Electrophoresis of bubbles

2014 ◽  
Vol 753 ◽  
pp. 49-79 ◽  
Author(s):  
Ory Schnitzer ◽  
Itzchak Frankel ◽  
Ehud Yariv
Keyword(s):  
Zeta Potential ◽  
Numerical Solutions ◽  
Nonlinear Flow ◽  
Coarse Grained ◽  
Bubble Velocity ◽  
Matched Asymptotic Expansion ◽  
Driving Mechanism ◽  
Analytic Approximation ◽  
Field Magnitude ◽  
Debye Layer

AbstractSmoluchowski’s celebrated electrophoresis formula is inapplicable to field-driven motion of drops and bubbles with mobile interfaces. We here analyse bubble electrophoresis in the thin-double-layer limit. To this end, we employ a systematic asymptotic procedure starting from the standard electrokinetic equations and a simple physicochemical interface model. This furnishes a coarse-grained macroscale description where the Debye-layer physics is embodied in effective boundary conditions. These conditions, in turn, represent a non-conventional driving mechanism for electrokinetic flows, where bulk concentration polarization, engendered by the interaction of the electric field and the Debye layer, results in a Marangoni-like shear stress. Remarkably, the electro-osmotic velocity jump at the macroscale level does not affect the electrophoretic velocity. Regular approximations are obtained in the respective cases of small zeta potentials, small ions, and weak applied fields. The nonlinear small-zeta-potential approximation rationalizes the paradoxical zero mobility predicted by the linearized scheme of Booth (J. Chem. Phys., vol. 19, 1951, pp. 1331–1336). For large (millimetre-size) bubbles the pertinent limit is actually that of strong fields. We have carried out a matched-asymptotic-expansion analysis of this singular limit, where salt polarization is confined to a narrow diffusive layer. This analysis establishes that the bubble velocity scales as the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2/3$-power of the applied-field magnitude and yields its explicit functional dependence upon a specific combination of the zeta potential and the ionic drag coefficient. The latter is provided to within an $O(1)$ numerical pre-factor which, in turn, is calculated via the solution of a universal (parameter-free) nonlinear flow problem. It is demonstrated that, with increasing field magnitude, all numerical solutions of the macroscale model indeed collapse on the analytic approximation thus obtained. Existing measurements of clean-bubble electrophoresis agree neither with present theory nor with previous models; we discuss this ongoing discrepancy.

scholarly journals The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170050 ◽  
Author(s):  
Christopher C. Green ◽  
Christopher J. Lustri ◽  
Scott W. McCue
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Bubble Velocity ◽  
Number Of Solutions ◽  
Branch Number ◽  
Single Solution ◽  
Countably Infinite ◽  
Prime Function ◽  
Hele Shaw Cell ◽  
Effect Of Surface

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

scholarly journals An individual-based mechanical model of cell movement in heterogeneous tissues and its coarse-grained approximation

2018 ◽  
Author(s):  
R. J. Murphy ◽  
P. R. Buenzli ◽  
R. E. Baker ◽  
M. J. Simpson
Keyword(s):  
Mechanical Model ◽  
Numerical Solutions ◽  
Cell Movement ◽  
Biological Tissues ◽  
Tissue Level ◽  
Cellular Level ◽  
Coarse Grained ◽  
Mechanical Heterogeneity ◽  
Single Scale ◽  
The Individual

AbstractMechanical heterogeneity in biological tissues, in particular stiffness, can be used to distinguish between healthy and diseased states. However, it is often difficult to explore relationships between cellular-level properties and tissue-level outcomes when biological experiments are performed at a single scale only. To overcome this difficulty we develop a multi-scale mathematical model which provides a clear framework to explore these connections across biological scales. Starting with an individual-based mechanical model of cell movement, we subsequently derive a novel coarse-grained system of partial differential equations governing the evolution of the cell density due to heterogeneous cellular properties. We demonstrate that solutions of the individual-based model converge to numerical solutions of the coarse-grained model, for both slowly-varying-in-space and rapidly-varying-in-space cellular properties. Applications of the model are discussed, including determining relative cellular-level properties and an interpretation of data from a breast cancer detection experiment.

scholarly journals Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hua Yang ◽  
Feng Jiang
Keyword(s):  
Diffusion Coefficients ◽  
Numerical Solutions ◽  
Delay Equations ◽  
Markovian Switching ◽  
Analytic Approximation ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Poisson Jump ◽  
And Diffusion

We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs). Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.

scholarly journals A Focus on Two Electrokinetics Issues

Micromachines ◽  
2020 ◽  
Vol 11 (12) ◽  
pp. 1028
Author(s):  
Cheng Dai ◽  
Ping Sheng
Keyword(s):  
Boltzmann Equation ◽  
Zeta Potential ◽  
Flow Field ◽  
Drag Coefficient ◽  
Surface Charge ◽  
Physical Picture ◽  
Debye Layer ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Mobile Ions

This review article intends to communicate the new understanding and viewpoints on two fundamental electrokinetics topics that have only become available recently. The first is on the holistic approach to the Poisson–Boltzmann equation that can account for the effects arising from the interaction between the mobile ions in the Debye layer and the surface charge. The second is on the physical picture of the inner electro-hydrodynamic flow field of an electrophoretic particle and its drag coefficient. For the first issue, the traditional Poisson–Boltzmann equation focuses only on the mobile ions in the Debye layer; effects such as charge regulation and the isoelectronic point arising from the interaction between the mobile ions in the Debye layer and the surface charge are left to supplemental measures. However, a holistic treatment is entirely possible in which the whole electrical double layer—the Debye layer and the surface charge—is treated consistently from the beginning. While the derived form of the Poisson–Boltzmann equation remains unchanged, the zeta potential boundary condition becomes a calculated quantity that can reflect the various effects due to the interaction between the surface charges and the mobile ions in the liquid. The second issue, regarding the drag coefficient of a spherical electrophoretic particle, has existed ever since the breakthrough by Smoluchowski a century ago that linked the zeta potential of the particle to its mobility. Due to the highly nonlinear mathematics involved in the electro-hydrodynamics inside the Debye layer, there has been a lack of an exact solution for the electrophoretic flow field. Recent numerical simulation results show that the flow field comprises an inner region and an outer region, separated by a rather sharp interface. As the inner flow field is carried along by the particle, the measured drag is that at the inner/outer interface rather than at the solid/liquid interface. This identification and its associated physical picture of the inner flow field resolves a long-standing puzzle regarding the electrophoretic drag coefficient.

scholarly journals Finite Element Analysis of Biot’s Consolidation with a Coupled Nonlinear Flow Model

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
Yue-bao Deng ◽  
Gan-bin Liu ◽  
Rong-yue Zheng ◽  
Kang-he Xie
Keyword(s):  
Finite Element ◽  
Flow Model ◽  
Numerical Solutions ◽  
Continuity Condition ◽  
Nonlinear Flow ◽  
Weighted Residual Method ◽  
Element Analysis ◽  
Vertical Drains ◽  
Soil Skeleton ◽  
Force Equilibrium

A nonlinear flow relationship, which assumes that the fluid flow in the soil skeleton obeys the Hansbo non-Darcian flow and that the coefficient of permeability changes with void ratio, was incorporated into Biot’s general consolidation theory for a consolidation simulation of normally consolidated soft ground with or without vertical drains. The governing equations with the coupled nonlinear flow model were presented first for the force equilibrium condition and then for the continuity condition. Based on the weighted residual method, the finite element (FE) formulations were then derived, and an existing FE program was modified accordingly to take the nonlinear flow model into consideration. Comparative analyses using established theoretical solutions and numerical solutions were completed, and the results were satisfactory. On this basis, we investigated the effect of the coupled nonlinear flow on consolidation development.

scholarly journals Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci

2018 ◽  
Vol 839 ◽  
Author(s):  
Toby L. Kirk
Keyword(s):  
Asymptotic Expansion ◽  
Shear Flow ◽  
Numerical Solutions ◽  
Large Range ◽  
Slip Length ◽  
Matched Asymptotic Expansion ◽  
Asymptotic Formulae ◽  
Relative Errors

This paper presents new asymptotic formulae for flow in a channel with one or both walls patterned with a longitudinal array of ridges and arbitrarily protruding menisci. Derived from a matched asymptotic expansion, they extend results by Crowdy (J. Fluid Mech., vol. 791, 2016, R7) for shear flow, and thus make no restriction on the protrusion into or out of the liquid. The slip length formula is compared against full numerical solutions and, despite the assumption of small ridge period in its derivation, is found to have a very large range of validity; relative errors are small even for periods large enough for the protruding menisci to degrade the flow and touch the opposing wall.

scholarly journals Evaluation of the Non-Darcy Effect of Water Inrush from Karst Collapse Columns by Means of a Nonlinear Flow Model

Water ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 1234 ◽  
Author(s):  
Yi Xue ◽  
Teng Teng ◽  
Lin Zhu ◽  
Mingming He ◽  
Jie Ren ◽  
...  
Keyword(s):  
Construction Projects ◽  
Numerical Solutions ◽  
Water Inrush ◽  
Nonlinear Flow ◽  
Karst Collapse ◽  
Viscous Resistance ◽  
Mining Operations ◽  
Flow Process ◽  
Mining Work ◽  
Forchheimer Flow

Karst collapse columns (KCCs) are naturally formed geological structures that are widely observed in North China. Given their influence on normal mining operations and the progress of mining work, collapse columns pose a hidden danger in coal mining under the influence of manual mining. By communicating often with the aquifer, the water inrush from KCCs poses a serious threat to construction projects. This paper adopts three flow field models, namely, Darcy aquifer laminar flow, Forchheimer flow, and Navier–Stokes turbulent flow, based on the changes in the water inrush flow pattern in the aquifer and laneway, and uses COMSOL Multiphysics software to produce the numerical solutions of these models. As the water inrush flow velocity increases, the Forchheimer flow shows the effect of additional force (inertial resistance) on flow in KCCs, in addition to the effect of viscous resistance. After the joint action of viscous resistance and inertial resistance, the inertial resistance ultimately dominates and gradually changes the water inrush from the KCCs to fluid seepage. Forchheimer flow can comprehensively reflect the nonlinear flow process in the broken rock mass of KCCs, demonstrate the dynamic process from the Darcy aquifer to the final tunnel turbulence layer, and quantitatively show the changes in the flow patterns of the water inrush from KCCs.

scholarly journals Modal Analysis of the Lysozyme Protein Considering All-Atom and Coarse-Grained Finite Element Models

2021 ◽  
Vol 11 (2) ◽  
pp. 547
Author(s):  
Gustavo Giordani ◽  
Domenico Scaramozzino ◽  
Ignacio Iturrioz ◽  
Giuseppe Lacidogna ◽  
Alberto Carpinteri
Keyword(s):  
Modal Analysis ◽  
Numerical Solutions ◽  
Dynamic Properties ◽  
Low Frequency ◽  
Coarse Grained ◽  
Representative Point ◽  
Dynamic Simulations ◽  
Quick Method ◽  
Egg White Lysozyme ◽  
General Terms

Proteins are the fundamental entities of several organic activities. They are essential for a broad range of tasks in a way that their shapes and folding processes are crucial to achieving proper biological functions. Low-frequency modes, generally associated with collective movements at terahertz (THz) and sub-terahertz frequencies, have been appointed as critical for the conformational processes of many proteins. Dynamic simulations, such as molecular dynamics, are vastly applied by biochemical researchers in this field. However, in the last years, proposals that define the protein as a simplified elastic macrostructure have shown appealing results when dealing with this type of problem. In this context, modal analysis based on different modelization techniques, i.e., considering both an all-atom (AA) and coarse-grained (CG) representation, is proposed to analyze the hen egg-white lysozyme. This work presents new considerations and conclusions compared to previous analyses. Experimental values for the B-factor, considering all the heavy atoms or only one representative point per amino acid, are used to evaluate the validity of the numerical solutions. In general terms, this comparison allows the assessment of the regional flexibility of the protein. Besides, the low computational requirements make this approach a quick method to extract the protein’s dynamic properties under scrutiny.

scholarly journals Recognition Dynamics in the Brain under the Free Energy Principle

2018 ◽  
Vol 30 (10) ◽  
pp. 2616-2659 ◽  
Author(s):  
Chang Sub Kim
Keyword(s):  
Free Energy ◽  
Numerical Solutions ◽  
Coarse Grained ◽  
Biophysical Model ◽  
Hierarchical Architecture ◽  
Bayesian Filtering ◽  
Energy Principle ◽  
Free Energy Principle ◽  
Least Action ◽  
The Brain

We formulate the computational processes of perception in the framework of the principle of least action by postulating the theoretical action as a time integral of the variational free energy in the neurosciences. The free energy principle is accordingly rephrased, on autopoetic grounds, as follows: all viable organisms attempt to minimize their sensory uncertainty about an unpredictable environment over a temporal horizon. By taking the variation of informational action, we derive neural recognition dynamics (RD), which by construction reduces to the Bayesian filtering of external states from noisy sensory inputs. Consequently, we effectively cast the gradient-descent scheme of minimizing the free energy into Hamiltonian mechanics by addressing only the positions and momenta of the organisms' representations of the causal environment. To demonstrate the utility of our theory, we show how the RD may be implemented in a neuronally based biophysical model at a single-cell level and subsequently in a coarse-grained, hierarchical architecture of the brain. We also present numerical solutions to the RD for a model brain and analyze the perceptual trajectories around attractors in neural state space.

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