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Magnetohydrodynamic flow of a binary electrolyte in a concentric annulus
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10.1063/1.3689187
/content/aip/journal/pof2/24/3/10.1063/1.3689187
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/3/10.1063/1.3689187
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

A schematic depiction of the flow cell and the cylindrical coordinate system (R, Θ, Z). The electrolyte is confined in a concentric annulus with an inner radius R 1 and an outer radius R 2. The electrodes coincide with the cylindrical surfaces.

Image of FIG. 2.
FIG. 2.

The azimuthal velocity v b (normalized with its maximum value) as a function of x when r 1 = 0, 0.2, 0.5, 1, 2, and ∞. The vertical lines at x = 1/e and x = 1/2 correspond, respectively, to the positions of the velocity maximum when the curvature is large and small.

Image of FIG. 3.
FIG. 3.

The electric current flux as a function of the potential difference between the electrodes. The dimensionless inner radius r 1 = 2. The electrolyte is binary and symmetric, z 1 = −z 2 = 1. The dimensionless diffusion coefficients D 1 = D 2 = 1. Exchange current density j e = 103. The symbols (squares) and the line correspond, respectively, to the analytical solution and the finite element simulation.

Image of FIG. 4.
FIG. 4.

The ratio between the outward and the inward limiting current fluxes () as a function of the radii ratio δ. D 1 = D 2 = 1, z 1 = −z 2 = 1, and j e = 103. The dashed line (2ln δ − 1) corresponds to the large δ asymptote. The dotted line is the small δ asymptote .

Image of FIG. 5.
FIG. 5.

The concentrations c 1b as functions of the radius r under limiting current conditions when the current is directed outwardly (dashed line) and inwardly (solid line). All other conditions are the same as in Figure 3.

Image of FIG. 6.
FIG. 6.

The current flux as a function of the potential difference between the electrodes when the exchange current density j e = 10−3, 10−2, 10−1, 1, 10, 102, and 103. The curvature is neglected. Binary, symmetric electrolyte, z 1 = −z 2 = 1, and D 1 = D 2 = 1. Butler-Volmer electrode kinetics are used with α = 0.5. The solid line corresponds to predictions obtained with the Nernst equation.

Image of FIG. 7.
FIG. 7.

The disturbance growth rate σ as a function of the Dean number when k = 1 (dashed line), 2.39 (solid line), and 4 (dashed-dotted line). Binary electrolyte, z 1 = −z 2 = 1, , , R 1 = 0.5 m, R 2 = 0.505 m, B = 0.4 T, ρ = 103 kg/m3, and μ = 10−3 Pa · s.

Image of FIG. 8.
FIG. 8.

The critical Dean number Dn0 at the onset of instability, predicted by linear stability analysis, as a function of the wave number k. The electrodes’ current is controlled. The white and gray regions correspond, respectively, to stable (σ < 0) and unstable (σ > 0) states. The symbols correspond to finite element solutions of the nonlinear equations. The solid and hollow symbols correspond, respectively, to subcritical (Dn) and supercritical (Dn+) cases. The symbols are located at {k, Dn, Dn+} = {2.39, 4.0, 8.0}, {3.77, 5.6, 7.2}, {5.05, 6.4, 9.7}, and {7.12, 10.5, 14.5}. The other conditions are the same as in Figure 7.

Image of FIG. 9.
FIG. 9.

The concentration distribution of c 1 when (a) j* = −0.1, Dn = 8.04; (b) j* = 0.05, Dn = 4.02; and (c) j* = 0.1, Dn = 8.04. The black solid lines in (c) are the streamlines associated with the secondary flow in the rz plane. The arrow shows the flow direction. (d) Electric current flux distribution for case (c). All the other parameters used are the same as in Figure 7.

Image of FIG. 10.
FIG. 10.

The critical Dean number Dn0 at the onset of instability as a function of the wave number. The annular conduit is infinitely long and electric potential is applied across the electrodes. The white and gray areas correspond, respectively, to stable (σ < 0) and unstable (σ > 0) cases. The hollow (Dn+) and solid (Dn) symbols correspond, respectively to subcritical and supercritical cases. The symbols are located at {k, Dn, Dn+} = {1, 3.93, 4.69}, {2.5, 4.46, 5.45} and {4, 5.73, 7.34}. j e = 6 × 10−3, α = 0.5, , , R 1 = 0.5 m, R 2 = 0.505 m, B = 0.4 T, ρ = 103 kg/m3, and μ = 10−3 Pa · s.

Image of FIG. 11.
FIG. 11.

The critical Dean number at the onset of instability as a function of the wave number. Controlled potential case. The white and gray areas correspond, respectively, to stable (σ < 0) and unstable (σ > 0) states. The dotted line with solid squares corresponds to the solution of the linear stability problem with Nernst boundary conditions. The solid line corresponds to the Butler-Volmer boundary conditions with j e = 103 and α = 0.5. All the electrolyte properties are the same as used for Figure 10.

Image of FIG. 12.
FIG. 12.

The kinetic energy of the secondary flow ‖u 2 as a function of r 1. D 1 = D 2 = 1. j* = 0.15. l = 2. The dotted line with circles corresponds to results of the simplified, two-dimensional model. The dashed line with crosses shows results of the axisymmetric model. The inset depicts the relative difference between the approximate model and exact model predictions as a function of r 1.

Image of FIG. 13.
FIG. 13.

MHD flow in an annulus of height l = 2π/2.39. (a) Concentration distribution c 1 and the (u, w) streamlines (solid lines) when j* = −0.1. (b) Concentration distribution c 1 and the (u, w) streamlines (solid lines) when j* = 0.1. (c) Intensity of electric current flux distribution in case (a). (d) Intensity of electric current flux distribution in case (b). All the other parameters are the same as used in Figure 7.

Image of FIG. 14.
FIG. 14.

The intensity of the secondary flow ‖u 2 (Eq. (61)) as a function of the potential difference between the electrodes (ΔV ext ). The dashed line with squares, the dashed line with circles, and the dashed line with triangles correspond, respectively, to capped conduits with heights π, π/2, and π/4. The solid line with crosses corresponds to an infinitely long, annular conduit with periodic boundary conditions in the axial (z) direction and wave number k = 4. All the other parameters are the same as in Figure 10.

Image of FIG. 15.
FIG. 15.

The intensity of the secondary flow ‖u 2 (a), the average azimuthal velocity (b), and the average current flux (c) as functions of the aspect ratio l when the potential is controlled. The dashed line and the hollow circles correspond, respectively, to positive (ΔV ext = 15) and negative (ΔV ext = −15) currents. (d) The relative difference between the intensity of the secondary flow ‖u 2 (dashed line), average azimuthal velocity (solid line), and average current flux as functions of the aspect ratio l.

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/content/aip/journal/pof2/24/3/10.1063/1.3689187
2012-03-05
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Magnetohydrodynamic flow of a binary electrolyte in a concentric annulus
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/3/10.1063/1.3689187
10.1063/1.3689187
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