Numerical investigation of a third-order ODE from thin film flow

Meccanica

Volumn 46, Issue 2, 2011, Pages 313-323

  Momoniat, E ^a  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

Author keywords

0 stability; Finite differences; Thin film; Third order ODE; Von Neumann stability

Indexed keywords

0-STABILITY; BACKWARD DIFFERENCE; CENTRAL DIFFERENCE APPROXIMATIONS; DIFFERENCE SCHEMES; FINITE DIFFERENCE; FINITE DIFFERENCE SCHEME; MODEL EQUATIONS; NUMERICAL INVESTIGATIONS; STABILITY PROPERTIES; SURFACE-TENSION-DRIVEN FLOW; THIN FILM FLOW; THIRD-ORDER; THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS; VON NEUMANN STABILITY;

BOUNDARY CONDITIONS; DIFFERENTIATION (CALCULUS); FINITE DIFFERENCE METHOD; ORDINARY DIFFERENTIAL EQUATIONS; STABILITY; SURFACE TENSION;

THIN FILMS;

EID: 79955958392     PISSN: 00256455     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11012-010-9310-3     Document Type: Article

References (40)

1. Greenspan HP (1978) On the motion of a small viscous droplet that wets a surface. J Fluid Mech 84:125-143 (Pubitemid 8574495)
2. Myers TG (1998) Thin films with high surface tension. SIAM Rev 40:441-462 (Pubitemid 128619816)
3. Boatto S, Kadanoff LP, Olla P (1993) Traveling-wave solutions to thin film equations. Phys Rev E 48:4423-4431
4. Almgren R (1996) Singularity formation in Hele-Shaw bubbles. Phys Fluids 8:344-352 (Pubitemid 126609152)
5. Constantin P, Dupont TF, Goldstein RE, Kadanoff LP, ShelleyMJ, Zhou S-M (1993) Droplet breakup in a model of the Hele-Shaw cell. Phys Rev E 47:4169-4181
6. Dupont TF, Goldstein RE, Kadanoff LP, Zhou S-M (1993) Finite-time singularity formation in Hele-Shaw systems. Phys Rev E 47:4182-4196
7. Goldstein RE, Pesci AI, Shelley MJ (1998) Instabilities and singularities in Hele-Shaw flow. Phys Fluids 10:2701-2723 (Pubitemid 128599798)
8. Pesci AI, Goldstein RE, ShelleyMJ (1999) Domain of convergence of perturbative solutions for Hele-Shaw flow near interface collapse. Phys Fluids 11:2809-2811 (Pubitemid 129654534)
9. Greenspan HP, McCay BM (1981) On the wetting of a surface by a very viscous fluid. Stud Appl Math 64:95-112 (Pubitemid 12514873)
10. Hocking LM (1981) Sliding and spreading of thin two dimensional drops. Q J Mech Appl Math 34:37-55 (Pubitemid 11500704)
11. Lacey AA (1982) The motion with slip of a thin viscous droplet over a solid surface. Stud Appl Math 67:217-230 (Pubitemid 13488076)
12. Tanner LH (1979) The spreading of silicone oil drops on horizontal surfaces. J Phys D, Appl Phys 12:1473-1484 (Pubitemid 10414577)
13. Bernis F (1996) Finite speed of propagation for thin viscous flow when 2 < n < 3. C R Acad Sci I, Math 322:1169-1174
14. Bernis F, Peletier LA (1996) Two problems from draining flows involving third-order ordinary differential equations. SIAM J Math Anal 27:515-527
15. Troy WC (1993) Solutions of third-order differential equations relevant to draining and coating flows. SIAM J Math Anal 24:155-171
16. Bertozzi AL (1996) Symmetric singularity formation in lubrication-type equations for interface motion. SIAM J Appl Math 56:681-714 (Pubitemid 126610893)
17. Bertozzi AL (1998) The mathematics of moving contact lines in thin liquid films. Not Am Math Soc 45:689-697
18. Howes FA (1983) The asymptotic solution of a class of third-order boundary value problems arising in the theory of thin film flows. SIAM J Appl Math 43:993-1004 (Pubitemid 14469726)
19. Tuck EO, Schwartz LW (1990) Numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev 32:453-469 (Pubitemid 20734631)
20. Duffy BR, Wilson SK (1997) A third-order differential equation arising in thin-film flows and relevant to Tanner's Law. Appl Math Lett 10:63-68 (Pubitemid 127378745)
21. Ford WF (1992) A third-order differential equation. SIAM Rev 34:121-122
22. Buckingham R, Shearer M, Bertozzi A (2003) Thin film traveling waves and the Navier slip condition. SIAM J Appl Math 63:722-744
23. Momoniat E, Selway TA, Jina K (2007) Analysis of Adomian decomposition applied to a third-order ordinary differential equation from thin film flow. Nonlinear Anal, Ser A, Theory Methods Appl 66:2315-2324 (Pubitemid 46281495)
24. Momoniat E (2009) Symmetries, first integrals and phase planes of a third order ordinary differential equation from thin film flow. Math Comput Model 49:215-225
25. Lambert JD (1991) Numerical methods for ordinary differential systems. Wiley, Chichester
26. Butcher JC (2008) Numerical methods for ordinary differential equations. Wiley, Chichester
27. Aceto L, Trigiante D (2007) The stability problem for linear multistep methods: old and new results. J Comput Appl Math 210:2-12 (Pubitemid 47488677)
28. Hairer E, Nørsett SP, Wanner G (1993) Solving ordinary differential equations I: nonstiff problems, 2nd edn. Springer, Berlin
29. Lax PD, Richtmyer RD (1956) Survey of the stability of linear finite difference equations. Commun Pure Appl Math 9:267-293
30. Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York
31. Geršgorin SA (1931) Über die Abgrenzung der Eigenwerte einer Matrix. Dokl Akad Nauk (A), Otd Fiz-Mat Nauk 749-754
32. Varga RS (2004) Geršgorin and his circles. Springer, New York
33. Myers TG, Charpin JPF (2000) The effect of the Coriolis force on axisymmetric rotating thin film flows. Int J Nonlinear Mech 36:629-635 (Pubitemid 32092434)
34. Bornside DE, Macosko CW, Scriven LE (1987) On the modeling of spin coating. J Imag Technol 13:122-130
35. Forcada ML, Mate CM (1993) The flow of thin viscous liquid films on rotating disks. J Colloid Interface Sci 160:218-225
36. Dandapat BS, Ray PC (1994) The effect of thermocapillarity on the flow of a thin liquid film on a rotating disc. J Phys D, Appl Phys 27:2041-2045
37. Leshev I, Peev G (2003) Film flow on a horizontal rotating disk. Chem Eng Proc 42:925-929 (Pubitemid 37144526)
38. Huang K-H, Chou F-C, Yang C-P (2007) Visualization of the effect of liquid dispensing method during spin coating. Jpn J Appl Phys 46:5238-5244 (Pubitemid 47252595)
39. Afsar-Siddiqui AB, Luckham PF, Matar OK (2003) The spreading of surfactant solutions on thin liquid films. Adv Colloid Interface Sci 106:183-236
40. Gaver DP III, Grotberg JB (1990) The dynamics of a localized surfactant on a thin film. J Fluid Mech 213:127-148