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Potential flows with electromagnetically induced circulation in a Hele-Shaw cell

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

^*Contact author: kimckee@mit.edu

Phys. Rev. Fluids 10, 054103 – Published 7 May, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.054103

Abstract

In Hele-Shaw cells, pressure-driven viscous fluid motion between two closely-spaced plates gives rise to a two-dimensional potential flow with zero circulation. Here, we show how the introduction of electromagnetic effects enables the realization of potential flows with circulation. We present canonical Hele-Shaw experiments with circulation prescribed by the electromagnetic configuration, and rationalize the observed flows theoretically. We also draw an analogy between this new class of circulatory potential flows and a class of electrostatic systems.

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References (37)

H. S. Hele-Shaw, The flow of water, Nature (London) 58, 34 (1898).
G. G. Stokes, Mathematical and Physical Papers (University Press, 1922), Vol. 3, p. 278.
L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Special issue of the Verhandlungen des III. Internationalen Mathematiker-Kongresses Heidelberg, 1904.
G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, UK, 2000).
M. Van Dyke, An Album of Fluid Motion (Parabolic Press, Stanford, 1982).
See [33] for a recent attempt to generalize the Kutta condition.
G. Magnus, Ueber die abweichung der geschosse, und: Ueber eine auffallende erscheinung bei rotirenden körpern, Annalen der physik 164, 1 (1853).
Note that potential flows have “slip” on the boundary while, in reality, a no-slip boundary condition is satisfied on any obstacle in the Hele-Shaw cell. Thus, within a boundary layer around each obstacle on the order of the cell thickness $h$ , the Hele-Shaw approximation breaks down. Nevertheless, the potential flow provides a good description of the flow outside of this boundary layer.
H. H. Bau, Applications of magneto electrochemistry and magnetohydrodynamics in microfluidics, Magnetochemistry 8, 140 (2022).
K. I. McKee, Magnetohydrodynamic flow control in Hele-Shaw cells, J. Fluid Mech. 993, A11 (2024).
Cy S. David, E. W. Hester, Y. Xu, and J. M. Aurnou, Magneto-stokes flow in a shallow free-surface annulus, J. Fluid Mech. 996, A33 (2024).
R.-J. Yang, L.-M. Fu, and Y.-C. Lin, Electro-osmotic flow in microchannels, J. Colloid Interface Sci. 239, 98 (2001).
H. Ohshima and T. Kondo, Electrokinetic flow between two parallel plates with surface charge layers: electro-osmosis and streaming potential, J. Colloid Interface Sci. 135, 443 (1990).
E. B. Cummings, S. K. Griffiths, R. H. Nilson, and P. H. Paul, Conditions for similitude between the fluid velocity and electric field in electroosmotic flow, Anal. Chem. 72, 2526 (2000).
S. Ghosal, Fluid mechanics of electroosmotic flow and its effect on band broadening in capillary electrophoresis, Electrophoresis 25, 214 (2004).
E. Boyko, V. Bacheva, M. Eigenbrod, F. Paratore, A. D. Gat, S. Hardt, and M. Bercovici, Microscale hydrodynamic cloaking and shielding via electro-osmosis, Phys. Rev. Lett. 126, 184502 (2021).
M. Mirzadeh, T. Zhou, M. A. Amooie, D. Fraggedakis, T. R. Ferguson, and M. Z. Bazant, Vortices of electro-osmotic flow in heterogeneous porous media, Phys. Rev. Fluids 5, 103701 (2020).
H. Darcy, Les Fontaines Publiques De La Ville De Dijon: Exposition Et Application Des Principes à Suivre Et Des Formules à Employer Dans Les Questions De Distribution D'eau (Victor dalmont, Paris, France, 1856), Vol. 1.
G. O. Brown, Henry darcy and the making of a law, Water Resour. Res. 38, 11 (2002).
S. Whitaker, Flow in porous media I: A theoretical derivation of Darcy's law, Transp. Porous Media 1, 3 (1986).
R. C. Smith and R. A. Greenkorn, An investigation of the flow regime for Hele-Shaw flow, Soc. Pet. Eng. J. 9, 434 (1969).
G. M. Homsy, Viscous fingering in porous media, Annu. Rev. Fluid Mech. 19, 271 (1987).
K. Singh, M. Jung, M. Brinkmann, and R. Seemann, Capillary-dominated fluid displacement in porous media, Annu. Rev. Fluid Mech. 51, 429 (2019).
R. Juanes, Y. Meng, and B. K. Primkulov, Multiphase flow and granular mechanics, Phys. Rev. Fluids 5, 110516 (2020).
Y. Qiu, K. Xu, A. A. Pahlavan, and R. Juanes, Wetting transition and fluid trapping in a microfluidic fracture, Proc. Natl. Acad. Sci. USA 120, e2303515120 (2023).
W. M. Haynes, CRC Handbook of Chemistry and Physics, 95th ed. (CRC Press, Boca Raton, Florida, 2014).
The hole in the top plate of the cell, through which the obstacles were placed, was made marginally larger than the obstacle dimensions to allow for bubbles to exit the cell in the event of electrolysis or chemical reactions taking place on the metal surface. Note that the voltage was never raised above 1 volt during the experiments to avoid electrolysis.
The magnetic field across the fluid domain region above the magnet was measured to be $B_{0} = 210 \pm 20 mT$ . In our analysis, we assume the field magnitude to be constant over the entire domain and neglect the small ( $≲ 10 %$ ) fluctuations.
McKee [10] showed that the solution form (1) is valid in the limit of small Hartmann numbers; when the Hartmann number becomes too large, the parabolic profile in the vertical coordinate becomes spoiled owing to magnetic effects in the boundary layer [ [34], Fig. 5.10 on p. 153].
If $\hat{t} = (t_{x}, t_{y})$ , then $\hat{n} = (t_{y}, - t_{x})$ . It follows that $J \times \hat{z} \cdot \hat{t} = (J_{y} t_{x} - J_{x} t_{y})$ is equivalent to $- J \cdot \hat{n} = - J_{x} t_{y} + J_{y} t_{x}$ .
$G$ is essentially a lift coefficient. Using the lift formula, it is seen that $G \propto F_{L} / (\frac{1}{2} ρ U^{2} L) = 2 Γ / U L$ .
C. A. Klettner and F. T. Smith, The effect of inertia and vertical confinement on the flow past a circular cylinder in a Hele-Shaw configuration, J. Fluid Mech. 934, A8 (2022).
C. Gonzalez and H. E. Taha, A variational theory of lift, J. Fluid Mech. 941, A58 (2022).
P. A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge University Press, Cambridge, UK, 2001).
P. J. Baddoo, M. Kurt, L. J. Ayton, and K. W. Moored, Exact solutions for ground effect, J. Fluid Mech. 891, R2 (2020).
D. Crowdy, Solving Problems in Multiply Connected Domains (SIAM, Philadelphia, PA, 2020).
L. N. Trefethen, Series solution of Laplace problems, The ANZIAM Journal 60, 1 (2018).

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Potential flows with electromagnetically induced circulation in a Hele-Shaw cell

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