Interdisciplinary Applied Mathematics

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The additional    drag    force acting    on    a    control    volume    due    to    the    elec


trokinetic effects can be expressed as


Fek = / PeE dQ.


J CV


Substituting pe from the Poisson-Boltzmann equation and E obtain



Уф, we



Fek



еУ2ф



CV



дф дф дф


~к~еп + уггег + уг— е дп dl ds



S



dQ,


where n, l, and s are the normal, streamwise and spanwise coordinates, respectively, and dQ = dnds dl. This volume integral is complicated to evaluate in general. However, some simplifications can be made when AD/h ^ 1. Also, for a general complex geometry, we further assume that the radius of curvature R is much larger than the Debye length AD. The latter condition is required to exclude application of the forthcoming procedure in the vicinity of sharp corners. Based on these assumptions, У2ф can be approximated to be Ftf. Also, « 0 across the entire EDL, which is approximately valid due to the small EDL thickness and the no-penetration boundary condition of the externally applied electric field on the surfaces. This enables us to separate the volume integral into the following two components:


Г Г d2rK 1


1* W 1* L


дф дф


——ffrdn


Jq dn


oo


—e; 4—e„


[dl ds s

dl ds,


Fek


where L and W are the streamwise and spanwise lengths of the domain, respectively. Also, for a general geometry we assumed the separation distance between the two surfaces to be 2h. The second integral in the above equation can be obtained in the postprocessing stage from the solution of the electrostatic problem. Numerical solution for the first integral requires resolution of the EDL region, which requires enhanced near-wall resolution and results in the numerical stiffness. However, this integral can be evaluated analytically in the following form:

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