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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1281 UNSTABLE CAPILLARY WAVES ON SURFACE OF SEPARATION OF TWO VISCOUS FLUIDS* By V. A. Borodin and Y. F. Dityakin The study of the breakup of a liquid jet moving in another medium, for example, a jet of fuel from a nozzle, shows that for sufficiently large outflow velocities the jet breaks up into a certain number of drops of different diameters. At still larger outflow velocities, the continuous part of the jet practically vanishes and the jet immediately breaks up at the nozzle into a large number of droplets of varying diameters (the case of "atomization"). The breakup mechanism in this case has a very complicated character and is quite irregular, with the droplets near the nozzle forming a divergent cone. Rayleigh (reference 1) was the first to make a theoretical study of the jet and to establish the possibility of droplet formation. The disturbance of a jet of an ideal fluid flowing into a vacuum and having a wave length 4.4 times as large as the diameter of the jet is shown to grow more rapidly than other disturbances; eventually, the jet breaks up into droplets of the same diameter. Rayleigh succeeded in determining theoretically the drop diameter, the value of which agrees well with tests on jets issuing with very small velocities. Later, the viscosity of the jet was also taken into consideration. The viscosity is found to decrease the rate of amplitude increase of the disturbances but the ratio of the optimal length of the wave to the diameter of the jet remains unchanged. Other authors that studied the conditions of the axialsymmetrical breakup of a jet of a viscous liquid found that the ratio of the optimal wave length to the jet diameter was somewhat greater than that computed by Rayleigh. In addition to the viscosity, Tomotika (reference 2) took into account the density and viscosity of the medium surrounding the jet and obtained good agreement with tests on jets issuing with very small velocities for which droplets of the same diameter are formed. *"Neustoichivye Kapilliarnye Volny na Poverkhnosti Razdela Dvukh Vyazkikh Zhidkostei." Prikladnaya Matematika i Mekhanika. Vol. XIII, no. 3, 1949, pp. 267276. NACA TM 12Zl Neither of the aforementioned theories of the breakup of a liquid jet provided a basis for the phenomenon for the case of breakup into droplets of different diameter, a fact that is explained by the idealized conditions of the problem. This idealization consisted either in neglecting the viscosity of the jet, the density, and viscosity of the surrounding medium, or the inertial forces. Such simplifications were assumed in view of the complicated mathematical equation (generally transcendental) that determines the relation between the wavelength and the increment of the vibration amplitude. In the present paper, an attempt is made to provide a mathematical basis for the possibility of the appearance of droplets of different diameters as a result of the jet breakup on the basis of the considera tion of unstable capillary waves on the surface of separation of two viscous liquids. For simplification of the solution of the problem, particularly for obtaining the algebraic characteristic of the equation, the lengths of the capillary waves on the surface of the liquid jet are assumed to be so small in comparison with the jet radius that the jet may be considered infinitely large; study of the stability of the plane surface of separation of two infinitely extending viscous fluids can thus be made. This assumption represents a considerable degree of idealization but nevertheless permits a qualitative explanation of not one but several unstable capillary waves that, in passing through the jet, lead to the formation of droplets of differing diameters. The existence of several unstable capillary waves is demonstrated that can lead to the breakaway of several infinitely long strings of different dimensions from the partition surface. The problem investi gated gives a rough approximation of the disintegration pattern of a liquid jet in another medium and does not pretend to explain the com plicated mechanism of the limiting form of the disintegration of a jet, namely, atomization. Nevertheless, one of the peculiarities of atomization, the appearance of a dimension spectrum of the droplets, begins to appear even for the given idealized consideration of the stability of the partition surface. 1. Equations of small waves and their solution. A plane surface of separation of two infinitely extending viscous fluids (fig. 1) is considered. The viscosity and density of the lower fluid are denoted by tl and pi, respectively, and of the upper fluid by L2 and P2. The lower fluid is assumed to move with the velocity V1 and the upper fluid with the velocity V2, the direction of motion being the same and the velocities independent of y. TFACA 'T i281a 3 A study of the character of the equilibrium of the surface of sep aration under the action of the viscous forces and the forces of silrface tension that impsrt to both liquids small disturbances parallel to the xaxis is presented. The fluids shall be onsidered incompressible and weightless and shall cause certain disturbances to the components of the motion. X = V + v V = v p = P + p* It is further assumed that the velocities of the imposed disturb ances and their derivatives up to the third inclusive are small and that the magnitudes of the secondorder smallness may be neglected. Fromn the NaviorStct:es equations, the following equaLions of the imposed disturbances are obtained: OVx Vx 1 + V =+ = ot ox P x x (1.1) S+V v = I+ Iv, ot ox P oy where u = p/p is the cinematic viscosity. The equation of continuity is OVv OV,, ovx ovyI + 0 ? x 3y By introducing the stream function of the disturbance v v = (1.3) oy ox and by eliminating the pressure p* from equations (1.1), the idealized equation is thus obtained in the Helmholtz form 0 (1.4) 6x 6t NACA TM 1281 Let the stream function of the imposed disturbance be a periodic function of x and of the time t: Sf(y)ei(axt) (1 where a is the propagated circular frequency of the vibrations (the wave number), X is the wavelength of the imposed disturbance, P = Pr + iPi is the complex frequency of vibrations in time, pr is the real frequency of vibration in time, and Pi is the increment of the growth of vibration or the decrement of damping. The character of the wave motion on the surface of separation after the imparting of disturbances to both surfaces will thus depend on the sign of the imaginary part of the frequency .i* If Pi is positive, there will be an increase in the wave amplitude with time; if Bi is negative, there will be a damping of the wave amplitude; finally, if Pr = 0, there will be an periodic increase (Si > 0) or a damping (Pi < 0) of the wave amplitude. By substituting expres sion (1.5) in equation (1.4), the following equation is obtained: fIV (2, ip) f" (ipa2 a4) f iVa (f" a2f) = 0 ( The problem of the characteristic values of a homogeneous system of equations of the fourth order will be considered. By setting f" a2f = p, a system of equations of the second order is obtained. p" + Pi 2 = 0 f a2f= (l Hereinafter, the following notations are introduced: SP VlM P Va ( i i2=ul i =2 (1 D1 V2 M The solution of the first of equations (1.7) has the form S= CleiY + C2eim1Y (1 By substituting expression (1.9) in the second of equations (1.7), a nonhomogeneous equation is obtained for which the solution is NACA TI 1281 f = Cl C2 + e' C, + ea C4 (1.10) m2j + 2 2 .+ a' The streacu furctinn for the lower and upper liquids according to equation (1.5) will be i ^(t) e/ iLmly S2i + a m1 + a2 4' S e (a.t) ( .em im2y ay \ S2 = ei(axt) ( 2 C2 5 e'2 2 C6 + eay 7 + e C (1.12) m2 + a. m22 + a2 / The arbitrary constants Ci must be determined from the conditions on the surface of separation and at infinity. 2. Boundary conditions. The boundary conditions of the problem will be as follows: 1. At infinity (y = +c), finite solutions must be maintained for 4lf and 29. Hence, the arbitrary constants of the terms with positive exponents for 1 and with negative exponents for W 2 must be equated to zero: C1 = C3 = C6 = Cg = C. Thus, equations (1.10) and (1.11) will have the form ei(camt) e C2 + eay C4 m12 + aOL 4) (2.1) / im\7 w2=ei(a't) f emi2 C+ + e Co7) y e Cr + I2 = .m92 + a? 7 2. On the surface of separation, there must be no slip, that is, (Vl) Y = ( 2)y=0 (Vy71) y= (v2) y=0 KNACA TM 1281 y =0 y )yy=o Ox =0 y=O x/y=0 (2.2) 3. The tangential stresses on the surface of separation are continuous l( Al)y=O = A2(A2)y=0 (2.3) 4. The difference between the normal stresses pyl and py2 on the surface of separation is equal to the pressure brought about by the surface tension; that is, I(V y2 (22 y2 yl Py2 = P + 2 P2 + 2 o2h = T 2 ox where T is the capillary constant of one liquid relative to the other and h is the rise in the surface of separation at the point x. By using equations (2.1), the boundary conditions (2.2) are obtained in the form iml im2 m12 2 2 +4 + 2 2 C5 aC7 m._ + a" m" = 0 (2.5) C2 mi + Ma C F5 + C4+ 2 2 C7 = 0 2 + a Similarly, the boundary condition (2.3) is obtained in the form C2 m12 + M2 + C4)a.2 C 7)M2 + 7 C .2 m2 25 ig22 + a2 S12 ml2 + a2 C2 + a2C4 2 i2 + 5 m2 + (2.4) (2 .C) iAlCA TPIM 1281 The pressures jl and p2 are computed from equations (i.1) and (2.1). Thus PL PI( I ' i =; + "'2 V mIC iiv CL i lt+ i:i} !lii f + i la2 VaM iula o e)e C4] (2.7) p2 = P2ei(axt) a '7+ m2 C5 ( 2) e y C, L 22 + a2 2 + m + 2 The rise of a point on the surface is a periodic function of x and t. h = Hei(xt) (2.8) where H is the maximal rise of a point on the surface of separation. The velocity of the raised point on the surface of separation is (2.9) oh oil + V1 ot ox '7=u After differenttiatin expressions (2.1) and (2.6) and b.r substi tuLing in expression (2.9), the following equation is obtained: H C2 H a C+ C2 ml + 9 (2.1'0) By substituting equation (2.10) in (2.9) and by differentiating equation (2.9), ,2 oh 62 ox ca C2 Sav1 P + a2 (2.11) m"l C? +  (v ) =  l^ ( c4 ei(axPt) NACA TM 1281 By computing the derivatives 8Vyl/By and 6vy2/Sy and tuting them simultaneously with expressions (2.7) and (2.11) the following boundary condition is obtained: substi in (2.4), a [La2 + iP1Via iPP ml2 + a2 m Ta2 + ml 1 + 0V ] C2  plVlao i2jla2 + T S" VP C4 + M m22 + a2 '2 M ip2V2co + iPP2 D+ m2 3m2p2 ] C5 + (P2P P2V2 + i22aa2) C7 = 0 (2.12) The following nondimensional parameters are then introduced: Z =  vl V2 A 1 2 N 2 K 1 2 (2.13) NACA TM 1281 9 where c = V/a is the complex wave velocity. Equations (1.8) can then be represented in the forms m = a/i(Z R) 1 m2 = a /i(ZA R2) 1 Equations (2.5), (2.6), and (2.12) are represented in nondimen sional parameters. The following notations are first introduced: i 2 N a1 1 z /i(zR) 1 Z la b* = ZR 2 + 1 = a bl "1 RCLP a 1 2(1 2R2 + 2AZ) CL* = 2 2=a2c, (ZAP,2) (ZP2) ;2 a 1 i a3 a3* 2  2 ZR1 a.2 (2.14) d6* = ,.2 (ZAPR + 2i) = 2a.2d 1 i c3 a2* = =  S4 i(ZAR) 1 c2 2 a ZAR2 a, where al, a2, a3, bl, cl, c2, c3, and dl are likewise nondimen sional magnitudes. NACA TM 1281 The following system of equations is then obtained for the con stants C2, C4, C5, and C7: al*C2 bl*C4 + cl*C5 + dl*C7 = 0 a2*C2 aC4 + c2*C5 a C7 = 0 (2.15) a%*C2 + C4 + c3*C5 C7 = 0 K C2 C5 = 0 This system of homogeneous equations has solutions different from zero if its determinant is equal to zero. By setting up the determinant and expanding 2K(al + cl) + (d1 Kbl)(a2 + Kc2) + (Kb1 + dl)(Kc3 a3) = 0 By solving this equation for Z, the following wave equation of the 18th degree with complex coefficients is obtained: rlgZl8 + (rl7 + isl7) Z17 + ..: + (rl + isl) Z + (r0 + is0) = 0 (2.16) The real and imaginary parts of the coefficients depend on the five nondimensional parameters: R1, R2, A, N, and K. 3. Investigation of roots of characteristic.equation. The increase in oscillation, that is, the loss of stability of the sur face of separation, arises from those waves for which the imaginary part of the frequency is positive (Pi > 0). Hence, the investiga tion of the roots of equation (2.16) should determine those ranges of the parameter N or the wave number a in which the complex roots of the equation lie in the upper halfplane. By the Rayleigh hypothesis, the further development of an unstable deformation, that is, the form and dimensions of the parts breaking away, is determined by the critical (or optimal) disturb ances. The critical disturbances may be defined as those that develop more rapidly than the others or that correspond to the maximum increment of the growth Pi. This principle of deter mining the character of the unstable deformations by the character of the maximum unstable disturbance has been experimentally confirmed by a number of investigators (reference 3). NACA TM 12:1 In the case considered, the growth in the amplitudes of the oscillations will lead tc breakaway of infinitely long strings from the surface of 3eoaration. similar to the formation and brea':awa.y of wave crests. The separation will take place for such values of a or wavelengtl. A for which i has the maximal value. If a spectrum of smallperiod disturbances that cen be developed into a Fou'rier series can be assumed to be imposed on both liquids, the. harm.onics w;ih the waveengths equal to the wselengths of the maximal, unstable disturbances bring about a separation of infinitely lon strings from the partition surface. Because the cnaracterietic dimension (for example, the diameter of the transverse string) is connected with the length of re::.,iisl unstable disturbance, str.nges of different dimensions will breai: awash from the surface of separation. In fi'.ure 2, the scheme of formation of such strings for three successive instants of time is shown. Investi..jatorn of the roots o.f the simplest particular case of equation (2.16) is presented. Let bc.th fluids be stationary and their kinetic viscosities the same. In this case, V V = 0 P1 = U2, mi = m2, A = 1, P, = F2 = 0, and equation (2.16) goes over into an equation of the 8th degree whose coefficients depend only on the two parameters K and N: A 8 + (A iB)Z + iB)Z6 + ( + iB3)5 (A + B)4 + A (A + E)2 (A+ i)Z + 4 (A5 + iB)Z (A6 + iB)Z2 + (A7 + iB)Z + A8 = 0 (7.1) 12 NACA TM 1281 where A0 = (1 K)2 A2 = 2K (K 1) N K4 + 2K3 4K2 + 6K + 13 A1 = 2K (K 1)2 A3 4K2 (K 1) N 2K (3K2 + 13) A7 = 2K3N2 A4 = K2N2 + 2 (K4 K3 + 3K2 + 5K) N 12K3 + 26K2 10K 9. AS = 2K3N2 + 12K3N 8K3 8K2 A8 = K2 (1 + 2K) H2 A6 = (1 K2) K2N2 + (4K4 10K3 4K2 SK) N 8K3 + 12K2 8K + 4 B1 = 2 (K2 + 2K 3) B3 = 3E4 14K3 + 13K2 + 18K + 13 8KN B2= 4K (K 1)2 B4 = 8K3 42 20K 2K3N B= 4K2 (1 K) B5 = 2K2N2 + [2K (1 + K) (1 + K K2) + 83 + 4K2 + 4K]N + 4(K 1 K2) (1 + K K2) + 20K3 4(K 1 K2)2 B7 = K2 (1+ K)2 2K4 N2 + [4K4 + 4K (1+ K) (K 1 K2) N (3.2) NACA TM 1281 The characteristic equation (3.1) is a polynomial whose coefficients depend nonlinearly on the two parameters K and N. Each pair of values of the parameters E and N' or each point of the plane Ki correspond to the completely defined polynomial (3.1), that is, completely determined values of the eight roots of the nolyinomial. In the plane KI, it is evidently possible to find a curve, each point of which corresponds to the polynomial (3.1), that has at least one root located on the real axis so that only in crossing this curve is a crou.iig of tre roots through the real axis possible. This curve breaks up the plane Kn into regions, the points of which each correspond to polynomials (3.1), that have the same number of roots with positive imaginary part. These curves are constructed by making use of the method of Y. I. Neimark (reference 4) that permits a breakup of the plane of the parameters for the roots of the polynomial lying in the left or right halfplane. The substitution Z = i( is made. The upper halfplane of the roots of equation (3.1) is transformed into the left halfplane of the roots of the equation  A0 r i(A1 + iB1) 7 + (A2 + iB2) 6 i(A3 + iB3)5 (A4 + iB4) + i(A5 + iB5)( + (A6 + iB6)2 (A7 + iB7) + A8 = 0 (3.5) By substituting = it/1 in the preceding equation and multi plying the result by 98, equation (3.3) is reduced to the form F(;,f) + iG(C,T) = 0 (3.4) where F(,q) = AOE + A Al7 + A2 6 + As53 + A44q4 + A5~3q5 + A6g2q6 + A7 67 + AT8 (3.5) G(t,q) = B~7r + B262 + B3 53 + B4 44 + B5 3r5 + B62+ 6 + B7+ 7 NACA TM 1281 If R2n is the space of complex polynomials of degree n and D(k,n k) is the manifold of polynomials R2n having k roots to the left and n k roots to the right of the imaginary axis of the complex sphere, then by setting up the following table: A0 Al A2 A3 A4 A5 A6 A7 Ag 0 B B2 B3 B4 B B6 B7 B8J and by making the transformation AO + X1B AI + XB2 A2 + lB3 A3 + lB4 ...Ag S 0 B B2 B3 ...0 (3.7) table (3.7) is found to correspond to a polynomial of the same type with respect to the distribution of the roots relative to the imaginary axis, as in equation (3.4). From table (3.6), an inequality is obtained that defines the region in the plane KN corresponding to the presence of the first root of equation (3.1) in the upper halfplane: A 0B < 0 (3.8) By setting X1 = AO/B1 in table (3.7) AB1 A B2)/B1 (AZB1 AOB3)/B1 (A3B1 AOB4)/B1 ..A7 A B1 2 B3 ...B7 0 (3.9) Because AIB1 AoB2 = 16K(K 1)3< 0 for K>1, by multi plying the elements of the first rows of (3.9) by B12/(A1B1 A B2) and changing signs in the second row SI_ D1 D2 D3 D4 D5 D6 D71 (3.10) B2 B3 B4 B5 . B B7 0 NACA Til 1231 15 where E1(A B A Bj) BI(A3B1 AC24) AiBi A 2 Ei(A4B ABD5) AiB1 AOB2 BI(A5Bi AeB6) ____ = D4 = (3.11) A1Bl A 72 Bi(A6B1 AB7) AlB1 AOB2 B A, D A1Bi AOB2 The first row of table (3.10) is left unchanged but to the second row is added the first row. Thus i D1 D2 D3 D4 D5 D6 D 0 D1 B2 D2 B3 D B4 D4 B5 D5 B6 D6 B7 C (3.12) From the preceding calculations, an inequality is obtained that defines the region in the plane IK that corresponds to the presence of the second root of equation (3.1) in the upper halfplane (3.13) B1(DI B2)<0 0 NACA TM 1281 By carrying out a transformation, similar to (3.7) of table (3.12), + X2(D1 B2) 0 D1 + 2(D2 B3) D1 B2 D2 + 2(D3 B4) .."D ) D2 B3 ...C By setting; 2 = B1/(D1 B2) and substituting in (3.14) D1(D1 B2) B1(D2 B3) D1 B2 D1 B2 D3(Dl B2) B1 D1 B2 D3 B4 D2(DI B2 B(D B4) D B2 D2 B3 (3.15) (D4 B5) Because DI(D B2) B1 (D2 B3) > 0 plying the elements of the first row of (3.15) [DI(D B2) BI(D2 B3)] for K>1, by multi by (DI B2)2/ SB 2(Dl B2) Bl(D3 B4) (D B2) D1 2 D1(DI B2) B1(D2 B) DI B2 D2~ B3 (3.16) The elements of the first row are subtracted from the elements of the second row of table (3.16). [(D2 (D1 B2) B(D3 B4) (D B2) 0 (D2 B3) D2(D1 B B Bl(D3 B)] (DI B2) D1(D1 B2) B1(D2 B3) From the preceding table, an inequality is obtained that defines the region in the plane of the parameters KN that corresponds to the presence of the third root of equation (3.1) in the upper halfplane. (D2 B3)  [D2(D1 B2) B1(D3 B)] (D B2 DI(D1 B2) B1(D2 B3) (3.18) (D1 B2) (3.17) * NACA TM 1281 Similar conditions can be obtained for all the remaining roots of equation (3.1). This investigation has been limited to the three conditions that are sufficient for proving the existence of several unstable waves. By replacing inequalities (3.8), (3.13), and (3.18) by equations, the equations of the curves determining the breakup of the FN plane into regions are obtained. The most interesting case of large K = pj/p2>>1 is considered. From inequalities (3.8), (3.13), and (3.18) and by considering equations (3.2) and (3.11) and neg lecting small powers of K, the following equations are obtained: 2(K l)3(K 4 3) = 0 eC3 + elN2 + e2N + e3 = 0 4(K2 1)(K + 3)N + K(K l)(K3 + 17K2 96K + 99) = 0 where (3.19) e = 128 (K4 K5 23K2 39K 18) el = 592K(K5 + 8.4K4 + 3.18K3 96K2 20.3K + 0.98) e2 = 9K2(6 + 8.4K5 97.3K4 2045K3 + 1700K2 + 390K + 363) e3 = 24K5(K5 + 12.3K4 + 306K3 4100K2 + 12,300K 7000) By plotting the curves (3.19) in the KN plane and separating by hatched 14nes the regions corresponding to the signs of the inequalities (3.8), (3.13), and (3.18), the diagram shown in figure 3 is obtained. This diagram shows that for K>0 and NT>0 a region of values of K and I exists that corresponds to the presence of three roots with positive imaginary part, that is, of three unstable waves on the surface of separation. The division of the KN plane for the remaining roots could establish regions with a still greater number of roots with posi tive imaginary part. The given incomplete diagram already shows, however, the existence of several unstable waves. In the presence of a maximum Pi or ci, several infinitely long strings will NACA TM 1281 break away from the surface of separation, the crosssectional dimensions of which will depend on the wavelength of the critical disturbance. Translated by S. Reiss, National Advisory Committee for Aeronautics. REFERENCES 1. Rayleigh: The Theory of Sound. Dover Pub., 2d ed., 1945. 2. Tomotika, S.: On the Instability of a Viscous Liquid Surrounded by Another Roy. Soc. London, vol. CL, no. A870, pp. 322337. Cylindrical Thread of a Viscous Fluid. Proc. ser. A, June 1935, 3. Petrov, G. I.: On the Stability of Turbulent Layers. Rep. No. 304, CAHI, 1937. 4. Neimark, Y. I.: On the Problem of the Distribution of the Roots of Polynomials. DAN, T. 58, No. 3, 1947. NACA TM 1281 ~.v.po V/ 0 iu 1 Figure 1. Figure 2. Figure 3. NACALangley 41851 875 ,I,,,L,,_ __ '_ Sroot 3 2,, I root 2 0 Z 3 V 5 6 N 43 0 *8 rI r *il *H s) *p l 0 4, CH a) C) (1 ril t. CH *H 0 P 0 H C a r, q o8 H H E1 O ^ & ii 4 CH a) 0 1 0 S10i I ( C( P 0 40 < '> !o 0 "1 k C) ai > rQ Sr o o o o 4 4' 40 0 0 w ::1 0 4 a qi 1 ) $ a ) 4 0 c0 c0 (I P40 H 0 po 0 0 s3 O ,I , O O EI D i 0 CC ,A a 4 l Cd 4 m CD ,i 0 0o O !J 43 0V)) o 0) 02P 0P 0 0 0) (D 4 a) 3 4 0 4p 0 r km r4 H P0 0*H M 4:O 4i 0 IsxiD 0 ci0 300l > *( od o amock o P4i 0 PPPci P *H 43 r c30 0 a q4 14 P P cC 0 O cd o JR O I 0 > & *H C) 0 o m k 9a 3 ao DtH O m 0 &D (O m tDb c O o3 o F 3 ri H 0 H taD 0) 0 0 P 0 0 rq r d.) tP 4 m +Ir O O +rQ 0 rim o 8 a pq 0 a> 0 03 r4 (0 Da) (D rt i 0 P c 3 (i !P4 > P4  0o 0) 0 m q 00 4) 0 ? OH CH P i 0 41 Is W 0* O 4 40 4 0a) SP 0 0 9 0 X{ ! 0 0 '0 d i 0 : S h Eio m a H ( O a 0H O 0 0 IC 0 4, 0 ?rI >a 0 0 4OP 0 1 m ci O 0 0HP + >4i C 04 0 4) 0 ) 0 i d 4 o ia r riP ci oM d 0 S i O LD ) H 4 0 I 41 d oD r 04 Um.of. 0.. o m .....0 Oam a ?O OriMD ip m (Dir p Oi a UNIVERSITY OF FLORIDA S1262 08105 058 41111 3 1262 08105 0584 