'IEEE... AD-A THE SHAPE OF A LIQUID DROP IN THE FLOW OF A PERFECT in1 I FLUID(U) HARRY DIAMOND LASS ADEIPHI RD C A MORRISON FEB 83 HDL-TL-83-2

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AD-A125 873 THE SHAPE OF A LIQUID DROP IN THE FLOW OF A PERFECT in1 I FLUID(U) HARRY DIAMOND LASS ADEIPHI RD C A MORRISON FEB 83 'IEEE... HDL-TL-83-2 IUNCLASSIFIED F/G 28/4 NL

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1.... N,... "4L-TL-83-2, February 1983 The Shape of a Liquid Drop in the Flow of a Perfect Fluid by Clyde A* 14orrison DTIC EECTE MAR 9J S.S "I ( Abstract D A method of analysis previously used to determine the shape of a liquid drop in an electric field has been applied to determine the shape of a drop in the flow of a perfect fluid. The surface of the drop is expanded in a series of Legendre polynomials with arbitrary coefficients. inimization of the total energy with respect to the coefficients then gives equations for their determination. Due to the similarity of the equations for determining the coefficients to previous results, it is expected that the shape of the drops will deviate considerably from the usually assumed oblate spheroids. I Asession For MIS ORAiij ". DTIC?AB Umnnounced 0 od.0 Justificati Distribut0ion/ C.- Approved for public release; distribution unlimited. Availability Codes CDvail and/or For further copies of this technical letter, Commander L.a. contact the author at the following address: Harry Olamond Laboratorles ATTN DELHD-RT-CA 2800 Powder MIN Road Ade phl, MD 20783 83 03 001 121

HDL-TL-83-2, February 1983 1 I ODMUCTION In this report we determine the shape of a liquid drop in the flow of an ideal fluid (no viscoity). The drop is assumed to be held together by the interface surface tension between the liquid inside the drop and the external fluid. In previous analyses of this problem, the shapes of the drops have, been assumed to be oblate spheroids with a minor axis parallel to the direction of flow of the external fluid. 1 ' 2 In our earlier analysis of the shape of liquid drops in an electric field, the commonly accepted assumption that the drops formed prolate spheroids with the major axis parallel to the electric field was shown to be generally invalid. 3 "' The spheroidal approximation for drops was only valt- for small drops or low electric fields. Since the spheroidal approximation was invalid for drops in an electric field, we suspected that perhaps the same approximation would be invalid for falling liquid drops in air or other fluids. We have made the assumption that the external fluid has no viscosity, otherwise, the problem is extremely complicated* Here we shall use the technique used in our previous analysis.4 That is, we express the total energy of the system, the kinetic energy of the external fluid, and the potential energy due to surface tension, in term of the coefficients, an, in the expansion a r (0) - a + ap n lcs O),1(1) *ri1 / ' 2 L:.--*e Y37 ' *.. -... *. **...

'7S HOL-TL-83-2, February 1983 which relates an arbitrary point on the surface of the drop to the angle 8 measured from the direction of the fluid flow (z axis) in equation (1) the P n(coo 8) are Legendre polynomials* Minimizing the total energy with respect to the a's gives a set of equations for these coefficients. The solutions of this set of equations then are used in equation (1) to determine the shape of the drop. 2. THEORY In equation (1), we have made the physically realistic assumption that the drop is symmetric about the direction of the external fluid flow. in standard texts on fluid mechanics,56 it is shown that the velocity in a perfect incompressible fluid can be derived from a velocity potential, #, which is a solution of Laplace's equation. That is, V2 0 o(2) and -.(3) The solution to equation (2) appropriate for the region external to the drop 44i 2 -- --. *,,',,'. -.,~,,. -...-... -.-,.-...-.... *. *...... -....,..,

E DL-TL-83-2. February 1983 n rn+ -Vrl? 1, (4) where we have dropped the arguments of the Legendro polynomals (Pn = Pn(Cos 0) ] and aumed that far fron the drop the velocity of the external fluid is given by 3A arecti(5) and e is a unit vector in the 3 hrreetw)on. As in car previous analyses 3 o we shall rewrite equation (1) a r() - a + anp a n (6) where 6 is an order parameter. We then expand the A n in equation (4) in powers of 8 as and retain the term through 6 2 A a A(0) + 8A (1) + 62A(2) + (7) nt n n.. The A n of equation (4) are determined by the boundary condition, which, for a perfect incompressible fluid, is that the velocity normal to the 3

HL-TL-83-2, February 1983 surf ace, vn, wst vanish. The velocity normal to the surface is given by equation (3.13) of Morrison et al 4 or 0 - u2 dr (8) n ""[Lr r2 dus.u( where p - coo 8 and r is given by equation (6). Using equations (6) and (7) in equation (9) and letting v - 0 we obtain -,n 2 'n1 A(2) 3a 3Van+1 Am-0 c lkln> 2 n 4(n + 1) Clnkk k Wan A (2) 0 T) aa ikjx 2 *31n>2C (21 - C n 8(n +. 1 I1, fta (9) where <Ikin> are Clebsch-Gordan coefficients " 7 (In Rose, 7 the...c(aknoo). - <Akln>), the Cab,, are given by Cab,c - a(a+l) + b(b+1) - c(c+1) and 6ni is the Kronecker delta function. * 4

HL-TL-83-2, February 1983 3. IUG For the change in the kinetic energy of the external fluid, we use T a f j pv 2 dt - f -1,V 2 dt (10) where the first integral covers the region r(8) C r < R o, and the second integral covers the region 0 4 r. R, and in the end result we let Ro become infinite. When equation (4) is used in equation (10), we obtain (R.. ).43 1 2WOV (11) Thus, we need only A, through second order to obtain the change in kinetic energy. All the Clebech-Gordan coefficients appearing in equation (9) are simple for n - 1, and algebraic expressions for them can be found in Rose. 7 Thus, A 1 2 + 1 2 nlanan +2 (12) where -,

m! " & "':...,.,.,.! 2,. -,,.,, HDL-TL-83-2, February 1983 'a ~(n- 1) G+ n,+ 3) hn " O(n - 1) (2n + 1)2 (2n + 3) gn1 n- 1 1(2 ) + 1) On + 3) "To obtain the r e sult given in equation (12), we have used the constraint of 5constant volume as given in Page. 5, a- (13 V: a = a -- O[ n 013) 0? a in + I The surface energy of the drop is simp'ly the product of the surface tension, y, and the surface area of the drop and is given by n (n - M1n + 2 )a () and the constraint given in equation (13) as used in Page 5 to obtain this result. We now have the total energy of the system expressed in term of the aa1 2 4. IM SQ $TIONS POR a. The condition for minimum energy can be written 0.6 I - b* '.~*'.. *... ".......,,,.. "~~~~. " - 5 "" '" "" "' 2'' - ;.n,,,".. '*,' -,. ; " "... "..............

:77..4 HDL-TL-83-2, February 1983 V;. n (T + U) 015 The derivative of U s is simple; from equation (14), this is ;U s 4ry(n - 1)1n + 2)a a n 1On + 1) Using equation (12) in equation (11), we have the kinetic energy; from this result, we obtain 3T 2 wpav 2 [ (g a a T - IaV2-2 [hnan 2 n+lan+2 + gn-an- 2 )] If the results given in equations (16) and (17) are substituted into equation (15), we get :" (18) [(n - 11n + 2) - y h] 3 2n + 1 Xn + [n+n+2 + gn-lxn-2 ] " 2 n,2 a n V2 " where x n - a and y- 2y 2 The result given in equation (18) can be used to deteraine the x n in the same manner as was used in Page.5 The resulting xn can then be used in equation 7 *. ** -**.......-...-...

3- _...... WIN HDL-TL-83-2, February 1983 (1) ( (n, ax an ) to determine the shape of the drop as a 'function of fluid velocity, V. Because of the similarity of equation (18) to the corresponding equation in Page, 5 it is expected that the drop will deviate considerably from an oblate spheroid for moderate fluid velocities or for large drop radii a. LITERATURE CITED (1) A. W. Green, J. Appl. Meteorology 14 (1975), 1578. (2) A. F. Spilhaus, J. Meteorology 5 (1948), 108. (3) C. A. Morrison, R. P. Leavitt, and D. E. Wortman, The Extended Rayleigh Theory of the Oscillation of Liquid Droplets, Harry Diamond Laboratories, HDL-TR-1924 (May 1980). (4) C. A. Morrison, R. P. Leavitt, and D. E. Wortman, J. Fluid Mech. 104 (1981), 295. (5) L. Page, *Introduction to Theoretical Physics" (New Yorks D. Van Nostrand, 1952). (6) L. D. Landau and E. 4. Lifshitz, Fluid Mechanics. Volume 6 of a Course of Theoretical Physics (New York: Pergamon Press 1959). (7) H. E. Rose, Elementary Theory of Angular Momentum New York: John Wiley and Sons (1957). '4-2,.:. 2..:-,-,,.,-..... -........,......

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