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

2021 ◽  
Vol 44 (9) ◽  
Author(s):  
Yaw Akyampon Boakye-Ansah ◽  
Paul Grassia
Keyword(s):  
Early Stage ◽  
Early Time ◽  
Similarity Solutions ◽  
Richards Equation ◽  
Linear Diffusion ◽  
Time Flow ◽  
Foam Drainage Equation ◽  
Boundary Flux ◽  
Moisture Profiles ◽  
Foam Drainage

Abstract 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

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

2021 ◽  
Author(s):  
Y. A. Boakye-Ansah ◽  
P. Grassia
Keyword(s):  
Moisture Content ◽  
Travelling Wave ◽  
Richards Equation ◽  
Gradual Change ◽  
Travelling Wave Solutions ◽  
Soil Material ◽  
Wave Solutions ◽  
Foam Drainage Equation ◽  
Gradual Approach ◽  
Foam Drainage

AbstractLiquid 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.

Numerical solutions of the fractal foam drainage equation

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Siddra Habib ◽  
Asad Islam ◽  
Amreen Batool ◽  
Muhammad Umer Sohail ◽  
Muhammad Nadeem
Keyword(s):  
Numerical Solutions ◽  
Foam Drainage Equation ◽  
Fractal Foam ◽  
Foam Drainage

scholarly journals Morphological evolution of microscopic dewetting droplets with slip

2017 ◽  
Vol 828 ◽  
pp. 271-288 ◽  
Author(s):  
Tak Shing Chan ◽  
Joshua D. McGraw ◽  
Thomas Salez ◽  
Ralf Seemann ◽  
Martin Brinkmann
Keyword(s):  
Slip Length ◽  
Morphological Evolution ◽  
Early Time ◽  
Slip Boundary Condition ◽  
Similarity Solutions ◽  
Contact Angles ◽  
Equilibrium Contact Angle ◽  
Navier Slip ◽  
Slip Boundary ◽  
Quasistatic Regime

We investigate the dewetting of a droplet on a smooth horizontal solid surface for different slip lengths and equilibrium contact angles. Specifically, we solve for the axisymmetric Stokes flow using the boundary element method with (i) the Navier-slip boundary condition at the solid/liquid boundary and (ii) a time-independent equilibrium contact angle at the contact line. When decreasing the rescaled slip length $\tilde{b}$ with respect to the initial central height of the droplet, the typical non-sphericity of a droplet first increases, reaches a maximum at a characteristic rescaled slip length $\tilde{b}_{m}\approx O(0.1{-}1)$ and then decreases. Regarding different equilibrium contact angles, two universal rescalings are proposed to describe the behaviour of the non-sphericity for rescaled slip lengths larger or smaller than $\tilde{b}_{m}$. Around $\tilde{b}_{m}$, the early time evolution of the profiles at the rim can be described by similarity solutions. The results are explained in terms of the structure of the flow field governed by different dissipation channels: elongational flows for $\tilde{b}\gg \tilde{b}_{m}$, friction at the substrate for $\tilde{b}\approx \tilde{b}_{m}$ and shear flows for $\tilde{b}\ll \tilde{b}_{m}$. Following the changes between these dominant dissipation mechanisms, our study indicates a crossover to the quasistatic regime when $\tilde{b}$ is many orders of magnitude smaller than $\tilde{b}_{m}$.

Exact solutions for nonlinear foam drainage equation

2016 ◽  
Vol 91 (2) ◽  
pp. 209-218 ◽  
Author(s):  
E. M. E. Zayed ◽  
Abdul-Ghani Al-Nowehy
Keyword(s):  
Exact Solutions ◽  
Foam Drainage Equation ◽  
Foam Drainage

Asymptotic behaviour of the thin film equation in bounded domains

2001 ◽  
Vol 12 (2) ◽  
pp. 135-157 ◽  
Author(s):  
M. BOWEN ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Early Time ◽  
Similarity Solutions ◽  
Bounded Domains ◽  
Extinction Time ◽  
Degenerate Diffusion ◽  
Thin Film Equation ◽  
Order Degenerate ◽  
Mass Increase

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

scholarly journals Theoretical analysis of the performance of a foam fractionation column

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130625 ◽  
Author(s):  
S. T. Tobin ◽  
D. Weaire ◽  
S. Hutzler
Keyword(s):  
Theoretical Analysis ◽  
Surfactant Solution ◽  
Model System ◽  
Foam Fractionation ◽  
Fractionation Column ◽  
Foam Drainage Equation ◽  
Limiting Case ◽  
Alternative Set ◽  
Theory And Experiment ◽  
Foam Drainage

A model system for theory and experiment which is relevant to foam fractionation consists of a column of foam moving through an inverted U-tube between two pools of surfactant solution. The foam drainage equation is used for a detailed theoretical analysis of this process. In a previous paper, we focused on the case where the lengths of the two legs are large. In this work, we examine the approach to the limiting case (i.e. the effects of finite leg lengths) and how it affects the performance of the fractionation column. We also briefly discuss some alternative set-ups that are of interest in industry and experiment, with numerical and analytical results to support them. Our analysis is shown to be generally applicable to a range of fractionation columns.

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