Foam Drainage Equation Research Articles

Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region below an interface to the bottom in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle. Moreover, the existence of its steady solution and its stability are shown.

Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.

New exact solutions of the conformable space-time two-mode foam drainage equation by two effective methods

In this article, the two-mode foam drainage equation in terms of time and space conformable sense has been investigated. Two effective methods, the generalized exponential rational function method (GERFM) and the improved version of the Bernoulli sub-equation function method (IBSEFM), are used to get new solutions of underlying equation. The fractional travelling wave transformation is applied to convert nonlinear partial differential equations to nonlinear ordinary differential equations. Proposed methods successfully extract trigonometric, hyperbolic and exponential solutions. Some of the obtained solutions are visualized to understand the effect of fractional orders of time and space derivatives on the wave profile and the dynamic behavior of the solutions.

Solution of the foam-drainage equation with cubic B-spline hybrid approach

This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L 1 formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L 2, and L ∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulations that the proposed method gives quite accurate solutions.

Modified optimal auxiliary functions method for approximate-analytical solutions in fractional order nonlinear Foam Drainage equations

The study aims to obtain new approximate analytical solutions for the fractional order nonlinear Foam Drainage equation through a novel modification of the well-known Optimal Auxiliary Functions Method (OAFM). This method systematically constructs auxiliary functions, and by optimizing their parameters, it provides approximate solutions closely mirroring the original behavior of the problem. Numerical simulations, presented in tabular form, demonstrate the effectiveness of the proposed approach. Graphical representations of the solutions are utilized to further validate the applicability and accuracy of the suggested technique. Additionally, the results affirm that the modified OAFM stands out as one of the most effective tools for addressing nonlinear problems in the field of science and technology, thereby contributing valuable addition to the field.

Foam drainage modeling of vertical foam column and validation with experimental results

The understanding of the mechanisms behind foam generation and the structure of foam itself form the basis of foam-related experiments for its application in Enhanced Oil Recovery and overcoming gas injection limitations. Novel insights in this paper towards the theory of foam generation can help explain experimental results and lead to improved formulas of the applied substances and concentrations. This study aims to investigate the mechanisms behind foam generation and the structure of foam by specific laboratory experiments and theoretical analyses. The liquid drainage through interconnected Plateau borders was found to be the most critical foam decay mechanism for this particular research. The justification of the foam drainage equation was demonstrated by comparing the numerical solution with the outcome of a few bulk experiments. The discrepancies were described according to the limitations of both the theory and the experimental settings. Foam modelling gives more profound knowledge in more detail of the different stages in foam drainage than experimental data can deliver, which is because of the lack of continuous measurement of foam conductivity for the foam bulk test. Therefore, a comprehension of foam modelling investigation and comparison is required to gain a deeper understanding of foam behaviour.

Fractional-order view analysis of Fisher’s and foam drainage equations within Aboodh transform

PurposeThe purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.Design/methodology/approachIn this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.FindingsWe develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.Originality/valueOwing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

The nonlinear wave dynamics of fractional foam drainage and Boussinesq equations with Atangana’s beta derivative through analytical solutions

The space–time fractional foam drainage and the Boussinesq equations are valuable mathematical models to describe different complex physical phenomena, including the drainage of soap films in bubbles and foams, the behavior of surface water waves in oceans and rivers, the propagation of long waves, etc. In this research, we explore several unrivaled and some existing analytical soliton solutions to the above-stated equations using an effective expansion scheme, known as the G′/(G′+G+A)-expansion scheme within the framework of Atangana’s beta derivative. A fractional composite transformation is used to interpret the considered equations into nonlinear equations of a single independent variable. The wave dynamics of the outcomes are illustrated through three- and two-dimensional graphical representations and contour plots. The solutions include kink, flat kink-shaped, the parabolic-shaped, the bell-shaped, and the compacton solitons. The study reveals that the fractional-order derivative in the projected equations influences the wave behavior, providing deeper insights into wave composition when the fractional derivative approaches larger. The comparison table demonstrates the effectiveness of the method across various fractional orders, with results approaching accuracy as the fractional-order approaches integer values. This study emphasizes the computational robustness and adaptability of the proposed method for investigating various phenomena across a wide range of physical science and engineering disciplines.

Foam drainage equation in fractal dimensions: breaking and instabilities

Foam drainage equation in fractal dimensions: breaking and instabilities

Analytical Methods for Fractional Differential Equations: Time-Fractional Foam Drainage and Fisher’s Equations

In this research, we employ a dual-approach that combines the Laplace residual power series method and the novel iteration method in conjunction with the Caputo operator. Our primary objective is to address the solution of two distinct, yet intricate partial differential equations: the Foam Drainage Equation and the nonlinear time-fractional Fisher’s equation. These equations, essential for modeling intricate processes, present analytical challenges due to their fractional derivatives and nonlinear characteristics. By amalgamating these distinctive methodologies, we derive precise and efficient solutions substantiated by comprehensive figures and tables showcasing the accuracy and reliability of our approach. Our study not only elucidates solutions to these equations, but also underscores the effectiveness of the Laplace Residual Power Series Method and the New Iteration Method as potent tools for grappling with intricate mathematical and physical models, thereby making significant contributions to advancements in diverse scientific domains.

Fractional-order modeling: Analysis of foam drainage and Fisher's equations

Abstract In this study, we use a dual technique that combines the Laplace residual power series method (LRPSM) and the new iteration method, both of which are combined with the Caputo operator. Our primary goal is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional Fisher’s equation. These equations, which are crucial in modeling complex processes, confront analytical complications, owing to their fractional derivatives and nonlinear behavior. We develop exact and efficient solutions by merging these unique methodologies, which are supported by thorough figures and tables that demonstrate the precision and trustworthiness of our methodology. We not only shed light on the solutions to these equations, but also demonstrate the prowess of the LRPSM and the new iteration method as powerful tools for grappling with complex mathematical and physical models, significantly contributing to advancements in various scientific domains.

Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Dynamics Investigation and Solitons Formation for (2+1) -Dimensional Zoomeron Equation and Foam Drainage Equation

The prime goal of this study is to investigate novel solutions of two well-known nonlinear models namely the (2+1)-dimensional Zoomeron equation and the foam drainage equation by utilizing a powerful technique; the extended left(frac{G'}{G^{2}}right)-expansion method. Using this methodology, the hyperbolic function solutions, the trigonometric function solutions and the rational function solutions are constructed. Abundant soliton solutions are retrieved from the obtained results. The dynamical structures of the solutions are illustrated graphically through 3-dimensional graphs and the corresponding contour plots. The reported results depict the effectiveness and capability of the extended left(frac{G'}{G^{2}}right)-expansion method for handling different nonlinear partial differential equations.

Propagations of symmetric bidirectional nonlinear waves in two-mode foam drainage model

In this paper, a new two-mode version of the foam drainage equation is presented. The new model describes the motion of instantaneous symmetric double-waves and their interaction depends on two involved parameters known as nonlinearity and phase-velocity. Two recent methods are used to obtain solutions with interesting variety of physical structures such as kink-type and periodic shape. In addition, the influence of both nonlinearity and phase-velocity parameters are investigated by using graphical analysis. Finally, all the obtained solutions to the TMFD are confirmed by direct substituting and verifying the correctness of the governing equation.

Solutions of fractional foam drainage and Zakharov-Kuznetsov equations using a new algorithm

The Daftardar-Gejji Jafari Method (DGJM) has been used extensively in the recent one and a half decades to solve various non-linear equations such as algebraic equations, integral equations, partial differential equations, ordinary and fractional differential equations and so on. In this paper, we present a new time-efficient algorithm for DGJM for solving non-linear fractional foam drainage and Zakharov-Kuznetsov equations. We compare the DGJM solutions with those obtained by the Adomian decomposition method and the homotopy perturbation method. The obtained results reveal that the new algorithm of DGJM is more effective and time-efficient for predicting the solutions to such problems that too without calculating tedious calculations such as finding Adomian's polynomials and constructing homotopy function.

Novel approximate solution for fractional differential equations by the optimal variational iteration method

In this work, the optimal variational iteration method (OVIM) is used to solve partial and ordinary fractional differential equations. The standard variational iteration method (VIM) is reassessed by introducing a parameter that accelerates convergence. The value of the convergence acceleration coefficient is determined by calculating the residual of the parameter within the L2 norm. The numerical results proved that the proposed method gives much better and more accurate results than the standard VIM method, as the effect of adding the parameter to the VIM method and its calculation method was clear in improving the method and increasing the convergence of the solution, where we solved several examples of ordinary and partial fractional differential equations. In addition, the fractional Kawahara and foam drainage equations have also been solved. All results were obtained using the Mathematica®12 program. The results were also compared with the optimal Adomian decomposition method (ADM), and the proposed method showed higher efficiency and better accuracy.

A Numerical Strategy for the Approximate Solution of the Nonlinear Time-Fractional Foam Drainage Equation

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions.

Computational study of non-linear integer and time-fractional foam drainage equations using radial basis functions

In this article, radial basis function (RBFs) based method is suggested to study non-linear Foam Drainage and Fractional Foam Drainage Equations. For this purpose, θ-weighted scheme is implemented along with the quasi-linearization technique on the spatial part to linearize the non-linear terms. Finite difference along with Crank–Nicholson is used for time discretization. Numerical results are computed to validate the technique’s accuracy while solving the fractional and classical foam drainage equations. A comparative analysis with other methods proved it to be one of the efficient and practical approaches. An algorithm is applied to get the optimal value of the involved shape parameter “c” to get better accuracy. Stability analysis is done to check how far the model remains stable. The efficiency and accuracy of the proposed scheme are further examined via mass and energy conservation laws.

Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism

Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism

Dynamical behaviour of the foam drainage equation

In this study, we examine the well-known foam drainage equation in the form of the nonlinear partial differential equation using a new extended direct algebraic method. This remarkable powerful method transforms the equation under study into an ordinary differential equation using a wave transformation which provides an exact form to the problem considered in the paper. Graphical representation for the acquired solution is presented through 3-D plots which variable parameters to show the efficiency and simplicity of the method used. From these plots, the methods prove to be a reliable and effective approach for solving nonlinear similar problems and can be accounted for in the near future for possible application to similar models.