Similarity solutions for early-time constant boundary flux imbibition in foams and soils

The foam drainage equation and Richards equation are transport equations for foams and soils, respectively. Each reduces to a nonlinear diffusion equation in the early stage of infiltration during which time, flow is predominantly capillary driven, hence is effectively capillary imbibition. Indeed such equations arise quite generally during imbibition processes in porous media. New early-time solutions based on the van Genuchten relative diffusivity function for soils are found and compared with the same for drainage in foams. The moisture profiles which develop when delivering a known flux into these various porous materials are sought. Solutions are found using the principle of self-similarity. Singular profiles that terminate abruptly are obtained for soils, a contrast with solutions obtained for node-dominated foam drainage which are known from the literature (the governing equation being now linear is analogous to the linear equation for heat transfer). As time evolves, the moisture that develops at the top boundary when a known flux is delivered is greater in soils than in foams and is greater still in loamy soils than in sandstones. Similarities and differences between the various solutions for nonlinear and linear diffusion are highlighted. Graphic abstract

Comparing and Contrasting Travelling Wave Behaviour for Groundwater Flow and Foam Drainage

Liquid drainage within foam is generally described by the foam drainage equation which admits travelling wave solutions. Meanwhile, Richards’ equation has been used to describe liquid flow in unsaturated soil. Travelling wave solutions for Richards equation are also available using soil material property functions which have been developed by Van Genuchten. In order to compare and contrast these solutions, the travelling waves are expressed as dimensionless height, $$ {\hat{\xi }} $$ ξ ^ , versus moisture content, $$ \varTheta $$ Θ . For low moisture content, $$ {{\hat{\xi }}} $$ ξ ^ exhibits an abrupt change away from the dry state in Richards equation compared to a much more gradual change in foam drainage. When moisture content nears saturation, $$ {\hat{\xi }} $$ ξ ^ reaches large values (i.e. $$ {\hat{\xi }} \gg 1 $$ ξ ^ ≫ 1 ) for both Richards equation and foam drainage, implying a gradual approach of $$ \varTheta $$ Θ towards the saturated state. The $$ {\hat{\xi }} $$ ξ ^ values in Richards equation tend, however, to be larger than those in the foam drainage equation, implying an even more gradual approach towards saturation. The reasons for the difference between foam drainage and Richards equation solutions are traced back to soil material properties and in particular a soil specific parameter “m” which is determined from the soil-water retention curve.

Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

In this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.

A numerical approach for the nonlinear temporal conformable fractional foam drainage equation

In this paper, we examine a novel method called the residual power series method (RPSM) which is used in finding an analytic approximate solution to the nonlinear temporal foam drainage equation of fractional order. The solution is obtained in a uniformly convergent series form. Also, 3D graphs for the solution are provided for different values of [Formula: see text] which proves that the method is an effective and straightforward approach of providing excellent results for similar problems.

MULTIPLE EXACT SOLUTIONS OF THE GENERALIZED TIME FRACTIONAL FOAM DRAINAGE EQUATION

In this paper, we investigate the exact solutions of the generalized time fractional foam drainage equation. The Lie-group scaling transformation method and improved [Formula: see text]-expansion method are adopted here. The equation describes the evolution of the vertical density profile of a foam under gravity. New exact solutions and maple diagrams of the generalized time fractional foam drainage equation can help us better understand the physical phenomena.