Food and Public Health 2012, 2(1): 6-10 DOI: 10.5923/j.fph.20120201.02 Traveling Wave Solutions for Foam Drainage Equation by Modified F-Expansion Method M. T. Darvishi 1,* , Maliheh Najafi 2 , Mohammad Najafi 2 1 Department of Mathematics, Razi University, Kermanshah, 67149, Iran 2 Department of Mathematics, Anar Branch, Islamic Azad University, Anar, Iran Abstract In this paper, using modified -expansion method, we present some explicit formulas of exact traveling wave solutions for the foam drainage equation. A modified -expansion method is proposed by taking full advantages of -expansion method and Riccati equation in seeking exact solutions of non-linear partial differential equations. Keywords Foam Drainage Equation, Exact Solution, Modified -expansion Method 1. Introduction Most scientific problems and physical phenomena occur nonlinearly. Except in a limited number of these problems, finding the exact analytical solutions of such problems are rather difficult. Recently, many kinds of powerful methods have been proposed to find exact solutions of nonlinear partial differential equations, e.g., the homogeneous balance method[1], homotopy analysis method[2, 3], three-wave method[4, 5, 6], extended homoclinic test approach[7, 8, 9], the ( ) G G ′ − expansion method[10, 11] and the exp-function method[12, 13, 14]. In this paper, we consider the following foam drainage equation 2 ( ) = 0, 2 t x x ψ ψ ψ ψ ∂ ∂ ∂ + − ∂ ∂ ∂ (1) where ψ is the cross section of a channel formed where three films meet, usually indicated as Plateau border and x and t are scaled position and time coordinates, respectively. 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. Liquid foam is an example of soft matter (or complex fluid) with a very well-defined structure, first clearly described by Joseph plateau in the 19th century. Foam is important in a number of everyday activities, both natural and industrial. This is why foam has been of great interest for academic research. Because of the everyday occurrence of foams, they are very well known to scientists as well as to common people[15, 16]. Foams are common in foods and * Corresponding author: darvishimt@yahoo.com (M. T. Darvishi) Published online at http://journal.sapub.org/fph Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved personal care products such as lotions and creams and foams often occur during cleaning of clothes and scrubbing (see, e.g.[17]). They have important applications in food and chemical industries, mineral processing, fire fighting and structural material sciences (see, for example[18]). Everyday experiences put us in direct contact with foams. Shampooing hair, washing dishes, eating chocolate bars and chocolate mousse desserts are only a few examples. History connects foams with a number of famous scientists and foam continue to excite imaginations[19]. There are now many applications of polymeric foam[20] and more recently metallic foams which are foams made out of metals such as aluminum[21]. Some popular mentioned applications include the use of foams for reducing the impact of explosions and for cleaning up oil spills. Uniformity of the foam is important for the designer interested in these applications. Gravitational drainage of the liquid is one mechanism leading to non uniformity. Polymeric foams are used in cushions, packing and structural materials[20]. Glass, ceramic and metal foams[22] can also be made. Recent research in foams has centered on three topics which are often treated separately, but are, in fact, interdependent: drainage, coarsening and rheology. We concentrate on a quantitative description of the coupling of drainage. The flow of liquid through Plateau borders (the liquid-filled channels) and intersections of four channels between the bubbles, driven by gravity and capillarity, is called foam drainage. Foams’ drainage plays a very important role in foam stability. In fact, when a foam dries, its structure becomes fragile (see, for example[23]). In spite of many applications and numerous scientific investigations of properties and mechanics of foams, dynamics of foam drainage have only recently been examined in detail. Verbist studied the main features of both free drainage[24, 25]. In force drainage, a solitary wave of constant velocity is generated when liquid is added at a constant rate[26]. Hellal and Mehanna[27] used a