluid mixing, Phys. Rev. Lett. 81, 3391 (1998).

14. R Banerjee, L Mandal, M Khan, M R Gupta, Effect of viscosity and shear flow on the nonlinear two fluid interfacial structures, Phys. Plasmas 19, 122105 (2012).

15. S Chandrasekhar, Hydrodynamic and hydro-magnetic stability, Dover, New York (1961).

16. K O Mikaelian, Rayleigh-Taylor instability in finite-thickness fluids with viscosity and surface tension, Phys. Rev. E 54, 3676 (1996).

17. Sung-Ik Sohn, Effects of surface tension and viscosity on the growth rates of Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Phys. Rev. E 80, 055302(R) (2009).

18. D I Pullin, Numerical studies of surface tension effects in nonlinear KelvinHelmholtz and RayleighTaylor instability, J. Fluid Mech. 119, 507 (1982).

19. J Garnier, C Cherfils-Clerouin, A P Holstein, Statistical analysis of multimode weakly nonlinear Rayleigh-Taylor instability in the presence of m surface tension, Phys. Rev. E 68, 036401 (2003).

20. R Banerjee, L Mandal, S Roy, M Khan, M R Gupta, Combined effect of viscosity and vorticity on single mode RayleighTaylor instability bubble growth, Phys. Plasmas 18, 022109 (2011).

21. M R Gupta, R Banerjee, L K Mandal, R Bhar, H C Pant, M Khan, M K Srivastava, Effect of viscosity and surface tension on the growth of RayleighTaylor instability and Richt-myerMeshkov instability induced two fluid interfacial nonlinear structure, Indian J. Phys. 86, 471 (2012).

22. R Banerjee, L Mandal, M Khan, M R Gupta, Bubble and spike growth rate of Rayleigh Taylor and Richtmeyer Meshkov instability in finite layers, Indian J. Phys. 87, 929 (2013).

23. R Banerjee, L Mandal, M Khan, M R Gupta, Spiky Development at the Interface in Rayleigh- Taylor Instability: Layzer Approximation with Second Harmonic, J. Morden Phys. 4, 174 (2013).