Potential Flows with Electromagnetically-Induced Circulation in a Hele-Shaw Cell

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.

Figures (4)

• Figure 1: The Hele-Shaw cell is filled with saltwater and placed atop a permanant neodymium magnet, whose field is $\boldsymbol{B}=B_0\hat{\boldsymbol{z}}$. Aluminum obstacles are placed into the cell from above. A pressure gradient drives flow from left to right. Circulation is induced by applying an electric current between the anode and cathode in the presence of the magnetic field.
• Figure 2: Experimental images of Hele-Shaw flow past an aerofoil with: (a) zero circulation ($V_{\mathrm{app}}=0\mathrm{V}$), (c) circulation near the Kutta value ($Q=140\mathrm{mL/min}$ and $V_{\mathrm{app}}=0.51\mathrm{V}$), and (e) a circulation significantly larger than the Kutta value ($Q=100\mathrm{mL/min}$ and $V_{\mathrm{app}}=0.51\mathrm{V}$). The voltage was held constant and the flow rate adjusted so that $\Gamma/Q$ in (e) is 1.4 times larger than in (c). (b,d,f) Theoretical streamlines (blue) deduced from the analysis in Appendix A are overlaid on the experiments of (a,b,c). In (e), streamlines are fit to the experiments by choosing $\Gamma=1.22\Gamma_K$. In (f), theoretical streamlines corresponding to a 40% increase in the parameter $\Gamma/Q$ relative to panel (d).
• Figure 3: Experimental images of Hele-Shaw flow past an circle with: (a) zero circulation and (c) non-zero circulation induced by applying an electric current between the circle and the rectangular electrode. (b,d) Theoretical streamlines deduced from the analysis in Appendix A are overlaid on the experiments (a,c). In (d), the circulation was computed using (\ref{['eq:circpred']}). The vertical magnetic field magnitude was measured to be $B_0=210\pm20\mathrm{mT}$, and the electrical current was $I=1.75\pm0.05\mathrm{mA}$.
• Figure 4: The aluminum aerofoil was cut using a CNC machine according to the points labelled by black circles connected by a thin black line. The fit of the experimental aerofoil to the analytical aerofoil expression described in equations (\ref{['eq:zmap']})-(\ref{['eq:muval']}) is given in red. The axes are labelled in units of millimeters. Despite its apparently complex form, the analytical map is parametrized according to the single parameter $\mu$, which we take to be equal to 0.78.