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Research Papers: Flows in Complex Systems

Effect of Relaxation on Drag Forces and Diffusivities of Particles Confined in Rectangular Channels

[+] Author and Article Information
Panadda Dechadilok

Department of Physics,
Faculty of Science,
Chulalongkorn University,
Payathai Road, Pathumwan,
Bangkok 10330, Thailand
e-mail: panadda.D@chula.ac.th

Chakrapong Intum

Department of Physics,
Faculty of Science,
Chulalongkorn University,
Payathai Road, Pathumwan,
Bangkok 10330, Thailand
e-mail: pi_wan_narak@hotmail.com

Sasipan Manaratha

Department of Physics,
Faculty of Science,
Chulalongkorn University,
Payathai Road, Pathumwan,
Bangkok 10330, Thailand
e-mail: fxsasipan@gmail.com

Umnart Sathanon

Department of Physics,
Faculty of Science,
Chulalongkorn University,
Payathai Road, Pathumwan,
Bangkok 10330, Thailand
e-mail: Umnart.S@chula.ac.th

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 15, 2016; final manuscript received June 3, 2016; published online August 17, 2016. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 138(12), 121105 (Aug 17, 2016) (15 pages) Paper No: FE-16-1037; doi: 10.1115/1.4033914 History: Received January 15, 2016; Revised June 03, 2016

When a particle is moving inside a channel, its hydrodynamic interaction with channel walls increases its drag coefficient, causing a diffusivity reduction. For charged particles moving in an electrolytic solution, there is an additional drag due to the distortion of an electrical double layer caused by particle motion known as the relaxation effect. Effects of relaxation on drag forces on spheres confined in rectangular channels are computed employing perturbations involving particle Peclet number and surface charge densities. Results indicate that confinement amplifies electrokinetic retardation; increasing the relative particle size or decreasing the channel cross section area enhances the relaxation effect. With the relative particle size kept constant, the relaxation effect on the drag exerted on charged spheres in cylindrical pores with its smaller cross section area is stronger than that on charged spheres in rectangular channels and slit pores. However, for certain values of Debye length and particle size, the ratio between excess drag due to relaxation on confined charged spheres and hydrodynamic drag on uncharged spheres confined at the same location is higher for particles in rectangular channels, resulting in higher percentages of diffusivity reduction. Diffusivity reduction due to relaxation of charged particles in square ducts displays a maximum as a function of relative particle size, whereas that of charged particles in rectangular channels with higher cross section aspect ratio increases monotonically as particle size increases.

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Figures

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Fig. 1

Schematic drawing of a negatively charged sphere (radius a) moving axially at speed U either in a negatively charged rectangular channel or between parallel plates. If the channel cross section is rectangular, the width and length of its cross section are 2H and 2L. If the sphere is moving between parallel plates, the distance between the plates is 2H. Electrical double layers are represented by “+” charges and shading.

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Fig. 2

Excess drag due to relaxation on a charged particle in a charged rectangular channel compared to hydrodynamic drag on an uncharged sphere in an uncharged channel (Frelaxation/Funcharged) as a function of relative particle size for (a) a charged sphere in a charged rectangular channel with L/H = 1 and 2, and (b) a charged sphere in a charged channel with L/H = 5 and 10 and a charged sphere confined in a charged slit pore. The particle is located at x = y = 0; qs  =  qc = −1.

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Fig. 3

Charge-related force coefficients in Eq. (52) as a function of dimensionless particle size (λ) for a sphere confined in a rectangular channel with (a) L/H = 1 and (b) L/H = 5. fs and fc describe the excess force on a charged sphere in an uncharged channel and that on an uncharged sphere in a charged channel. fsc is an additional term due to an interaction between particle and channel surface charges; κH  = 0.25. Also presented is the dimensionless drag on an uncharged sphere in an uncharged channel (Funcharged).

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Fig. 4

Charge-related force coefficients, (a) fs, (b) fsc, and (c) fc, as a function of dimensionless particle size (λ) for a sphere confined in a rectangular channel with L/H = 2. Results are shown for κH  = 0.25, 0.5, and 1.

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Fig. 5

Diffusive hindrance factor (Kd) as a function of dimensionless solute size (λ) for a charged sphere (solid lines with symbols) confined in (a) a charged square duct, (b) a charged rectangular channel with L/H = 2, (c) a charged channel with L/H = 5, and (d) a charged channel with L/H = 10 and slit pore; qs = qc = −5. Also shown is the upper-bound estimation of Kd; the particle–wall electrostatic interaction biased the particle toward the centerline, estimated as (1/Funcharged) of an uncharged sphere located at a centerline of an uncharged channel (dashed lines).

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Fig. 6

Electric force contribution and hydrodynamic force contribution to the force coefficients, fs and fc, as a function of dimensionless particle size (λ). Shown results are (a) an electric force contribution (|fes|) and a hydrodynamic force contribution (|fVs|) to fs, and (b) an electric force contribution (|fec|) and a hydrodynamic force contribution (|fVc|) to fc; L/H = 5 andκH  = 0.25.

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Fig. 7

Electric force contribution (|fesc|: empty squares), and hydrodynamic force contribution (|fVsc|: filled squares) to fsc, force coefficient corresponding to interaction between particle and channel surface charges, as a function of relative particle size (λ) for spheres confined in (a) a charged square duct and (b) a charged rectangular channel with L/H = 5; κH  = 0.5 (dashed lines) and 1 (solid lines)

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Fig. 8

Electric force contribution (FE) and hydrodynamic force contribution (FV) to the excess drag due to relaxation effect (Frelaxation) as a function of relative particle radius (λ) for spheres confined in (a) a charged square duct and (b) a charged channel with L/H = 5; κH  = 1 and qs  =  qc  = −1

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Fig. 9

Excess drag due to relaxation (Frelaxation) as a function of particle radius divided by Debye length (κa) for an unconfined particle, and for charged particles confined in uncharged cylindrical and slit pores as well as in an uncharged channel with L/H = 1, 2, and 5; λ = 0.2. The minimum distance between the particle and channel wall is the same for all confined spheres; q̂s  = qsλ = −1.

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Fig. 10

Excess drag due to relaxation (Frelaxation) as a function of particle radius divided by Debye length (κa) for charged particles confined in uncharged cylindrical as well as in an uncharged rectangular channel with L/H = 1, 2, and 5. The particle radius, a, is chosen as a length scale and is kept constant. The dimensional channel cross section area is 25πa2 for all channels; q̂s  = qsλ = −1.

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