Basavaraja, S

Mathad, R D

Sankarappa, T

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Indian./. Phys. 76B (2), 193-196 (2002)U P  Ban international journalEffect of an aligned magnetic field on thf growth of Rayleigh-Taylor instability in a viscous and conducting fluidT Sankarappa*, S Basavaraja and R D MathadDepartment of Physics, Gulbarga University, Oulbarg| -^585 106, Karnataka. IndiaifE-mail. sankarappa@rcdifl'mai|iCom Received 25 May 2001, accepted 8 Fetruaiy 2002\bslract The inlUicnee of an aligned magnetic field on the Raylcigh-Taylor instability in a thin layer of finite electrically conducting and liiglilt VISCOUS fluid has been investigated using tlic linear theory m the creeping flow limit It is for the first time that the cfrect of an aligned field on the Raylcigh-Taylor instability in a layer of finite electrically conducting and highly viscous fluid is studied The growth rate of the instability IS found to be controlled by the ratio of surface tension to the stress gradient, thickness of the fluid layer and the Chandrashekhar number which IS a function of magnetic field It is also noted that an aligned magnetic field prevent the growth of finger instabilities which otherwise exist for thicker filmsKeywords ■ Rayleigh-Taylor instability, finite conducting fluid, aligned magnetic field PACS Nos. 52.35 Py, 47 65 +aI. Introductionliie Rayleigh-Taylor (RT) instability is an instability of iiiierfacc between two fluids of different densities, which occurs when the light fluid is pushing the heavy fluid [1], Ihc study of RT in.stabilities have been a subject of considerable interest due to their applications in the areas of menial fusion target design, astrophysics, plasma physics and physics of fluids [2-6], The instabilities are usually considered in a viscous creeping flow situations. The mathematical models of the instabilities are widely used in the areas of failure of metallic glasses [7,8], grain boundary failure in metals [7], non-cross linked polymers [9] and cracking of cross linked polymers [10]. Brown [11] theoretically investigated RT instability in a finite thickness layer of viscous fluid making creeping flow approximation. The RT like instabilities in the presence of magnetic field arise in many astrophysical and geophysical situations, however, a particular area of interest is the inertial fusion "'here the application of magnetic field is used to control the growth of the instabilities. Davalos-Orozco [12] studied the effect of horizontal magnetic field on RT instability of a twofluid layer for an unstable density stratification and concluded that in the case of an adverse density stratification, the horizontal magnetic field gives stability and propagate the perturbation as traveling waves. Sharma and Chhajlani [13] and Chandrasekhar [14] have carried out the instability studies on ideal plasma in the presence of magnetic field. The aim of this Brief Communication is to examine the stability of the interface of a fluid layer of finite thickness. The particular concerns arc the effects of the thickness of the layer and the strength of the magnetic field on the growth rates of instability.2. TheoryWe shall consider a situation where an electrically finite conducting lighter fluid (layer of finite thickness) accelerates towards a heavier fluid causing the interface between the fluids to be unstable (Figure 1). The other interface of the lighter fluid is assumed to be bounded by a rigid material. The fluid Is considered to be highly viscous, so the inertial terms can be ignored. The flow is taken to be steady and two dimensional. The relevant parameters are the fluid viscosity,-^'wesponding Author © 2002 lACS194 T Sankarappa, S Basavaraja and R D Mathad///, magnetic permeability, //, electrical conductivity, a, fluid layer thickness, A, surface tension, y, stress gradient, <5, and the applied magnetic field, Hx-Heavier fluid InterfaceXdu du I dP /.i~p~K p dy- ( 0 ^ 0 (2)ch' dwM-r-+a x  a v p  Py p y  Px ^ Py { P^w , P^w^[px^ c y  J ’Pu Pw P x ^  PyPH,= 0PH,(3 )(4 )du^ Pv” 1, Py^ J>Px Py Px ^ ^ '  Py+ Vn(P^Hy P^HyPx^PH, PHy Px Py ’(6)(7 )and hx and hy are the induced magnetic fields In the x and y  directions respectively.Substituting eq. (8) Into the eqs. (2-7) and linearizing, we get after omitting the primes for simplicity, the required linearized equations :i ^ + £ «p P x  p ^ ' “ Pxdh^Figure 1. The geometry of the perturbationThe basic system can be defined asu -  - 0, H, -  IIx. Hy -  0. ( 1)The basic equations of motion of the system in the absence of body force are given byp  p  Px [ Px'  ^ Py'^  ] ~ ' P i p w„  PuPx^It ^I  fox ^  +  nP^h,^ 2d^hdx^ ^j . o ,' r hd ^  d y  ’(9)( 10)( 11)( 12)(13)( 14)The boundary conditions and the interface conditions shown in Figure 1 areVi; = /)vv -  0; By -  Dhy = 0 at y = 0, (15)where D refers to d/dy.The free surface is at y = A and we shall consider the development of small perturbations in this interface. The actual deviation of the interface from y  -  h is given by rj. The continuity of tangential stress at the interface is given bydx dy(5) The continuity of normal stress at the interface givesd^t] dwn i y ^ h . (17)The kinematic condition with>> = A -  7 as the interface, after the linearization givesPt] . ,w = a t y - h . (18)where P = Po + (pW (^2) is the total pressure, Po is the hydrostatic pressure, (m, w) are the velocity components in X andy directions respectively, H{H„ Hy) is a time independent magnetic field, p  is the fluid density, v(= pf/p ) is the kinematic viscosity and v„(= \/pd) is the magnetic viscosity.We superimpose on the basic state given in eq. ( I) a small symmetric disturbance of the formu ^  u', w ^  w*, P ~ P*, Hox ^  Hx hx. Hoy ~ hy, (8) where the primes indicate the perturbed quantities, H ^  and Hoy are the total fields (sum of applied and induced fields)The continuity of magnetic tangential stress at y = A is given byPhv Phv Px Pyh. (19)The continuity of normal magnetic stress is given by~ 2 ----- “ T ' - ^  + p T /o r -^ ,  a ty  “ /i. (20)By eliminating the pressure gradient from the eqs. (9) and (10) and using (11-14) we getp H ^  Pw  ^ p^u  ^ p^u P ^w ______p w „  P x  p x ^ p y  P y^ Px^ P y ^ P x= 0 . (21)Effect o f an aligned magnetic field on the growth o f Rayleigh-Taylor instability etc 195Assuming the variations of u, w, P and rj in the x-direction to be of the form‘u iyYuwpw(y)p(y).^Cv)]exp(/ajr + «r)^ (22)where n is the growth rate of the RT instability and a  is the wave number.By using eqs. (9), (11) and (22) in (21) we get D'*w-2 a^D^w+a^{ l  + a l ) w ^ 0 ,  (23)where a^ = { j u f f l f p w „ a ^ ) .The general solution of eq. (23) can be written asw = exp(aaiy)[y4 sin(fl5a2y) + Rcos(afl2>’)]+exp(-QW|>’)[Csin(Qa2y) + £ 005(002^^ )]. (24)where «i - ( l+ a g )V 2 and «2 = w 0 V2By imposing the boundary conditions given in eqs. (15) and (16) on the solution given in eq. (24), we obtain theconstants B, C and E to beR)(a2/2 ax\ C -  -- v4/? and £  -  /i( 1 -  R){ail2ax\whereexp( A/)j^ (<3^  + ~ a\ )cos(A )^ la^ (l + )sin(A^ )J-  cxp(-A/)[(a| -  a} aj -02)cos(A^)-2apsin(A^)j/r=cxp(A/)j^ (<7| - 0|^ fl2 - 02)cos(A^)+2fl2a] sin(A^ )j f exp(-A/)^(a| + 3flj^ a2 -  af)cQs(N)-  2 a \ (oj^  + l)sin(A7)j(25)M =  aa\h  and h  = aoih.On substituting HP- = {Hox + hxY + h f  in the total pressure P and using eq. (8), we get1p  dx (26)^ dy^Making use of eqs. (10), (17), (18), (22) and (26), the dispersion relation is obtained to be„ = _ ( £ r « W ! ^ a t y  = /i.P f{ 2 a^ D w- D^ w)(27)On making this equation dimensionless using the relations=(”/ ^ / V V ^ .  « ’ = («^*X A* = (^^*)and D'  = Dh and for simplicity neglecting the asterisks,become( Ih ^ a ’^ D w-D ^w) ' (2 8 )In the eq. (24), tlie quantity oo in the dimensionless form isigiven as ~[0/(/?^or^)]2, with Chandrashekhar number, 3. Discfission and conclusionThe eq^  (28) is a function of the dimensionless wave number!42, the ratio of surface tension to the stress gradient A, the |5lm thickness h and the Chandrashekhar number Q whiefc is a function of an applied horizontal magnetic field The growth rate n is numerically computed for difTercnl values of h and Q and, plotted in Figures 2-5.Figure 2. A plot of growth rate n versus wave number a  at diiTcrent Chandrashekhar number, Q, for film thickness, h « O.I.Figure 3. A plot of growth rate n versus wave number a  at different Chandrashekhar number, Q, for ftlm thickness, h \.196 T Sankarappa, S Basavaraja and R D MathadThese plots reveal that the size scale of the RT instabilities of the fluid of finite layer thickness is controlled by A, h and Q. Further, it can be seen that the increase in the film thickness h, produce a finger type of instability but the effectFigure 4. A plot of growth rate n versus wave number a  at different Chandrashckhar number Q, for film thickness h =  \0.Figure 5. A plot o f growth rate n versus wave number a  at diflerent Chandrashekhar number Q, for film thickness h "  20.of an aligned magnetic field is to suppress the growth rates When Q  is made zero in the dispersion relation (28), thi hydrodynamic results of Brown [11] are recovered. This cat be easily realized by substituting the value of appiTed fieli to be zero in eq. (23). These results may be of great use it inertial control of fiision (ICF). The RT instabilities grow it imploding ICF targets when the heavy fluid is acceleratec by a lighter fluid. All the ICF targets undergo such at instability at some time or the other during the implosion The outer ablating surface may be stabilized in the presence of an aligned magnetic field.It is for the first time that the RT instability in a finite thickness layer of highly viscous, electrically finite conducting fluid has been studied in the presence of ar aligned magnetic field under the approximation of creeping flow limit. It is found that the growth rate of the RI instabilities is controlled by the ratio of surface tension tc the stress gradient, fluid layer thickness and the Chandrashekhar number. The magnetic field is also found to prevent the finger instabilities which otherwise exist foi thick films. The results tend to hydrodynamic domain in the absence of an applied field.AcknowledgmentAuthors acknowledge the help received from Professor >. Rudraiah and Professor B S Krishnamurthy.ReferencesI I ]  D H Sharp Physica 12D 3 (1984)[2] K 0  Mikaelian Phys. Rev. A26 2140 (1982)[3] R Menikoff, R C MjolsBessm, D H Sharp and C Z  Zemach Phys Fluids 20 2000 (1977)[4] H W Emmons, 1 F Cham and B C Waston Fluid M eek 7 171 (1960)[5] K O Mikaelian Phys. Rev A26 2140 (1982)[6] K O Mikaelian PIq>s. Rev. E54 2 3676 (1996)[7] A S  Argon and M Salama Mater. Set. Eng. 23 219 (1976)18] C A Pampillo /  Mater Set. 10 1194 (1975)[9] E J Kramer Adv. Polym. Sci. 52/3 1 (1983)[10] A J Kinloch, D Gilbert and S J Shaw Pofym. Commun. 26 290 (1985)[11] H R  Brown Phys. Fluids A t 5 895 (1989)[12] L A Davalos-Orozco Astrophys. Space Sci. 243 291 (1996)[13] P K Sharma and R K Chhajlani Pl^. Plasmas S 2203 (1998)[14] S Chandrasekhar Hydrodynamic and Ifydromagnetic Stabilll} (Oxford : Oxford University Press) (1961)

Effect of an Aligned Magnetic Field on the Growth of Rayleigh-Taylor Instability in a Viscous and Conducting Fluid

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Effect of an Aligned Magnetic Field on the Growth of Rayleigh-Taylor Instability in a Viscous and Conducting Fluid

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