×

Symmetries, first integrals and phase planes of a third-order ordinary differential equation from thin film flow. (English) Zbl 1175.34047

The author investigates the third-order ODE
\[ y^ny'''=1 \]
subject to
\[ y(0)=1,\;y'(0)=0\text{ and }y''(0)=\lambda<0, \]
where \(n\) is a positive rational number and the constant \(\lambda\) is to be determined so that the solution satisfies a certain geometrical condition. This problem admits two generators of Lie point symmetries and therefore can be reduced to a first-order OPD. Phase planes for different values of the parameter \(n\) are analyzed.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
76A20 Thin fluid films

Software:

DIMSYM; LIE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Almgren, R., Singularity formation in HeleShaw bubbles, Phys. Fluids, 8, 344-352 (1996) · Zbl 1023.76522
[2] Baumann, G., Symmetry Analysis of Differential Equations With Mathematica (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0898.34003
[3] Bernis, F., Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, (Diaz, J. I.; etal., Free Boundary Problems: Theory and Applications. Free Boundary Problems: Theory and Applications, Pitman Research Notes in Mathematics, vol. 323 (1995), Longman: Longman Harlow), 40-56 · Zbl 0839.35102
[4] Bernis, F., Finite speed of propagation for thin viscous flow when \(2 < n < 3\), Comptes Rendus Acad. Sci. I - Math., 322, 1169-1174 (1996) · Zbl 0853.76018
[5] Bernis, F.; Peletier, L. A., Two problems from draining flows involving third-order ordinary differential equations, SIAM J. Math. Anal., 27, 515-527 (1996) · Zbl 0845.34033
[6] Bertozzi, A. L.; Brenner, M. P.; Dupont, T. F.; Kadanoff, L. P., Singularities and similarities in interface flows, (Sirovich, L., Trends and Perspectives in Applied Mathematics. Trends and Perspectives in Applied Mathematics, Applied Mathematical Sciences, vol. 100 (1994), Springer: Springer Berlin), 155-208 · Zbl 0808.76022
[7] Bertozzi, A. L., The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45, 689-697 (1998) · Zbl 0917.35100
[8] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0718.35004
[9] Boatto, S.; Kadanoff, L. P.; Olla, P., Traveling wave solutions to thin film equations, Phys. Rev. E, 48, 4423-4431 (1993)
[10] Buckingham, R.; Shearer, M.; Bertozzi, A., Thin film traveling waves and the navier slip condition, SIAM J. Appl. Math., 63, 722-744 (2003) · Zbl 1024.35038
[11] Constantin, P.; Dupont, T. F.; Goldstein, R. E.; Kadanoff, L. P.; Shelley, M. J.; Zhou, Su-Min, Droplet breakup in a model of the HeleShaw cell, Phys. Rev. E, 47, 4169-4181 (1993)
[12] Dresner, L., Phase-plane analysis of nonlinear, second-order, ordinary differential equations, J. Math. Phys., 12, 1339-1348 (1971) · Zbl 0266.34034
[13] Duffy, B. R.; Wilson, S. K., A third-order differential equation arising in thin-film flows and relevant to Tanner’s Law, Appl. Math. Lett., 10, 63-68 (1997) · Zbl 0882.34001
[14] Dupont, T. F.; Goldstein, R. E.; Kadanoff, L. P.; Zhou, Su-Min, Finite-time singularity formation in HeleShaw systems, Phys. Rev. E, 47, 4182-4196 (1993)
[15] Ford, W. F., A third-order differential equation, SIAM Review, 34, 121-122 (1992)
[16] Greenspan, H. P., On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84, 125-143 (1978) · Zbl 0373.76040
[17] Greenspan, H. P.; McCay, B. M., On the wetting of a surface by a very viscous fluid, Stud. Appl. Math., 64, 95-112 (1981) · Zbl 0474.76099
[18] Goldstein, R. E.; Pesci, A. I.; Shelley, M. J., Instabilities and singularities in HeleShaw flow, Phys. Fluids, 10, 2701-2723 (1998) · Zbl 1185.76632
[19] Head, A. K., LIE, a PC program for Lie analysis of differential equations, Comput. Phys. Comm., 77, 241-248 (1993) · Zbl 0854.65055
[20] Hocking, L. M., Sliding and spreading of thin two dimensional drops, Q. J. Mech. Appl. Math., 34, 37-55 (1981) · Zbl 0487.76094
[21] Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations (1999), J. Wiley and Sons: J. Wiley and Sons Chichester · Zbl 1047.34001
[22] Kara, A. H.; Mahomed, F. M., A Basis of Conservation Laws for Partial Differential Equations, J. Nonlinear Math. Phys., 9, 2, 6072 (2002) · Zbl 1362.35024
[23] Lacey, A. A., The motion with slip of a thin viscous droplet over a solid surface, Stud. Appl. Math., 67, 217-230 (1982) · Zbl 0505.76112
[24] Myers, T. G., Thin films with high surface tension, SIAM Rev., 40, 441-462 (1998) · Zbl 0908.35057
[25] Noether, E., Invariante Variationsprobleme, König. Gesell. Wissen Göttingen, Math.- Phys. Kl. Heft, 2 (1918)
[26] Ovsiannikov, L. V., Group Analysis of Differential Equations (1982), Academic Press: Academic Press New York · Zbl 0485.58002
[27] Pesci, A. I.; Goldstein, R. E.; Shelley, M. J., Domain of convergence of perturbative solutions for HeleShaw flow near interface collapse, Phys. Fluids, 11, 2809-2811 (1999) · Zbl 1149.76512
[28] Sherring, J.; Head, A. K.; Prince, G. E., Dimsym and LIE: Symmetry determining packages, Math. Comput. Modelling, 25, 153-164 (1997) · Zbl 0918.34007
[29] Smyth, N. F.; Hill, J. M., High-order nonlinear diffusion, IMA J. Appl. Math., 40, 73-86 (1988) · Zbl 0694.35091
[30] Tanner, L. H., The spreading of silicone oil drops on horizontal surfaces, J. Phys. D: Appl. Phys., 12, 1473-1484 (1979)
[31] Troy, W. C., Solutions of third-order differential equations relevant to draining and coating flows, SIAM J. Math. Anal., 24, 155-171 (1993) · Zbl 0807.34030
[32] Tuck, E. O.; Schwartz, L. W., Numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows, SIAM Rev., 32, 453-469 (1990) · Zbl 0705.76062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.