Time-dependent electrophoresis of a dielectric spherical particle embedded in Brinkman medium

Saad, E. I.; Faltas, M. S.

doi:10.1007/s00033-018-0939-4

Cited by 17 publications

(17 citation statements)

References 38 publications

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“…When σ → ∞ , the electrophoretic mobility vanishes for two models. In the limiting situation as γ = 0, the Happel and Kuwabara unit cell models lead to the same starting electrophoretic velocity, which can be analytically obtained in a closed form As ν t / a 2 → ∞ , the electrophoretic velocity attains its steady value, ie,

\begin{matrix} alignleft \\ align-1 \frac{μ U}{ε ζ E_{0}} = & align-2 - \frac{18 sech α δ}{(γ + 2) {normalΔ}_{5}} ((α^{4} + 3 γ^{1 / 3} (2 γ^{1 / 3} - 1) α^{2} - 6 γ) \sinh [α (1 - γ^{- 1 / 3})] \\ align-1 & align-2 + ((3 γ^{1 / 3} - 1) α^{3} + 6 (γ - γ^{2 / 3}) α) \cosh [α (1 - γ^{- 1 / 3})] - 2 α^{3} γ), \end{matrix}

for the Happel model, and

\frac{μ U}{ε ζ E_{0}} = - \frac{18 normalsech α δ}{false(γ + 2 false) Δ_{6}} ((), false(α^{2} - γ^{1 false/ 3} false) \sinh false[α false(1 - γ^{- 1 false/ 3} false) false] + α false(γ^{1 false/ 3} - 1 false) \cosh false[α false(1 - γ^{- 1 false/ 3})

…”

Section: Time‐dependent Electrophoresis Of a Sphere‐in‐cell Model Witmentioning

confidence: 93%

“…In addition, the particle velocity vanishes at the limit σ → ∞ . In the limit of γ = 0, both cases of the Happel and Kuwabara unit cell models result in the same starting particle velocity, whose analytical result available in the literature for limiting case, as reported by Saad and Faltas …”

Section: Time‐dependent Translation Of Spherical Particles Produced Bmentioning

confidence: 99%

“…The field equations governing an incompressible time‐evolving viscous fluid flow through porous media (including a modified Stokes/Brinkman equation) with the absence of body forces are

\nabla \cdot \overset{q}{true} = 0,

\frac{ρ}{ϕ} \frac{\partial \overset{q}{true}}{\partial t} = μ \nabla^{2} \overset{q}{true} - \frac{μ}{K} \overset{q}{true} - \nabla p,

where

\overset{q}{true}

is the volume‐averaged velocity vector, p is the pore average pressure, μ is the viscosity of the fluid, and ϕ represents the porosity of the porous medium. The porous medium is characterized by its permeability, which is a measure of the flow conductivity in the porous medium.…”

Section: Time‐dependent Electrophoresis Of a Sphere‐in‐cell Model Witmentioning

confidence: 99%

“…U ( t ) the start‐up of electrophoretic velocity of the spherical particle to be determined. The apparent velocity slip distribution denoted by the last term of Equation , which has been obtained from the local electroosmosis in the double layer and leads to the electrophoresis of the spherical particle, increases with increases in the elapsed time from zero t = 0 to the steady‐state values as t → ∞ and in the electrokinetic radius κ a to these steady values as κ a → ∞ , and also, it increases with decreases in the permeability α to the clear fluid case α = 0 . As indicated in Equations , Happel's unit cell model assumes that the radial velocity relative to the bulk electroosmotic flow and shear stress of the fluid vanish at the fictitious surface of the cell, whereas Kuwabara's unit cell model takes this radial velocity and the vorticity of the fluid to be zero there.…”

Section: Time‐dependent Electrophoresis Of a Sphere‐in‐cell Model Witmentioning

confidence: 99%

“…Recently, the unit cell model was employed to investigate the transient electrophoresis of a swarm of dielectric spheres with constant zeta potential after the application of a step function electric field for the case of thin but finite double layer . Saad and Faltas studied semitheoretically the time‐dependent electrophoresis of a charged spherical particle in an electrolyte solution saturated in a charged porous medium after the sudden application of an external electric field.…”

Section: Introductionmentioning

confidence: 99%

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Start‐up Brinkman electrophoresis of a dielectric sphere for Happel and Kuwabara models

Saad

2018

Math Methods in App Sciences

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The starting electrophoresis in a charged porous continuum response of a homogeneous suspension of spherical particles to the sudden application of an external electric field is analyzed semianalytically through the use of a unit cell model. The electric double layer in the porous Brinkman medium surrounding each sphere is assumed to be thin but finite, and the effect of dynamic electroosmosis within it is included. The field equation for the fluid outside the double layers is solved based on the unit cell model under the start-up Brinkman model. Expressions of the time-dependent electrophoretic and settling velocities of the particle in the Laplace transform as functions of the permeability, the relative mass density, the electrokinetic particle radius, and the volume fraction of the particles are performed for two different boundary conditions at the fictitious surface of the cell. Numerical results indicate that the timescale for the development of electrophoresis and sedimentation is small for a high permeability, an assemblage of spheres through a dielectric porous medium with a higher particle volume fraction and a smaller particle to porous fluid density ratio; the electrophoretic mobility is a monotonically increasing function of the electrokinetic radius of the sphere at any instant. The start-up of electrophoretic mobility decreases with an increase in the particle volume fraction at a small value of the particle-to-medium density ratio, but it may increase as the particle volume fraction increases at a large value of this density. The particle interaction effect in a swarm on the time-dependent electrophoresis is much smaller than that on the starting sedimentation of the particles. The detailed comparison of the results of the cell model with different boundary conditions at the virtual surface of the cell is obtained and illustrated graphically.

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\begin{matrix} alignleft \\ align-1 \frac{μ U}{ε ζ E_{0}} = & align-2 - \frac{18 sech α δ}{(γ + 2) {normalΔ}_{5}} ((α^{4} + 3 γ^{1 / 3} (2 γ^{1 / 3} - 1) α^{2} - 6 γ) \sinh [α (1 - γ^{- 1 / 3})] \\ align-1 & align-2 + ((3 γ^{1 / 3} - 1) α^{3} + 6 (γ - γ^{2 / 3}) α) \cosh [α (1 - γ^{- 1 / 3})] - 2 α^{3} γ), \end{matrix}

for the Happel model, and

\frac{μ U}{ε ζ E_{0}} = - \frac{18 normalsech α δ}{false(γ + 2 false) Δ_{6}} ((), false(α^{2} - γ^{1 false/ 3} false) \sinh false[α false(1 - γ^{- 1 false/ 3} false) false] + α false(γ^{1 false/ 3} - 1 false) \cosh false[α false(1 - γ^{- 1 false/ 3})

…”

Section: Time‐dependent Electrophoresis Of a Sphere‐in‐cell Model Witmentioning

confidence: 93%

Section: Time‐dependent Translation Of Spherical Particles Produced Bmentioning

confidence: 99%

“…The field equations governing an incompressible time‐evolving viscous fluid flow through porous media (including a modified Stokes/Brinkman equation) with the absence of body forces are

\nabla \cdot \overset{q}{true} = 0,

\frac{ρ}{ϕ} \frac{\partial \overset{q}{true}}{\partial t} = μ \nabla^{2} \overset{q}{true} - \frac{μ}{K} \overset{q}{true} - \nabla p,

where

\overset{q}{true}

Section: Time‐dependent Electrophoresis Of a Sphere‐in‐cell Model Witmentioning

confidence: 99%

Section: Time‐dependent Electrophoresis Of a Sphere‐in‐cell Model Witmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Start‐up Brinkman electrophoresis of a dielectric sphere for Happel and Kuwabara models

Saad

2018

Math Methods in App Sciences

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Transient electrophoresis of a charged porous particle

Lai

Keh

2020

Electrophoresis

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Transient electrophoresis of a charged porous particleThe starting electrophoretic motion of a porous, uniformly charged, spherical particle, which models a solvent-permeable and ion-penetrable polyelectrolyte coil or floc of nanoparticles, in an arbitrary electrolyte solution due to the sudden application of an electric field is studied for the first time. The unsteady Stokes/Brinkman equations with the electric force term governing the fluid velocity fields are solved by means of the Laplace transform. An analytical formula for the electrophoretic mobility of the porous sphere is obtained as a function of the dimensionless parameters κa, λa, ρ p /ρ, and νt/a 2 , where a is the radius of the particle, κ is the Debye screening parameter, λ is the reciprocal of the square root of the fluid permeability in the particle, ρ p and ρ are the mass densities of the particle and fluid, respectively, ν is the kinematic viscosity of the fluid, and t is the time. The electrophoretic mobility normalized by its steady-state value increases monotonically with increases in νt/a 2 and κa, but decreases monotonically with an increase in ρ p /ρ, keeping the other parameters unchanged. In general, a porous particle with a high fluid permeability trails behind an identical porous particle with a lower permeability and a corresponding hard particle in the growth of the normalized electrophoretic mobility The normalized electrophoretic acceleration of the porous sphere decreases monotonically with an increase in the time and increases with an increase in λa from zero at λa = 0.

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Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

Lai

Keh

2021

Electrophoresis

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The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.

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Time-dependent electrophoresis of a dielectric spherical particle embedded in Brinkman medium

Cited by 17 publications

References 38 publications

Start‐up Brinkman electrophoresis of a dielectric sphere for Happel and Kuwabara models

Start‐up Brinkman electrophoresis of a dielectric sphere for Happel and Kuwabara models

Transient electrophoresis of a charged porous particle

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

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