TECHNICAL PHYSICS VOLUME 43, NUMBER 9 SEPTEMBER 1998 Capillary oscillations and stability of a charged, viscous drop in a viscous dielectric medium A. I. Grigor’ev, S. O. Shiryaeva, and V. A. Koromyslov Yaroslavl State University, 150000 Yaroslavl, Russia ~Submitted March 28, 1997! Zh. Tekh. Fiz. 68, 1–8 ~September 1998! The scalarization method is used to obtain a dispersion relation for capillary oscillations of a charged, conducting drop in a viscous, dielectric medium. It is found that the instability growth rate of the charged interface depends substantially on the viscosity and density of the surrounding medium, dropping rapidly as they are increased. In the subcritical regime the influence of the viscosity and density of both media leads to a nonmonotonic dependence of the damping rate of the capillary motions of the liquid on the viscosity or density of the external medium for a fixed value of the viscosity or density of the internal medium. The falloff of the frequencies of the capillary motions with growth of the viscosity or density of the external medium is monotonic in this case. © 1998 American Institute of Physics. @S1063-7842~98!00109-3# INTRODUCTION ]u~a! 1 52 ¹p~a!1n~a!Du~a!; a51,2, ~1! ]t ~a! A study of the electrostatic instability of a charged drop r of viscous liquid suspended in another viscous liquid is of ~where D is the Laplacian operator!, the equation of conti- significant interest in connection with numerous applications nuity, reduced in the case of an incompressible liquid to the in which such an object figures: in technological applications condition involving the uniform mixing of immiscible liquids, in prac- ~a! tical applications involving the combustion of liquid fuels in ¹•u 50, ~2! regard to the mixing of the fuel and oxidizer, and in geo- physical experiments ~see, e.g., Refs. 1–4 and the literature kinematic boundary conditions at the interface of the two cited therein!. Nevertheless, many questions associated with liquids r5R for the tangent components this problem have so far been only scantily investigated be- u~1!5u~2! , ~3! cause of the complicated technique of the experiments and u u the difficulty of the theoretical calculations. This also per- u~1!5u~2! , ~4! tains to the effect of the viscosity and density of the media w w on regularities of realizations of the instability of the charged and normal component surface of such drops. 1. We consider a system consisting of two immiscible ]j u~1!5u~2!5 ~5! incompressible liquids with densities r(1) and r(2) and ki- r r ]t netic viscosities n(1) and n(2). As a result of the action of the forces of surface tension, whose coefficient we denote as s, of the velocity field, dynamic boundary conditions for the the inner liquid, to which we assign the index ~1!, takes the tangent components form of a spherical drop with radius R. The outer liquid, 1 ]u~1! ]u~1! 1 identified by the index ~2!, is assumed to be unbounded. A ~1! ~1! r u ~1! r n 1 2 uu charge Q is found on the interface of the two liquids. Further F r ]u ]r r G we assume that the liquid in the drop is a perfect conductor, ~2! ~2! 1 ]ur ]uu 1 and the liquid of the medium is a perfect insulator with di- 5r~2!n~2! 1 2 u~2! , ~6! F r ]u ]r r u G electric constant «. We will find the spectrum of normal vibrations of the interface of the two liquids. ~1! ~1! 1 ]ur ]uw 1 In the spherical coordinate system r, u, w with origin at r~1!n~1! 1 2 u~1! r sin u ]w ]r r w the center of the unperturbed drop the equation of the inter- F • G face of the two liquids has the form r5R1j(u,w,t). The 1 ]u~2! ]u~2! 1 ~2! ~2! r w ~2! system of hydrodynamic equations, linearized about the 5r n 1 2 uw , ~7! Fr•sin u ]w ]r r G equilibrium spherical state, consists of the Navier–Stokes equation and normal component 1063-7842/98/43(9)/8/$15.00 1011 © 1998 American Institute of Physics 1012 Tech. Phys. 43 (9), September 1998 Grigor’ev et al. ~1! one of the densities or a combination of them with the units ]ur 2p~1!12r~1!n~1! 1p 2p of density equal to unity in all the equations of the system. ]r s E We set R51 and s51 at once, but we will choose the ~2! ]ur quantity with dimensions of density later from considerations 52p~2!12r~2!n~2! ~8! ]r of convenience. To transform back to the old basis, it is necessary to divide each quantity encountered in the equa- of the stress fields, the condition of constant volume of the tion by its corresponding dimensionality. two liquids 2. We now scalarize the problem using a procedure de- scribed in detail in Ref. 5. The velocity field u, as an arbi- E j~u,w,t!dV50 ~9! trary vector field, can expanded into a sum of three orthogo- V nal fields and the condition of immobility of the center of mass of the u r,t!5Nˆ C r,t!1Nˆ C r,t!1Nˆ C r,t!, ~13! system relative to the chosen system of coordinates ~ 1 1~ 2 2~ 3 3~ where Ci ~i51,2,3! are scalar fields. By virtue of orthogo- ˆ j~u,w,t!erdV50. ~10! nality of the fields NiCi EV ˆ ˆ Here j, u, and p denote the perturbations of the shape of the E ~NiCi!*•~NjC j!dV50 ~iÞj!, interface, the velocity field, and pressure field, respectively; V ps is the perturbation of the pressure of the surface tension where the symbol * denotes the complex conjugate, and the forces integral is taken over all space, the vector operators should satisfy the relations Nˆ 1 Nˆ 50 for i h), where Nˆ 1 is the s i • j ~ Þ i p ~j!52 ~21D !j, ~11! ˆ s 2 V Hermitian conjugate of the operator Nj . For the problem R under consideration, it is convenient to choose these opera- pE is the perturbation of the pressure of the electric field tors in the form associated with capillary deformation of the interface ~see Nˆ 5¹, Nˆ 5¹3r, Nˆ 5¹3~¹3r!, Appendix A! 1 2 3 2 ˆ 1 ˆ 1 ˆ 1 Q N1 52¹, N2 5r3¹, N3 5~r3¹!3¹. p ~j!52 j E 2p« Substituting expansion ~13! into the condition of incom- ` pressibility ~2!, we obtain Q2 1 ~a! 1 ~m11!Pn~m! jPn~m!dm, DC 50. ~14! 4p« m(50 E21 1 ~12! Substituting expansion ~13! into the Navier–Stokes equation ~1!, allowing for commutativity of the operators Nˆ is the angular part of the Laplace operator in spherical i DV and D, gives coordinates, Pn(m) are the eigenfunctions of the operator D , and dV is the solid angle element. ]C~a! 1 ]C~a! V ˆ 1 ~a! ~a! ~a! ˆ 2 In order to simplify the solution of the problem, it is N1 2n DC1 1 p 1N2 S ]t r~a! D S ]t useful to transform from the MTL-dimensional basis, where M, L, and T represent units of mass, length, and time, re- ~a! ]C3 2n~a!DC~a! 1Nˆ 2n~a!DC~a! 50. spectively, to another, more suitable basis which decreases 2 D 3S ]t 3 D the number of parameters of the problem. It is convenient to choose the rRs basis. The formulas connecting these two Acting on this equation successively with the operators ˆ 1 bases have the form Ni , we obtain three independent equations @r#5ML23, @R#5L, @s#5MT22, 1 ]Ca ˆ 1 ˆ ~a! i ~a! ~a! Ni •Ni p d1i1 2n DCi 50, ~15! @M#5rR3, @L#5R, @T#5R3/2r1/2s21/2. S r~a! ]t D We write the dimensions of quantities in the old and in where dij is the Kronecker delta. ˆ 1 ˆ the new basis Since the operators Ni •Ni and D commute, they possess a general system of eigenfunctions. Here the eigenvalues of 21 22 21 @r#5@j#5L5R, @p#5ML T 5R s, these operators corresponding to one or another eigenfunc- @u#5LT215R21/2r21/2s1/2, tion are, general speaking, different. Equation ~15! implies that either the fields pd1i and Ci should be expanded in the @n#5L2T215R1/2r21/2s1/2, eigenfunctions corresponding to the zero eigenvalue of the operators N1 N or the following equation is valid: @t#5T5R3/2r1/2s21/2. i • i 1 ]C~a! To transform to the rRs basis, it is sufficient to set the ~a! i ~a! ~a! p d1i1 2n DCi 50. ~16! radius of the drop R, the coefficient of surface tension s, and r~a! ]t Tech. Phys. 43 (9), September 1998 Grigor’ev et al. 1013 It is obvious that the first possibility would be to great a ] ] C~1!1 rC~1! 2 C~2!1 rC~2! 50, ~23! restriction; therefore we assume that Eq. ~16! follows from F 1 ]r 3 G F 1 ]r 3 G Eq. ~15!. Taking condition ~14! into account, we obtain the scalar analog of Eqs. 1 and 2 , which consists of the fol- ~1! ~2! ~ ! ~ ! C2 2C2 50. ~24! lowing set of independent scalar equations of the fields (a) Conditions ~23! and ~24! give the scalar analog of DCi , boundary conditions ~3! and ~4!.