Symmetry-Breaking Bifurcations of Charged Drops Article First Online: 15 January 2004 Accepted: 13 October 2003 DOI :
10.1007/s00205-003-0298-x

Cite this article as: Fontelos, M. & Friedman, A. Arch. Rational Mech. Anal. (2004) 172: 267. doi:10.1007/s00205-003-0298-x
20
Citations
152
Downloads
Abstract. It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient γ is larger than some critical value (i.e., when γ<γ_{c} ). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where γ_{2} =γ_{c} . We further prove that the spherical drop is stable for any γ>γ_{2} , that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as t ρ∞ provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at γ=γ_{2} which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.

References 1.

Borisovich , A., Friedman , A.: Symmetry-breaking bifurcations for free boundary problems. To appear

2.

C

randall , M.G., R

abinowitz , L.H.: Bifurcation from simple eigenvalies.

J. Functional Analysis
8 , 321–340 (1971)

MATH Google Scholar 3.

C

randall , M.G., R

abinowitz , L.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability.

Arch. Rat. Mech. Anal.
52 , 161–180 (1973)

MATH Google Scholar 4.

Duft , D., Achtzehn , T., Müller , R., Huber , B.A., Leisner , T.: Rayleigh jets from levitated microdroplets. Nature , vol. 421 , 9 January 2003, p. 128

5.

F

ontelos , M.A., F

riedman , A.: Symmetry-breaking bifurcations of free boundary problems in three dimensions.

Asymptotic Analysis
35 , 187–206 (2003)

Google Scholar 6.

F

riedman , A., H

u , B., V

elázquez , J.J.L.: A Stefan problem for a protocell model with symmetry-breaking bifurcation of analytic solutions.

Interfaces Free Bound
3 , 143–199 (2001)

MathSciNet MATH Google Scholar 7.

F

riedman , A., R

eitich , F.: On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic tumors.

Math. Models & Methods in Applied Science
11 , 601–625 (2001)

Google Scholar 8.

F

riedman , A., R

eitich , F.: Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth.

Trans. Am. Math. Soc.
353 , 1587–1634 (2000)

MATH Google Scholar 9.

Friedman , A., Reitich , F.: Nonlinear stability of quasi-static Stefan problem with surface tension: a continuation approach. Ann. Scu. Norm. Super. Pisa (4 ), 30 , 341–403 (2001)

10.

F

riedman , A., R

eitich , F.: Quasi-static motion of a capillary drop, II: the three-dimensional case.

J. Differential Equations
186 , 509–557 (2002)

MathSciNet MATH Google Scholar 11.

Jackson , J.D.: Classical Electrodynamics . John Wiley & Sons, New York; 3rd edition, 1999

12.

Koshlyakov , N.S., Smirnov , M.M., Gliner , E.B.: Differential equations of mathematical physics . John Wiley & Sons, New York; 3rd edition, 1964

13.

M

iksis , M.J.: Shape of a drop in an electric field.

Phys. Fluids
24 , 1967–1972 (1981)

MATH Google Scholar 14.

Nirenberg , L.: Topics in Nonlinear Functional Analysis . Courant Inst. of Math. Sci., New York, 1974

15.

L

ord R

ayleigh : On the equilibrium of liquid conducting masses charged with electricity.

Phil. Mag.
14 , 184–186 (1882)

Google Scholar 16.

Tikhonov , A.N., Samarskii , A.A.: Equations of Mathematical Physics . Dover Publications, New York, 1990

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations 1. Departamento de Matemática Aplicada Universidad Rey Juan Carlos Madrid Spain 2. Department of Mathematics The Ohio State University W Columbus U.S.A