# zbMATH — the first resource for mathematics

The tanh method and Adomian decomposition method for solving the foam drainage equation. (English) Zbl 1120.76051
Summary: Foaming occurs in many distillation and absorption processes. The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. In this paper, we use a semi-analytic method, the Adomian decomposition method, and an analytic method, the tanh method to handle the foam drainage equation. The powerful tanh method gives the solution in a closed form. However, Adomian decomposition method computes the solution in a rapidly convergent infinite series. The comparison between the two approaches is conducted to illustrate the performance of each method.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76S05 Flows in porous media; filtration; seepage 76D45 Capillarity (surface tension) for incompressible viscous fluids
Full Text:
##### References:
 [1] Stoyanov, S.D.; Paunov, V.N.; Basheva, E.S.; Ivanov, I.B.; Mehreteab, A.; Broze, G., Motion of the front between thick and thin film: hydrodynamic theory and experiment with vertical foam films, Langmuir, 13, 1400-1407, (1997) [2] Weaire, D.; Hutzler, S.; Cox, S.; Kern, N.; Alonso, M.D.; Drenckhan, D., The fluid dynamics of foams, J. phys. condens. matter, 15, S65-S73, (2003) [3] () [4] Weaire, D.L.; Hutzler, S., The physics of foams, (2000), Oxford University Press Oxford [5] Stone, H.A.; Koehler, S.A.; Hilgenfeldt, S.; Durand, M., Perspectives on foam drainage and the influence of interfacial rheology, J. phys. condens. matter, 15, S283-S290, (2003) [6] Hilgenfeldt, S.; Koehler, S.A.; Stone, H.A., Dynamics of coarsening foams: accelerated and self-limiting drainage, Phys. rev. lett., 20, 4704-4707, (2001) [7] Wilson, J.I.B., Essay review, scholarly froth and engineering skeletons, Contemp. phys., 44, 153-155, (2003) [8] Gibson, L.J.; Ashby, M.F., Cellular solids: structure & properties, (1997), Cambridge University Press Cambridge [9] J. Banhart, Metallschaüme, MIT-Verlag, Bremen, 1997. [10] Koehler, S.A.; Stone, H.A.; Brenner, M.P., Eggers J. dynamics of foam drainage, Phys. rev., 58, 2, 2097-2106, (1998) [11] Gibson, L.J.; Ashby, M.F., Cellular solids: structure and properties, (1999), Cambridge University Press Cambridge [12] Ashby, M.F.; Evans, A.G.; Fleck, N.A.; Gibson, L.J.; Hutchinson, J.W.; Wadley, H.N.G., Metal foams: A design guide, (2000), Society of Automotive Engineers Boston [13] Leonard, R.A.; Lemlich, R., Aiche j., 11, 18, (1965) [14] Gol’dfarb, I.I.; Kann, K.B.; Shreiber, I.R., Fluid dyn., 23, 244, (1988) [15] Weaire, D.; Hutzler, S., The physics of foams, (2000), Oxford University Press Oxford [16] Weaire, D.; Hutzler, S.; Verbist, G.; Peters, E.A.J., Adv. chem. phys., 102, 315, (1997) [17] Bhakta, A.; Ruckenstein, E., Adv. colloid interf. sci., 70, 1, (1997) [18] Duranda, M.; Langevin, D., Physicochemical approach to the theory of foam drainage, Eur. phys. J. E, 7, 35-44, (2002) [19] Verbist, G.; Weaire, D., Soluble model for foam drainage, Europhys. lett., 26, 631-634, (1994) [20] Verbist, G.; Weaire, D.; Kraynik, A.M., The foam drainage equation, J. phys. condens. matter, 8, 3715-3731, (1996) [21] Verbist, G.; Weuire, D., Eumphys. lett., 26, 631, (1994) [22] Weah, D.; Hutzler, S.; Pittet, N.; Pard, D., Phys. rev. lett., 71, 2670, (1993) [23] Weaire, D.; Findlay, S.; Verbist, G., Measurement of foam drainage using AC conductivity, J. phys. condens. matter, 7, L217-L222, (1995) [24] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1998) · Zbl 0671.34053 [25] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston, MA · Zbl 0802.65122 [26] Wazwaz, A.M., The decomposition for approximate solution of the Goursat problem, Appl. math. comput., 69, 299-311, (1995) · Zbl 0826.65077 [27] Wazwaz, A.M., A comparison between adomian’s decomposition methods and Taylor series method in the series solutions, Appl. math. comput., 97, 37-44, (1998) · Zbl 0943.65084 [28] Wazwaz, A.M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. math. comput., 111, 53-69, (2000) · Zbl 1023.65108 [29] Wazwaz, A.M., The modified decomposition method and pade approximants for solving the Thomas-Fermi equation, Appl. math. comput., 105, 9-11, (1999) · Zbl 0956.65064 [30] Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl. math. comput., 102, 77-86, (1999) · Zbl 0928.65083 [31] Cherruault, Y.; Saccomandi, G.; Some, B., New results for convergence of adomian’s method applied to integral equations, Math. comput. model., 16, 2, 85-93, (1992) · Zbl 0756.65083 [32] Mavoungou, T.; Cherruault, Y., Convergence of adomian’s method and applications to non-linear partial differential equations, Kybernetes, 21, 6, 13-25, (1992) · Zbl 0801.35007 [33] Repaci, A., Nonlinear dynamical systems: on the accuracy of adomian’s decomposition method, Appl. math. lett., 3, 35-39, (1990) · Zbl 0719.93041 [34] Cherruault, Y., Convergence of adomian’s method, Kybernetics, 18, 31-38, (1989) · Zbl 0697.65051 [35] Ngarhasta, N.; Some, B.; Abbaoui, K.; Cherruault, Y., New numerical study of Adomian method applied to a diffusion model, Kybernetes, 31, 61-75, (2002) · Zbl 1011.65073 [36] Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am. J. phys., 60, 7, 650-654, (1992) · Zbl 1219.35246 [37] Fan, E.; Hon, Y.C., Generalized tanh method extended to special types of nonlinear equations, Z. naturforsch, 57a, 692-700, (2002) [38] Wazwaz, A.M., The tanh method for travelling wave solutions of nonlinear equations, Appl. math. comput., 154, 3, 713-723, (2004) · Zbl 1054.65106 [39] Malfliet, W., The tanh method: I. exact solutions of nonlinear evolution and wave equations, Phys. scripta, 54, 563-568, (1996) · Zbl 0942.35034 [40] Malfliet, W., The tanh method: II. perturbation technique for conservative systems, Phys. scripta, 54, 569-575, (1996) · Zbl 0942.35035
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.