Application of the Homotopy Perturbation Method for Solving the Foam Drainage Equation

Foams are of great importance in many technological processes and applications, and their properties are subject of intensive studies from both practical and scientific points of view [26]. Liquid foam is an example of soft matter (or complex fluid) with a very well-defined structure that first clearly described by Joseph plateau in the 19th century. Weaire et al. [27] showed in their work simple answers to many such questions exist, but no going experiments continue to challenge our understanding. Foams and emulsions are wellknown to scientists and the general public alike because of their everyday occurrence [28, 29]. Foams are common in foods and personal care products such as creams and lotions, and foams often occur, even when not desired, during cleaning (clothes, dishes, scrubbing) and dispensing processes [30]. They have important applications in the food and chemical industries, firefighting, mineral processing, and structural material science [31]. Less obviously, they appear in acoustic cladding, lightweight mechanical components, and impact absorbing parts on cars, heat exchangers, and textured wallpapers (incorporated as foaming inks) and even have an analogy in cosmology. The packing of bubbles or cells can form both random and symmetrical arrays, such as sea foam and bees’ honeycomb. History connects foams with a number of eminent scientists, and foams continue to excite imaginations [32]. There are now many applications of polymeric foams [33] and more recently metallic foams, which are foams made of metals such as aluminum [34]. Some commonly mentioned applications include the use of foams for reducing the impact of explosions and for cleaning up oil spills. In addition, industrial applications of polymeric foams and porous metals include their use for structural purposes and as heat exchange media analogous to common “finned” structures [35]. Polymeric foams are used in cushions and packing and structural materials [36]. Glass, ceramic, and metal foams [37] 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 [30]. 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 [38–40]. 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 [30]. 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 [41, 42]. 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 [43]. 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 Sir W. 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 [32]. Recent theoretical studies by Verbist and Weaire describe the main features of both free drainage [44, 45], where liquid drains out of a foam due to gravity, and forced drainage [46], 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 [47]. 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. The model developed by Verbist and Weaire idealizes the network of Plateau borders, through which the majority of liquid is assumed to drain, as a set of N identical pipes of cross section A, which is a function of position and time [48] where x and t are scaled position and time coordinates, respectively. In the case of forced drainage, the solution can be expressed as [48] where c is the velocity of the wave front [46].

To explain this method, let us consider the following function: with the boundary conditions of where A is a general differential operator f(r) is a known analytic function, B is a boundary operator, and Γ is the boundary of the domain Ω. The operator A can be generally divided into two operators, L and N, where L is a linear and N a nonlinear operator. Equation (2.1) can be, therefore, written as follows: Using the homotopy technique, we constructed a homotopy v(r, p) : Ω × [0,1] → R, which satisfies or where p ∈ [0,1] is called homotopy parameter and u₀ is an initial approximation for the solution of (2.1), which satisfies the boundary conditions. Obviously, from (2.4) or (2.5), we will have We can assume that the solution of (2.4) or (2.5) can be expressed as a series in p as follows: Setting p = 1 results in the approximate solution of  (2.1)

In this section, we obtain an analytical solution of (1.1). We first use the transformation to convert (1.1) to with initial condition To solve (3.2) with the initial condition (3.3), according to the homotopy perturbation method, we construct the following He’s polynomials corresponding to (2.5): Substituting v  =  v₀  +  pv₁  +  p²v₂  +  ⋯ into (3.2) and rearranging the resultant equation based on powers of p-terms, one has with the following conditions: With the effective initial approximation for v₀ from the condition (3.6), the solutions of (3.5) are obtained as follows: In the same manner, the rest of components were obtained using the Maple package. According to the HPM, we can conclude that Therefore, we will have By substituting (3.10) into (3.1), we can find the solution of (1.1).