Papers in Physics, vol. 6, art. 060006 (2014) 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 www.papersinphysics.org ISSN 1852-4249 Influence of surface tension on two fluids shearing instability Rahul Banerjee,1∗ S. Kanjilal1 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 instabil- ities, 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 pen- etrates 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 abla- ∗E-mail:rbanerjee.math@gmail.com 1 St. Paul’s Cathedral Mission College, 33/1, Raja Ram- mohan Roy, Sarani, 700 009 Kolkata, India. tion region at compression front during the pro- cess of inertial confinement fusion [1, 2] to mix- ing of plasmas in space plasma systems, such as boundary of planetary magnetosphere, solar wind and cluster of galaxies [3]. In high energy den- sity physics(HEDP), formation of supernova rem- nant 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 non- linear structure of the interface of two constant density fluids under potential theory and the as- sociated 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 dimen- sional 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 low- ering in the growth rate is seen to increase with increase in the wave number k up to a critical wave 060006-1 Papers in Physics, vol. 6, art. 060006 (2014) / R. Banerjee et al. number kc = √ (ρh−ρl)g T , where T denotes surface tension, and ρh and ρl are the densities of the heav- ier 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 non- linear potential model. The nonlinear theory influ- ence 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 sig- nificantly reduced due to the surface tension. We observed an oscillatory stabilization of the inter- face for large surface tension. This oscillation de- pends on the relative velocity shear also. Section II deals with the basic hydrodynamical equations to- gether 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 character- istics of the tip of the bubble and the spike derived analytically and numerically. Finally, we have con- cluded the results in section IV. II. Basic mathematical model We have considered two incompressible fluids sep- arated by an interface located at y = 0 in a two- dimensional x − y plane, where x axis lying nor- mal to the unperturbed fluid interface. The fluid with density ρh is assumed to overlie the fluid with density ρl 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 sub- script l. After perturbation, the nonlinear interface is assumed to take up a parabolic shape, given by y = η(x,t) = η0(t) + η2(t)(x−η1(t))2 (1) The perturbed interface forms a bubble or spike according to η0(t) > 0, η2(t) < 0 or η0(t) < 0, η2(t) > 0. Functions η0(t) and η1(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 (η1(t),η0(t)) and η2(t) is related to the bubble curvature. In our previous works [14, 20–23], we have con- sidered η1(t) = 0 due to the absence of velocity shear parallel to the unperturbed interface. How- ever, in presence of streaming motion of the fluids, the tip of the bubble moves parallel to unperturbed interface with velocity η̇1(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 φh(x,y,t) = a1(t) cos (k(x−η1(t))e−k(y−η0(t)) + a2(t) sin (k(x−η1(t))e−k(y−η0(t)) −xUh (2) φl(x,y,t) = b0(t)y + b1(t) cos (k(x−η1(t))ek(y−η0(t)) + b2(t) sin (k(x−η1(t))ek(y−η0(t)) −xUl (3) where Uh and Ul are streaming velocities of up- per and lower fluids, respectively, and k is the per- turbed wave number. The evolution of the interface y = η(x,t) can be determined by the kinematical and dynamical boundary conditions. The kinematical boundary conditions are ∂η ∂t − ∂η ∂x ∂φh ∂x = − ∂φh ∂y (4) ∂η ∂x ( ∂φh ∂x − ∂φl ∂x ) = ∂φh ∂y − ∂φl ∂y (5) and the dynamical boundary condition (first inte- gral of the momentum equation) is of the form 060006-2 Papers in Physics, vol. 6, art. 060006 (2014) / R. Banerjee et al. −ρh(l) ∂φh(l) ∂t + 1 2 ρh(l)(~∇φh(l))2 + ρh(l)gy = −ph(l) + fh(l)(t) (6) The pressure boundary condition at two fluid in- terface including surface tension [17, 22] is ph −pl = T R (7) where T is the surface tension and R is the radius of curvature. Plugging the condition (7) at the interface y = η(x,t) in Eq. (6), we obtain the following equation. ρh[− ∂φh ∂t + 1 2 (~∇φh)2] −ρl[− ∂φl ∂t + 1 2 (~∇φl)2] +g(ρh −ρl)y = − T R + fh −fl (8) We have restricted our study near the peak of the perturbed structure where |k(x−η1(t))|� 1. Thus, we can neglect the terms of O(|x−η1|i) (i ≥ 3) [14]. With this point of view, we have 1 R = 2η2 ( 1 + 4η22(x−η1) 2 )−3 2 ≈ 2η2 ( 1 − 6η22(x−η1) 2 ) (9) We substitute all the parameters η, φh and φl in the kinematic and dynamic boundary conditions repre- sented by Eqs. (4), (5), (8) and (9), and equate co- efficients of (x−η1)i,(i = 0, 1, 2) and neglect terms O(|x−η1|i) (i ≥ 3). This yields the following equa- tions. dξ1 dτ = ξ4 (10) dξ2 dτ = Vh − ξ5(2ξ3 + 1) 2ξ3 (11) dξ3 dτ = − 1 2 (6ξ3 + 1)ξ4 (12) kb0√ kg = − 12ξ3ξ4 6ξ3 − 1 (13) k2b1√ kg = 6ξ3 + 1 6ξ3 − 1 ξ4 (14) k2b2√ kg = (2ξ3 + 1)ξ5 − 2ξ3(Vh −Vl) 2ξ3 − 1 (15) dξ4 dτ = N1(ξ3,r) D1(ξ3,r) ξ24 (6ξ3 − 1) + 2(1 −r)ξ3(6ξ3 − 1) D1(ξ3,r) ( 1 − 12ξ22 k2 k2c ) + N2(ξ3,r) D1(ξ3,r) (6ξ2 − 1)ξ25 2ξ3(2ξ3 − 1)2 + 2(4ξ3 − 1)(6ξ3 − 1) D1(ξ3,r)(2ξ3 − 1)2 × [(Vh −Vl)2ξ3 − (Vh −Vl)(2ξ3 + 1)ξ5] (16) and dξ5 dτ = − (2ξ3 − 1)rξ4ξ5 2ξ3D2(ξ3,r) + ξ4(6ξ3 + 1) 2D2(ξ3,r)(6ξ3 − 1)(2ξ3 − 1) (17) × [4(Vh −Vl)(4ξ3 − 1) − ξ5 ξ3 (28ξ23 − 4ξ3 − 1)] where r = ρh ρl ; ξ1 = kη0; ξ2 = kη1; ξ3 = η2 k ; ξ4 = k2a1√ kg ; ξ5 = k2a2√ kg ; τ = t √ kg; k2c = (ρh−ρl)g T and Vh(l) = kUh(l)√ kg are corresponding dimensionless quantities. The function N1,2(ξ3,r) and D1,2(ξ3,r) are given by N1(ξ3,r) = 36(1 −r)ξ23 + 12(4 + r)ξ3 + (7 −r); D1(ξ3,r) = 12(r − 1)ξ23 + 4(r − 1)ξ3 − (r + 1) (18) and N2(ξ3,r) = 16(1 −r)ξ33 + 12(1 + r)ξ 2 3 − (1 + r); D2(ξ3,r) = 2(1 −r)ξ3 + (r + 1) (19) The temporal development of the combined effect of Rayleigh–Taylor and Kelvin–Helmholtz instabil- ity is given by Eqs. (10)–(12), (16) and (17). 060006-3 Papers in Physics, vol. 6, art. 060006 (2014) / R. Banerjee et al. 0 10 20 30 40 1 2 3 4 5 1 0 10 20 30 40 5 10 15 20 25 2 0 10 20 30 40 0.18 0.16 0.14 0.12 0.10 0.08 0.06 - 0.04 3 0 10 20 30 40 0.1 0.2 0.3 0.4 4 0 10 20 30 40 0.020 0.015 0.010 - 0.005 0.000 5 ξ τ τ τ τ τ ξ ξ ξ ξ - - - - - - - - - - Figure 1: Bubble- variation of ξ1, ξ2, ξ3, ξ4 and ξ5 with τ. Initial value ξ1 =0.1, ξ2 =0, ξ3 =-0.05, ξ4 = 0, and ξ5 = 0 with ρh = 3, ρl = 2, Vh=0.5, Vl = 0.1, k2 k2c =0 (line), 0.5 (dot), 1 (dash), 3.9 (dash- dot). III. Numerical results and discus- sions i. Effect of surface tension on bubble growth In this section, we present the effect of surface tension on the nonlinear growth rate of the bub- ble tip for combined Rayleigh–Taylor and Kelvin– Helmholtz instability. To describe the dynamics of the bubble tip, it is essential to integrate Eqs. (10)–(12), (15) and (16) by numerical simulation. To obtain the initial conditions of the numerical integration, we assume that the initial interface is given by y = η0(t = 0)cos(kx). The expan- sion of the cosine function gives (ξ2)initial = 0 and (ξ3)initial = −12 (ξ1)initial, where (ξ1)initial is the arbitrary initial amplitude. Since the perturbation starts from rest, we may often choose (ξ4)initial = (ξ5)initial = 0. The non-dimensionalized time de- 0 10 20 30 40 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 0 10 20 30 0 5 10 15 20 2 0 10 20 30 40 - 0.14 - 0.12 - 0.10 - 0.08 - 0.06 - 0.04 3 0 10 20 30 - 0.10 - 0.05 0.00 0.05 0.10 4 0 10 20 30 40 0.020 0.015 0.010 - 0.005 0.000 5 ξ τ 40 τ τ τ τ ξ ξ ξ ξ 40 - - - Figure 2: Bubble- variation of ξ1, ξ2, ξ3, ξ4 and ξ5 with τ. Initial value ξ1 =0.1, ξ2 =0, ξ3 =-0.05, ξ4 = 0, and ξ5 = 0 with ρh = 3, ρl = 2, Vh=0.5, Vl = 0.1, k2 k2c =10 (line), 15 (dot), 20 (dash). velopment plots of ξ1, ξ2, ξ3,ξ4 and ξ5 are shown in Figs. 1, 2 and 3. Before we describe the nature of the bubble tip, consider the asymptotic behavior of the tip. As τ → ∞, the asymptotic values of ξ3, ξ4 and ξ5 for bubble are obtained by setting dξ3 dτ = 0, dξ4 dτ = 0 and dξ5 dτ = 0. Note that, if k2 < 3 ( 1 + 15 16 ρl ρh−ρl (∆V )2 ) k2c , where ∆V = Vh − Vl, the asymptotic values are [(ξ3)asymp]bubble = − 1 6 (20) [(ξ4)asymp]bubble (21) = √ 2A 3(1 + A) ( 1 − k2 3k2c ) + 5 16 1 −A 1 + A (∆V )2 and [(ξ5)asymp]bubble = 0 (22) 060006-4 Papers in Physics, vol. 6, art. 060006 (2014) / R. Banerjee et al. 0 10 20 30 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 0 10 20 30 40 0 5 10 15 20 2 0 10 20 30 40 - 0.16 - 0.14 - 0.12 - 0.10 - 0.08 0.06 0.04 3 0 10 20 30 40 0.2 0.1 0.0 0.1 0.2 4 0 10 20 30 40 - 0.07 - 0.06 - 0.05 - 0.04 - 0.03 - 0.02 - 0.01 5 ξ ξ ξξ ξ ττ τ τ τ - - - - 0.00 Figure 3: Bubble- variation of ξ1, ξ2, ξ3, ξ4 and ξ5 with τ. Initial value ξ1 =0.1, ξ2 =0, ξ3 =-0.05, ξ4 = 0, and ξ5 = 0 with ρh = 3, ρl = 2, k2 k2c =20, Vh=0, Vl = 0 (line), Vh=0.5, Vl = 0.1(dot), Vh=1, Vl = 0.1 (dash), Vh=1.5, Vl = 0.1 (dash-dot). where, A = ρh−ρl ρh+ρl is the Atwood number. It is clear form Fig. 1 that surface tension sup- presses the velocity and growth of the bubble tip significantly, provided surface tension is larger than a critical threshold, T < Tbubblec , where Tbubblec = 3 ( (ρh −ρl) + 15 16 ρl(∆V ) 2 ) g k2 (23) Here the critical value depends on the magnitude of relative velocity shear of two fluids and the growth and velocity of the tip reduced if T < Tbubblec . When there is no tangential velocity difference (i.e., Vh = Vl) between the two fluids initially, the fluids are purely prone to the Rayleigh–Taylor in- stability and the critical value becomes 3(ρh−ρl)g k2 . These results agree with the argument in Ref. [17]. In absence of surface tension, the asymptotic values coincide with the results obtained in our previous work [14]. Further, if T > Tbubblec , oscillatory state emerges even for r > 1. Figures 2 and 3 describe the os- cillatory state of the motion. The amplitude and the period of oscillation decrease monotonically for large surface tension (Fig. 2), while the ampli- tude of oscillation increases for large relative ve- locity shear (Fig. 3). In this respect, Figs. 2 and 3 show that there always exists a self generated os- cillatory transverse velocity component (−ξ5) due to perturbation and this depends upon surface ten- sion as well as the relative velocity shear ∆V at the two fluids interface. For negative velocity shear (i.e, ∆V < 0), the self generated oscillatory trans- verse velocity of the bubble peak acts opposite to the direction of Vh and the amplitude of oscillation increases for large surface tension. If T = Tbubblec , equilibrium is attained, i.e, ξ̇3 = ξ̇4 = ξ̇5 = 0 when ξ3 = − 1 6 and ξ4 = ξ5 = 0 (24) and the equilibrium becomes unstable. This feature is shown with a dot-dash line in Fig. 1. Thus, the combined Rayleigh–Taylor and Kelvin–Helmholtz instability is stabilized when k2 > 3 ( 1 + 15 16 ρl ρh −ρl (∆V )2 ) k2c, i.e., T > Tbubblec (25) while the instability however persists but with re- duced growth rate for k2 ≤ 3 ( 1 + 15 16 ρl ρh −ρl (∆V )2 ) k2c, i.e., T ≤ Tbubblec (26) According to the condition (25), for ρh = 3, ρl = 2, Vh = 0.5 and Vl = 0.1, the motion is stabi- lized when k 2 k2c > 3.9. These results are exhibited in Fig. 2, where k 2 k2c > 3.9. For k 2 k2c = 3.9, the growth rate of the instability is asymptotically diminished and becomes 0 (dash-dot line of Fig. 1). However, Fig. 1 shows the suppression of growth rate of the instability due to surface tension, when k 2 k2c < 3.9. 060006-5 Papers in Physics, vol. 6, art. 060006 (2014) / R. Banerjee et al. 0 10 20 30 40 - 8 - 6 - 4 - 2 0 1 0 10 20 30 40 0 1 2 3 4 2 0 10 20 30 40 0.06 0.08 0.10 0.12 0.14 0.16 3 0 10 20 30 40 0.0 0.1 0.2 0.3 0.4 0.5 4 0 10 20 30 40 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 5 ξ ξ ξ ξ τ τ ττ τ ξ Figure 4: Spike- variation of ξ1, ξ2, ξ3, ξ4 and ξ5 with τ. Initial value ξ1 =-0.1, ξ2 =0, ξ3 =0.05, ξ4 = 0, and ξ5 = 0 with ρh = 3, ρl = 2, Vh=0.5, Vl = 0.1, k2 k2c =0 (line), 0.5 (dot), 1 (dash), 4.35 (dash-dot). ii. Effect of surface tension on spike growth The temporal evolution of spike state is exhibited in Figs. 4, 5 and 6; the results follow from the numerical integration of Eqs. (10)–(12), (15) and (16) using the transformation ξ1 →−ξ1, ξ3 →−ξ3, g → −g, r → 1 r and Vh ⇀↽ Vl. The saturation curvature and velocity of the spike tip are given by [(ξ3)asymp]spike = 1 6 (27) [(ξ4)asymp]spike (28) = √ 2A 3(1 −A) ( 1 − k2 3k2c ) + 5 16 1 + A 1 −A (∆V )2 and [(ξ5)asymp]spike = 0 (29) provided k2 < 3 ( 1 + 15 16 ρh ρh−ρl (∆V )2 ) k2c . 0 10 20 30 40 - 0.5 - 0.4 - 0.3 - 0.2 -0.1 1 0 10 20 30 40 0 1 2 3 4 5 6 7 2 0 10 20 30 40 0.06 0.08 0.10 0.12 3 0 10 20 30 40 - 0.10 - 0.05 0.00 0.05 0.10 4 0 10 20 30 40 0.00 0.01 0.02 0.03 0.04 5 ξ ξ ξ ξ ξ τ τ ττ τ - Figure 5: Spike- variation of ξ1, ξ2, ξ3, ξ4 and ξ5 with τ. Initial value ξ1 =-0.1, ξ2 =0, ξ3 =0.05, ξ4 = 0, and ξ5 = 0 with ρh = 3, ρl = 2, Vh=0.5, Vl = 0.1, k2 k2c =10 (line), 15 (dot), 20 (dash). Figure 4 describes that large surface tension sup- presses the growth rate of the spike tip, as well as the bubble. The nonlinear oscillation of the spike tip is observed for k2 > 3 ( 1 + 15 16 ρh ρh−ρl (∆V )2 ) k2c and the equilibrium state arises when equality holds. The pattern of amplitude and period of os- cillation are identical to that for the bubble (Figs. 5 and 6). Figure 5 shows the oscillatory behavior of the spike structure for different values of surface tension while the dependency of the relative veloc- ity shear is demonstrated in Fig. 6. IV. Conclusion In this paper, we have studied a potential flow model to describe the nature of the nonlinear struc- ture of a two-fluid interface under the combined action of Rayleigh–Taylor and Kelvin–Helmholtz instabilities due to surface tension. The analytic expressions for bubble and spike growth rates at 060006-6 Papers in Physics, vol. 6, art. 060006 (2014) / R. Banerjee et al. 0 10 20 30 40 - 1.0 - 0.8 - 0.6 - 0.4 - 0.2 1 0 2 4 6 8 2 0.06 0.08 0.10 0.12 0.14 0.16 3 - 0.3 - 0.2 - 0.1 0.0 0.1 0.2 0.3 4 0.00 0.02 0.04 0.06 0.08 0.10 0.12 5 ξ ξξ ξ ξ τ τ τ τ τ 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Figure 6: Spike- variation of ξ1, ξ2, ξ3, ξ4 and ξ5 with τ. Initial value ξ1 =-0.1, ξ2 =0, ξ3 =0.05, ξ4 = 0, and ξ5 = 0 with ρh = 3, ρl = 2, k2 k2c =20, Vh=0, Vl = 0 (line), Vh=0.5, Vl = 0.1 (dot), Vh=1, Vl = 0.1 (dash), Vh=1.5, Vl = 0.1 (dash-dot). asymptotic stage are obtained for arbitrary Atwood number and velocity shear. Surface tension be- comes a stabilizing factor of the instability, pro- vided it is larger than a critical value. In this case, oscillatory behavior of motion described by numer- ical 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 dom- inates the growth and growth rate of the instability. This result is expected to improve the understand- ing 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). 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