CC BY
558 PHYSICAL AND MATHEMATICAL SCIENCES / «ШУШШНУМ-ЛЭУШаУ» #3И54)), 2022
PHYSICAL AND MATHEMATICAL SCIENCES
УДК 532.5.031
Koroleva Yu. O.
MIEM HSE
DOI: 10.24412/2520-6990-2022-31154-58-59 ESTIMATION OF THE PRESSURE IN THE FILTRATION PROBLEM THROUGH THE CATION-
EXCHANGE MEMBRANE
Abstract:
The filtration of a conducting fluid through a porous layer in considered. A porous medium is modeled as an assemblage of spherical cells. Each cell consists of a porous core and liquid shell. We derive apriori estimates for pressure which show the specific behavior of the fluid depending on non-zero Debye constant.
Key words: filtration through the membrane, porous medium, weak solution, Debye radius
The subject of the present paper is the flow of electrolyte in the filtration process ([12]). We present the apriori estimates for pressure depending on other flow characteristics. The models of Stokes and Brinkman's equation (see e.g. [3]-[5]) are used to describe the flow.
Consider a porous cell H = H' u H°, which is a spherical particle. Its boundary is denoted by 3H = r' u r0, where P is the boundary of the sphere of radius awhile r° is the boundary of the sphere of radius b, 0<a b.
Flow of fluid (electrolyte) in the outer domain H°can be described by Stokes equation under low Reynold's number and which involves also electromass force:
Vp° = (U°Au° - p°V<p° , where p° = F°(Z+C+° - Z-C-)
is the volumetric density of movable electric charges in a porous particle, Z± are charge modules of cations and anions of the electrolyte, C± are concentrations of cations and anions; Fo is Faraday constant. We assume that the liquid is incompressible:
div vo = 0. Here po is local pressure, vo is velocity vector, n° is a dynamic viscosity,
(p° is the electric potential which satisfies to the
„0
Poisson equation: =--
££0
where e is the relative permittivity of the medium, e° is the dielectric constant.
In general stationary case the charge conservation law is valid: div /± =0 = 0.
where /± are ions flux densities and which satisfies to the following Nernst representation: /± = v°C± -D± (vC± ± Z±C±V<p° Here D± is the coefficient of
ion's diffusion in the fluid, R is the gas constant, Tis the absolute temperature.
Fluid flow is subjected to the Brinkman's equation with mass electric force in the inner domain H :
Vp' = p'Aw' — p'V^' — fcw', where p1 = F0(Z+C+ — Z_C-), is the volumetric density of movable electric charges in a porous particle, k is the Brink-man's constant which is inversely proportional to the particle's permeability. Here is the coefficient of Brinkman's viscosity. Brinkman's liquid is assumed to be incompressible: div v = 0.
Electric potential satisfies to Poisson's equation:
=
(P'-Pv)
where pv is the bulk density of the fixed charges. We model our membrane such that the particle has negative charge, then pv >0. We assume the equality of permittivity for the liquid and Brinkman's medium in Poisson's equations. This condition lets us do not take into account the Maxwel's stress tensor in boundary conditions since it keeps continuous automatically under the considered case. The following charge conservation equations must be valid:div /± = 0 where /± are densities of ion's fluxes in the porous particle,
......F°
/± = vtC±±Z±C±V<^
Here Dm± are diffusion coefficients of electrolyte ions inside the porous particle. The continuity of velocity filed, stress tensor, continuity of electric potential is assumed on the common boundary Zero gradient for ion concentrations as well as electric potential are satisfied on the outer boundary The velocity filed is a known function on the outer boundary. Passing to the dimensionless variables, one can estimate the norms for the pressure depending on the norms of difference between concentrations. The following estimates are valid:
«ШУШ(ШШиМ-Ши©Ма1> #3Щ134)), 2022 / PHYSICAL AND MATHEMATICAL SCIENCES 59
on
r = -
where the positive constants Ci, C2 do not depend any flow cliaracteristics. Here
b F,c, \œaRT) r„ is the equivalent concentration of electrolyte equilibrated with a membrane.
The work was supported in part by Russian Science Foundation, project 20-19-00670.
Literature
1. Filippov A.N. A Cell Model of an Ion-Exchange Membrane. Electrical Conductivity and Elec-troosmotic Permeability, Colloid J. 2018. V. 80. P. 728-738.
2. Filippov A.N. A Cell Model of an Ion-Exchange Membrane. Electrodiffusion Coefficient and
Diffusion Permeability, Colloid J. 2021. V. 83. pp. 387398.
3. Koroleva Yu. O. Qualitative properties of the Solution to Brinkman-Stokes system modelling a filtration process, Mathematics and Statistics, 2017, V. 5 (4), pp. 143-150.
4. Filippov A. N., Koroleva Yu. O. Viscous flow through a porous medium filled by liquid with varying viscosity, Buletinul Academiei de §tiinje al Republicii Moldova, Matematica, 2017, vol.3, pp.74-87.
5. Koroleva Y. O. On some properties of solution to Herschel-Bulkley and Casson's models of blood flow, Вестник современных исследований, 2018, 23-8(1), С. 344-349.
