The use of homotopy methods for solving nonlinear foam drainage equation
Communications on Advanced Computational Science with Applications
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, the nonlinear foam drainage equation is solved by using the Adomian's decomposition method , modified Adomian's decomposition method , variational iteration method , modified variational iteration method, homotopy perturbation method, modified homotopy perturbation method and homotopy analysis method. The existence and
... The existence and uniqueness of the solution and convergence of the proposed methods are proved in details. Finally an example shows the accuracy of these methods. Keywords: Nonlinear foam drainage equation, Adomian decomposition method (ADM) , Modified Adomian decomposition method (MADM), Variational iteration method (VIM) , Modified variational iteration method (MVIM), Homotopy perturbation method (HPM), Modified homotopy perturbation method (MHPM), Homotopy analysis method (HAM). exchange media analogous to common finned structures  . Polymeric foams are used in cushions and packing and structural materials  . Glass, ceramic, and metal foams  can also be made and find an increasing number of new applications. In addition, mineral processing utilizes foam to separate valuable products by flotation. Finally, foams enter geophysical studies of the mechanics of volcanic eruptions  . Recent research in foams and emulsions has centered on three topics which are often treated separately but are, in fact, interdependent: drainage, coarsening, and rheology, see Figure 1 . We focus here on a quantitative description of the coupling of drainage and coarsening. Foam drainage is the flow of liquid through channels (plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity [13, 14, 15] . During foam production, the material is in the liquid state, and fluid can rearrange while the bubble structure stays relatively unchanged. The flow of liquid relative to the bubbles is called drainage. Generally, drainage is driven by gravity and/or capillary (surface tension) forces and is resisted by viscous forces  . Because of their limited time stability and despite the numerous studies reported in the literature, many of their properties are still not well understood, in particular the drainage of the liquid in between the bubbles under the influence of gravity [16, 17] . Drainage plays an important role in foam stability. Indeed, when foam dries, its structure becomes more fragile; the liquid films between adjacent bubbles being thinner, then can break, leading to foam collapse. In the case of aqueous foams, surfactant is added into water, and it adsorbs at the surface of the films, protecting them against rupture  . Most of the basic rules that explain the stability of liquid gas foams were introduced over 100 years ago by the Belgian Joseph Plateau who was blind before he completed his important book on the subject. This modern-day book by Weaire and Hutzler provides valuable summaries of plateaus work on the laws of equilibrium of soap films, and it is especially useful since the original 1873 French text does not appear to be in a fully translated English version.Weaire and Hutzler note that SirW. Thompson (Lord Kelvin) was simulated by Plateau's book to examine the optimum packing of free space by regular geometrical cells. His solution to the problem remained the best until quite recently. Why does this area of theoretical research, still active today, have connections with the apparently frivolous theme of bubbles? It is because the packing of free space involves the minimization of the surface energy of the structure (i.e., least amount of boundary material). Thus, one might ask why such an often-observed medium as a foam has not provided the optimum solution to this problem much earlier, perhaps, this shows that observation is often biased towards what one expects to see, rather than to the unexpected. Also, in nature, there are packing problems, such as the bees' honeycomb. Its shaped ends provide a nice example of Plateau's rules in a natural environment  . Recent theoretical studies by Verbist and Weaire describe the main features of both free drainage [19, 20] , where liquid drains out of a foam due to gravity, and forced drainage  , where liquid is introduced to the top of a column of foam. In the latter case, a solitary wave of constant velocity is generated when liquid is added at a constant rate  . Forced foam drainage may well be the best prototype for certain general phenomena described by nonlinear differential equations, particularly the type of solitary wave which is most familiar in tidal bores.