Solution Of Advection Research Articles

Competing thermal and solutal advection decelerates droplet evaporation on heated surfaces

We report the complex evaporation kinetics of saline sessile droplets on surfaces with elevated temperatures. Our previous studies show that on non-heated substrates, saline sessile droplets evaporate faster compared to the water counterparts. In the present study, we discover that on heated surfaces, the saline droplets evaporate slower than the water counterpart, thereby posing a counter-intuitive phenomenon. The reduction in the evaporation rates is directly dependent on the salt concentration and the surface wettability. Natural convection around the droplet and thermal modulation of surface tension is found to be inadequate to explain the mechanisms. Flow visualizations using particle image velocimetry (PIV) reveal that the morphed advection within the saline droplets is a probable reason behind the arrested evaporation. Infrared thermography is employed to map the thermal state of the droplets. A thermo-solutal Marangoni based scaling analysis is put forward and the major governing non-dimensional numbers have been accounted for in the analysis. It is observed that the Marangoni flow and internal advection borne of thermal and solutal gradients are competitive, thereby leading to the overall decay of internal circulation velocity compared to the equivalent pure water case, which reduces the evaporation rates. The theoretically proposed advection velocities conform to the experimental results. This study sheds rich insight on a novel species transport behaviour in saline droplets.

Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering

Abstract The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations.

Humidity-controlled direct ink writing for micro-additive manufacturing with water-based inks

Direct-ink-writing type additive manufacturing with water-based inks such as hydrogels or polymer solutions is broadly utilized for bioprinting applications. However, the resolution of this implementation is limited due to the rapid ink drying experienced at small size scales leading to nozzle clogging and low fidelity prints. To address this issue, a humidity-controlled direct-ink-writing (HCDIW) approach is presented, where the microenvironment around the printing nozzle is controlled through introduction of aerosolized water. Using this approach, the nozzle microenvironment can be varied between undersaturated yet high relative humidity (ambient to 100% RH) to oversaturated conditions where condensing droplets are deposited on the already printed ink filaments. The influence of the aerosolized water and the resultant micro-environment on the printing process was studied using water dissolved sodium carboxymethyl cellulose (NaCMC) ink, specifically focusing on nozzle clogging, printed filament geometry, surface quality and layer by layer stacking. These studies showed that the ink drying induced nozzle clogging issues can be mitigated through increased relative humidity at the nozzle exit and is virtually eliminated for ~100 μm nozzles in oversaturated conditions. The decelerated ink drying under the effect of aerosolized water leads to reduced solute advection and skin formation. This in turn improves surface roughness while reducing adverse effects such as coffee-ring and void formation during additive manufacturing of micro-scale structures. Furthermore, layer-to-layer fusion and associated side-surface quality were improved specifically in undersaturated conditions. Reduction of deposited ink filament concentration in oversaturated conditions leads to excessive spreading of the deposited inks leading to geometric distortions. These results indicate that the humidity control approach can significantly improve the resolution of the direct-ink-writing approaches, enabling processing of water-based inks at micro-scales.

Droplet plume emission during plasmonic bubble growth in ternary liquids

Plasmonic bubbles are of great relevance in numerous applications, including catalytic reactions, micro/nanomanipulation of molecules or particles dispersed in liquids, and cancer therapeutics. So far, studies have been focused on bubble nucleation in pure liquids. Here we investigate plasmonic bubble nucleation in ternary liquids consisting of ethanol, water, and trans-anethole oil, which can show the so-called ouzo effect. We find that oil (trans-anethole) droplet plumes are produced around the growing plasmonic bubbles. The nucleation of the microdroplets and their organization in droplet plumes is due to the symmetry breaking of the ethanol concentration field during the selective evaporation of ethanol from the surrounding ternary liquids into the growing plasmonic bubbles. Numerical simulations show the existence of a critical Marangoni number Ma (the ratio between solutal advection rate and the diffusion rate), above which the symmetry breaking of the ethanol concentration field occurs, leading to the emission of the droplet plumes. The numerical results agree with the experimental observation that more plumes are emitted with increasing ethanol-water relative weight ratios and hence Ma. Our findings on the droplet plume formation reveal the rich phenomena of plasmonic bubble nucleation in multicomponent liquids and help to pave the way to achieve enhanced mixing in multicomponent liquids in chemical, pharmaceutical, and cosmetic industries.

The Mass Transfer Index (MTI): A semi-empirical approach for quantifying transport of solutes in variably saturated porous media

The processes impacting solute transport through unsaturated porous media have been receiving renewed attention due to their relevance to the transport of emerging contaminants. A set of well-monitored and highly controlled experiments in sand columns were conducted to determine the effect of partial saturation on conservative solute breakthrough in porous media. The results suggest traditional transport parameter estimation methods inadequately account for the pore-scale processes of mass transfer to the immobile zones and the effects of partial saturation on advective transport, even for conservative tracers. Accurate estimation of these basic transport parameters is critical to evaluate the multi-phase partitioning of nonconservative solutes, as any errors in these parameters would bias the estimates of multi-phase partitioning parameters. Herein, we introduced the Mass Transfer Index (MTI), a semi-empirical approach for quantifying the impact of non-Fickian elements of pore-scale unsaturated solute transport (i.e. immobile water, tortuous flow paths, and non-uniform solute distribution), which become increasingly important as the wetting fluid saturation decreases. Importantly, this MTI was determined independently of chemically driven phase partitioning and is supported by experimental data. Based on this conceptualization, the 1-D equilibrium advection dispersion equation was modified to incorporate the MTI as a lumped parameter which quantifies resistance to (MTI > 1) or promotion of (MTI < 1) of advective solute flux. Analytical solutions to the modified advection-dispersion-reaction equation for pulse and step inputs were developed. Conservative tracer experiments were conducted in variably saturated sand columns to validate both the MTI conceptualization and the inversion method used to estimate the MTI. These experiments involved the use of X-ray absorption spectroscopy integrated with sensor-based measurements of soil moisture, temperature, and electrical conductivity for tracer breakthrough. The mathematical model developed herein adapts traditional macroscopic models of solute transport to account for the non-Fickian pore-scale transport behaviors observed in unsaturated porous media with significant advective flux.

The Local Singular Boundary Method for Solution of Two-Dimensional Advection–Diffusion Equation

This work focuses on the derivation of the local variant of the singular boundary method (SBM) for solving the advection-diffusion equation of tracer transport. Localization is based on the combination of SBM and finite collocation. Unlike the global variant, local SBM leads to a sparse matrix of the resulting system of equations, making it much more efficient to solve large-scale tasks. It also allows solving velocity vector variable tasks, which is a problem with global SBM. This paper compares the results on several examples for the steady and unsteady variant of the advection-diffusion equation and also examines the dependence of the accuracy of the solution on the density of the nodal grid and the size of the subdomain.

Macrotransport theory for diffusiophoretic colloids and chemotactic microorganisms

Abstract

A Review of Possible Advective Disk Structures around a Black Hole with Two Types of Gas Inflows

Abstract We studied general advective accretion solutions around a Kerr black hole (BH) by investigating two types of inflow gases at the outer accretion boundary (AB). We classified these two types of gases as cold-mode and hot-mode inflow gas at the outer AB on the basis of their temperatures and solutions. We found that the hot-mode gas is more efficient for angular momentum transport around the outer AB than the cold-mode gas. The hot-mode gas can give multiple global (popular as a shock solution) or single sonic point solutions, and the cold-mode gas can give a smooth global solution (popularly known as advection-dominated accretion flow) or two sonic point solutions. These solutions are also presented on a plane in energy and angular momentum (B ob−L 0) parameter space. For the first time, we explored theoretically the relation between the nature of accretion solutions and the nature of the initial accreting gas at the AB with a detailed computational and possible physical analysis. We also found that the surface density of the flow is highly affected by changes in the temperature at the AB, which can alter the radiative emissivities of the flow. The flow variables of various advective solutions are also compared. On the basis of those results, we plotted some inner disk structures around the BHs. By doing so, we conjecture on the persistent/transient nature of spectral states, soft excess, and timescales of variabilities around the BH X-ray binaries and active galactic nuclei.

Shaping the gradients driving phoretic micro-swimmers: influence of swimming speed, budget of carbonic acid and environment

pH gradient-driven modular micro-swimmers are investigated as a model for a large variety of quasi-two-dimensional chemi-phoretic self-propelled entities. Using three-channel micro-photometry, we obtain a precise large field mapping of pH at a spatial resolution of a few microns and a pH resolution of sim 0.02~hbox {pH} units for swimmers of different velocities propelling on two differently charged substrates. We model our results in terms of solutions of the three-dimensional advection–diffusion equation for a 1:1 electrolyte, i.e. carbonic acid, which is produced by ion exchange and consumed by equilibration with dissolved hbox {CO}_{2}. We demonstrate the dependence of gradient shape and steepness on swimmer speed, diffusivity of chemicals, as well as the fuel budget. Moreover, we experimentally observe a subtle, but significant feedback of the swimmer’s immediate environment in terms of a substrate charge-mediated solvent convection. We discuss our findings in view of different recent results from other micro-fluidic or active matter investigations. We anticipate that they are relevant for quantitative modelling and targeted applications of diffusio-phoretic flows in general and artificial micro-swimmers in particular.Graphic

Numerical treatment of the space fractional advection–dispersion model arising in groundwater hydrology

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature.

Analysis of Concentration of $$ {\text{Ca}}^{2 + } $$ Arising in Astrocytes Cell

In the present article we analyzed the concentration of cytosolic calcium ion $$ \left( {{\text{Ca}}^{2 + } } \right) $$ via fractional calculus. Here considered a mathematical model in which fractional advection–diffusion equation is uprise. The analytic solution of fractional advection–diffusion equation has been obtained by using Elzaki transform for analysis of concentration of $$ {\text{Ca}}^{2 + } $$ which is obtained in diffusion process in astrocytes cell. Numerical simulations are carried out for illustrating the obtained result.

An unconditionally stable algorithm for multiterm time fractional advection–diffusion equation with variable coefficients and convergence analysis

AbstractThis paper focuses on the numerical solution of the variable coefficient multiterm time fractional advection–diffusion equation via exponential B‐splines. We discretize the temporal part by using the Crank–Nicolson method and spatial part by the exponential B‐splines. The unconditional stability is obtained by the Von‐Neumann method. The convergence rates are also studied. Numerical simulations confirm the theoretically expected accuracy in both time and space directions. A comparative analysis with the other methods shows the superiority of the proposed algorithm.

Existence and uniqueness results and analytical solution of the multi-dimensional Riesz space distributed-order advection–diffusion equation via two-step Adomian decomposition method

In this article, we introduced for the first time the two-step Adomian decomposition method (TSADM) for solving the multi-dimensional Riesz space distributed-order advection–diffusion (RSDOAD) equation. The TSADM was successfully applied to obtain the analytical solution of the multi-dimensional (RSDOAD) equation. The analytical solution has been obtained without approximation/discretization of the Riesz fractional operator. Furthermore, new results for the existence are obtained with the help of some fixed point theorems, while the uniqueness of the solution was investigated employing the Banach contraction principle. Finally, we included a generalized example to demonstrate the validity and application of the proposed method. The obtained results conclude that the proposed method is powerful and efficient for the considered problem compared to the other existing methods.

The Impact of Chebyshev Collocation Method on Solutions of fractional Advection–Diffusion Equation

The present paper is primarily aimed at obtaining the numerical solution of space fractional advection–diffusion equation including two fractional space derivatives of order. At the first stage, a difference approach with the second-order accuracy is formulated to obtain a semi-discrete scheme. Unconditional stability and convergence analysis has been analyzed. At the second stage, the spatial discretization is accomplished by means of the second kind shifted Chebyshev polynomials. Two examples are investigated, and numerical results are reported to confirm the theoretical results.

New Semi-Analytical Solutions for Advection–Dispersion Equations in Multilayer Porous Media

A new semi-analytical solution to the advection-dispersion-reaction equation for modelling solute transport in layered porous media is derived using the Laplace transform. Our solution approach involves introducing unknown functions representing the dispersive flux at the interfaces between adjacent layers, allowing the multilayer problem to be solved separately on each layer in the Laplace domain before being numerically inverted back to the time domain. The derived solution is applicable to the most general form of linear advection-dispersion-reaction equation, a finite medium comprising an arbitrary number of layers, continuity of concentration and dispersive flux at the interfaces between adjacent layers and transient boundary conditions of arbitrary type at the inlet and outlet. The derived semi-analytical solution extends and addresses deficiencies of existing analytical solutions in a layered medium, which consider analogous processes such as diffusion or reaction-diffusion only and/or require the solution of complicated nonlinear transcendental equations to evaluate the solution expressions. Code implementing our semi-analytical solution is supplied and applied to a selection of test cases, with the reported results in excellent agreement with a standard numerical solution and other analytical results available in the literature.

Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

A pore-scale thermo–hydro-mechanical model for particulate systems

A pore scale numerical method dedicated to the simulation of heat transfer and associated thermo–hydro-mechanical couplings in granular media is described. The proposed thermo–hydro-mechanical approach builds on an existing hydro-mechanical model that employs the discrete element method for simulating the mechanical behavior of dense sphere packings and combines it with the finite volume method for simulating pore space fluid flow and the resulting hydro-mechanical coupling. Within the hydro-mechanical framework, the pore space is discretized as a tetrahedral network whose geometry is defined by the triangulation of discrete element method (DEM) particle centers. It is this discretization of DEM particle contacts and tetrahedral pore spaces that enables the efficient conductive and advective heat transfer models proposed herein. In particular, conductive heat transfer is modeled explicitly between and within solid and fluid phases: across DEM particle contacts, between adjacent tetrahedral pores, and between pores and incident particles. Meanwhile, advective heat transfer is added to the existing implicit fluid flow scheme by estimating mass–energy–flux from pressure induced fluid fluxes. In addition to the heat transfer model, a thermo-mechanical coupling is implemented by considering volume changes based on the thermal expansion of particles and fluid. The conduction and advection models are verified by presenting comparisons to an analytical solution for conduction and a fully resolved numerical solution for conduction and advection. Finally, the relevance of the fully coupled thermo–hydro-mechanical model is illustrated by simulating an experiment where a saturated porous rock sample is subjected to a cyclic temperature loading.

Derivation and numerical validation of the fundamental solutions for constant and variable-order structural derivative advection–dispersion models

Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental solutions for constant and variable-order structural derivative advection–dispersion models are achieved via scaling transformation and the generalized non-Euclidean Hausdorff fractal distance. Comparative numerical investigations of the structural derivative models have been conducted to reveal the influences of various kernels via the meshless method of fundamental solutions. Numerical results verify the validity of the derived fundamental solutions and the rationality of the employed numerical method for structural derivative advection–dispersion models.

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

AbstractIn this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection–diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter (ϵ) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black–Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L∞, L2, Lrms, and Lrel error norms as well as number of nodes N over space domain and time‐step δt. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill‐conditioning problem greatly, a major issue in the Kansa method.

Plugging nanoscale imperfections in the polyamide active layer of thin-film composite reverse osmosis membrane to inhibit advective solute transport

The objective of this study is to inhibit advective solute passage through reverse osmosis (RO) membranes by filtering a small volume of polyvinyl alcohol (PVA) aqueous solution for 1 min at 0.1 MPa to selectively plug nanoscale imperfections. The PVA plugging the nanoscale imperfections was stabilized by cross-linking with glutaraldehyde. Experimental data showed that PVA treatment with PVA concentrations of up to 20 ppm did not decrease the water permeability, but it improved the solute removal efficiencies significantly. Specifically, at an applied pressure of 2.0 MPa, the NaCl and Rhodamine-WT rejection improved from 97.4% to 98.8% (a 67% decrease in NaCl flux) and 99.74% to 99.96% (a 85% decrease in Rhodamine-WT flux), respectively. Modeling analysis using a solution-diffusion model coupled with unhindered advection through nanoscale imperfections demonstrated that these improvements in solute rejection were due to selective plugging of nanoscale imperfections by PVA without building up an additional resistive layer on the polyamide active layer. Experimental data also showed that the cross-linked PVA plugging the nanoscale imperfections was stable under repeated exposure to citric acid, sodium hydroxide, and ethylenediaminetetraacetic acid.