Experimental measurements.

<p>Experimental measurements.</p

Experimental and theoretical parameters.

<p>Experimental and theoretical parameters.</p

Contour lines of <i>f</i> as a function of the cuboidal magnet dimensions.

<p>The dashed black lines show <i>L</i>_<i>y</i>/<i>L</i>_<i>x</i> = 1/3 and <i>L</i>_<i>z</i>/<i>L</i>_<i>x</i> = 1/3, and the magnets used in this work are represented by the star (<i>Oz</i> being the magnet magnetization direction and <i>Ox</i> the mean flow velocity in the thruster).</p

Picture of the ship in the experimental tank.

<p>The LiPo 6s battery is placed on top of the boat and connected to the electrode using an XT60 plug.</p

Thruster working regimes (seawater, <i>k</i> = 1).

<p>Below (resp. above) the thick solid tilted line <i>K</i> = 1, the thruster behaves as a flow (resp. a voltage) generator. The dotted lines parallel to the thick line <i>K</i> = 1 show solutions for constant values of <i>K</i>. Our typical thruster (the star) evolves along the thick dashed curve when <i>B</i> is changed in the range <i>B</i> = 10⁻² − 1 T. The other thick dashed curve is similar, for a thruster size of 10 m and a magnetic field from <i>B</i> = 10⁻² T to <i>B</i> = 4 T.</p

Evolution of the conductivity with the NaCl concentration.

<p>Our data (circles) are in excellent agreement with the Kohlrausch law (solid line), given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0178599#pone.0178599.e017" target="_blank">Eq (4)</a>, using (<i>a</i>₀ ≈ 2071 ⋅ 10⁻⁴ S.m².kg⁻¹, <i>b</i>₀ ≈ 98.32 ⋅ 10⁻⁴ S.m^7/2.kg^−3/2). Note that the NaCl solubility in water is 360 kg.m⁻³ at 25°C.</p

Ship velocity as a function of <i>H</i>.

<p>For the intermediate battery 4s (the dashed line being the design chosen for our experiments), with <i>σ</i> ≈ 8.7 S.m⁻¹, <i>B</i>_<i>r</i> ≈ 1.26 T (skin friction modifications neglected).</p

Sketch of the thruster and the different velocities taken into account in this study.

<p>Four vortices are displayed to represent the singular head loss at the entrance and exit of the thruster.</p

Schematic representation of top (a) and side (b) view of the ship (length are in millimeters).

<p>c) Diagram of the thruster used to propel the ship. The black part represents the magnetic bridge, dark gray part the magnets, light gray part the electrodes and the white part the plastic used to isolate the electric circuit. Pictures (d) and (e) are aerial and underwater view of the ship respectively. The 6s LiPo Battery is visible on picture (d). Note that for aesthetics reasons, pictures have been taken in a swimming pool and not in the actual experimental tank.</p

Experimental and theoretical study of magnetohydrodynamic ship models

Magnetohydrodynamic (MHD) ships represent a clear demonstration of the Lorentz force in fluids, which explains the number of students practicals or exercises described on the web. However, the related literature is rather specific and no complete comparison between theory and typical small scale experiments is currently available. This work provides, in a self-consistent framework, a detailed presentation of the relevant theoretical equations for small MHD ships and experimental measurements for future benchmarks. Theoretical results of the literature are adapted to these simple battery/magnets powered ships moving on salt water. Comparison between theory and experiments are performed to validate each theoretical step such as the Tafel and the Kohlrausch laws, or the predicted ship speed. A successful agreement is obtained without any adjustable parameter. Finally, based on these results, an optimal design is then deduced from the theory. Therefore this work provides a solid theoretical and experimental ground for small scale MHD ships, by presenting in detail several approximations and how they affect the boat efficiency. Moreover, the theory is general enough to be adapted to other contexts, such as large scale ships or industrial flow measurement techniques