Journal of Mathematical Chemistry

, Volume 57, Issue 9, pp 2075–2081

# A variational principle for a thin film equation

Original Paper

## Abstract

Thin film arises in various applications from electrochemistry to nano devices, many mathematical tools were adopted to study the problem, e.g. Lie symmetries and conservation laws, however, the variational approach is rare. This paper shows that the semi-inverse method is an effective approach to establishment of a variational formulation for the thin film equation. A detailed derivation process is given, a special skill for construction of a heuristic trial-functional is elucidated.

## Keywords

Variational theory Special function Trial-functional Nanoscale adhesion Coating Wetting

## References

1. 1.
E. Recio, T.M. Garrido, R. de la Rosa et al., Conservation laws and Lie symmetries a (2 + 1)-dimensional thin film equation. J. Math. Chem. 57(5), 1243–1251 (2019)
2. 2.
X.-X. Li, D. Tian, C.-H. He, J.-H. He, A fractal modification of the surface coverage model for an electrochemical arsenic sensor. Electrochim. Acta 296, 491–493 (2019)
3. 3.
J. Fan, Y.R. Zhang, Y. Liu et al., Explanation of the cell orientation in a nanofiber membrane by the geometric potential theory. Results Phys. 15, 102537 (2019)
4. 4.
Z.P. Yang, F. Dou, T. Yu et al., On the cross-section of shaped fibers in the dry spinning process: physical explanation by the geometric potential theory. Results Phys. 14, 102347 (2019)
5. 5.
X.X. Li, J.H. He, Nanoscale adhesion and attachment oscillation under the geometric potential. Part 1: the formation mechanism of nanofiber membrane in the electrospinning. Results Phys. 12, 1405–1410 (2019)
6. 6.
A. Saeed, Z. Shah, S. Islam et al., Three-dimensional casson nanofluid thin film flow over an inclined rotating disk with the impact of heat generation/consumption and thermal radiation. Coatings 9(4), 248 (2019)
7. 7.
C.-J. Zhou, D. Tian, J.-H. He, What factors affect lotus effect? Therm. Sci. 22, 1737–1743 (2018)
8. 8.
J.H. He, From micro to nano and from science to technology: nano age makes the impossible possible. Micro Nanosyst. 12(1), 1–2 (2010)Google Scholar
9. 9.
J. Manafian, C.T. Sindi, An optimal homotopy asymptotic method applied to the nonlinear thin film flow problems. Int. J. Numer. Methods Heat Fluid Flow 28(12), 2816–2841 (2018)
10. 10.
N. Faraz, Y. Khan, Thin film flow of an unsteady Maxwell fluid over a shrinking/stretching sheet with variable fluid properties. Int. J. Numer. Methods Heat Fluid Flow 28(7), 1596–1612 (2018)
11. 11.
F. Ghani, T. Gul, S. Islam et al., Unsteady magnetohydrodynamics thin film flow of a third grade fluid over an oscillating inclined belt embedded in a porous medium. Therm. Sci. 21(2), 875–887 (2017)
12. 12.
Q.T. Ain, J.H. He, On two-scale dimension and its applications. Therm. Sci. 23(3B), 1707–1712 (2019)
13. 13.
J.H. He, F.Y. Ji, Two-scale mathematics and fractional calculus for thermodynamics. Therm. Sci. 57(8), 1932–1934 (2019)Google Scholar
14. 14.
J.H. He, F.Y. Ji, Taylor series solution for Lane–Emden equation. J. Math. Chem. (2019).
15. 15.
J.H. He, The simplest approach to nonlinear oscillators. Results Phys. 15, 102546 (2019)
16. 16.
J.H. He, Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 19(4), 847–851 (2004)
17. 17.
J.H. He, J. Zhang, Semi-inverse method for establishment of variational theory for incremental thermoelasticity with voids, in Variational and Extremum Principles in Macroscopic Systems, ed. by S. Sieniutycz, H. Farkas (Elsevier, Amsterdam, 2005), pp. 75–95
18. 18.
J.H. He, A modified Li–He’s variational principle for plasma. Int. J. Numer. Methods Heat Fluid Flow (2019).
19. 19.
J.H. He, Lagrange crisis and generalized variational principle for 3D unsteady flow. Int. J. Numer. Methods Heat Fluid Flow (2019).
20. 20.
Y. Wu, J.H. He, A remark on Samuelson’s variational principle in economics. Appl. Math. Lett. 84, 143–147 (2018)
21. 21.
J.H. He, Hamilton’s principle for dynamical elasticity. Appl. Math. Lett. 72, 65–69 (2017)
22. 22.
K. Libarir, A. Zerarka, A semi-inversevariational method for generating the bound state energy eigenvalues in a quantum system: the Dirac Coulomb type-equation. J. Mod. Opt. 65(8), 987–993 (2018)
23. 23.
J. Manafian, P. Bolghar, A. Mohammadalian, Abundant soliton solutions of the resonant nonlinear Schrodinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method. Opt. Quant. Electron. 49(10), 322 (2017)
24. 24.
O.H. El-Kalaawy, New variational principle-exact solutions and conservation laws for modified ion-acoustic shock waves and double layers with electron degenerate in plasma. Phys. Plasmas 24(3), 032308 (2017)
25. 25.
A. Biswas, Q. Zhou, S.P. Moshokoa et al., Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations. Optik 145, 14–17 (2017)
26. 26.
Y. Li, C.H. He, A short remark on Kalaawy’s variational principle for plasma. Int. J. Numer. Methods Heat Fluid Flow 27(10), 2203–2206 (2017)
27. 27.
Y. Wang, J.Y. An, X.Q. Wang, A variational formulation for anisotropic wave traveling in a porous medium. Fractals 27(4), 1950047 (2019)
28. 28.
J.H. He, A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys. 53(11), 3698–3718 (2014)
29. 29.
J.H. He, Fractal calculus and its geometrical explanation. Results Phys. 10, 272–276 (2018)
30. 30.
N. Anjum, J.H. He, Laplace transform: making the variational iteration method easier. Appl. Math. Lett. 92, 134–138 (2019)
31. 31.
J.H. He, Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)
32. 32.
D. Baleanu, H.K. Jassim, H. Khan, A modified fractional variational iteration method for solving nonlinear gas dynamic and coupled KdV equations involving local fractional operator. Therm. Sci. 22, S165–S175 (2018)
33. 33.
D. Dogan Durgun, A. Konuralp, Fractional variational iteration method for time-fractional nonlinear functional partial differential equation having proportional delays. Therm. Sci. 22, S33–S46 (2018)
34. 34.
M. Inc, H. Khan, D. Baleanu et al., Modified variational iteration method for straight fins with temperature dependent thermal conductivity. Therm. Sci. 22, S229–S236 (2018)
35. 35.
H. Jafari, H.K. Jassim, J. Vahidi, Reduced differential transform and variational iteration methods for 3-D diffusion model in fractal heat transfer within local fractional operators. Therm. Sci. 22, S301–S307 (2018)
36. 36.
Y. Wang, Y.F. Zhang, Z.J. Liu, An explanation of local fractional variational iteration method and its application to local fractional modified Kortewed-de Vries equation. Therm. Sci. 22, 23–27 (2018)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Ji-Huan He
• 1
• Chang Sun
• 2
1. 1.School of ScienceXi’an University of Architecture and TechnologyXi’anChina
2. 2.Qujing Normal UniversityQujingChina