This browser-side companion solves the low-Rm cross-section problem for fully developed duct or pipe flow in a transverse magnetic field. The tool keeps the coupled velocity and electric-potential equations, lets the Hartmann and Shercliff walls be either conducting or insulating, and compares rectangular and circular sections. The result is a compact way to see the Hartmann-layer scale δH ∼ a / Ha, the broader Shercliff scale δS ∼ a / √Ha, and the pressure gradient required to sustain a chosen bulk speed.
The preset only initializes the sliders and wall conditions; you can still fine-tune everything afterward.
These presets initialize representative density, viscosity, conductivity, and melting temperature values for quick estimates.
For the circular case, the top and bottom arcs are treated as the Hartmann sectors and the left and right arcs as the Shercliff sectors.
The Hartmann half-gap is a; the Shercliff half-span is b. Larger b/a separates the two layer scales more clearly.
The solver now reaches moderately larger Hartmann numbers while still keeping enough grid points near the walls to resolve the boundary layers honestly in the browser.
For rectangles, this is the half-gap between the Hartmann walls. For the circular case, it is the pipe radius.
The solver first computes the unit-forcing shape, then rescales it to this chosen bulk speed to report the dimensional pressure drop.
With a, ρ, and ν, this turns the Hartmann-number slider into an actual applied-field estimate.
Together with ρ, this sets μ = ρν and therefore the dimensional pressure gradient required for the target flow.
In this cross-section model the applied field points in the positive y direction, the flow is along z, and the only induced magnetic correction retained is the axial field bz(x,y). The panels below therefore separate the streamwise velocity, electrostatic potential, induced current closure, and reconstructed axial magnetic response rather than mixing them together.
Colors show u / Ū. The guide arrows mark the imposed magnetic field pointing in the positive y direction, while the chord overlays mark the vertical Hartmann cut, the horizontal Shercliff cut, and the diagonal comparison cut.
Colors show the dimensionless electrostatic potential φ. Conducting walls pin φ, while insulating walls let the potential float so that J·n = 0.
Colors show |J⊥|, while contours show the current-streamline family reconstructed from the same streamfunction that generates bz. Conducting walls pin the potential strongly enough to redirect those loops at the boundary; insulating walls force them tangent to the wall.
Colors show the reconstructed induced axial field bz / B0. This reduced model does not evolve any separate transverse or poloidal magnetic field; only the axial correction bz is reconstructed. Its contours are also streamlines of J⊥.
The vertical cut sees the Hartmann layers most directly; the horizontal cut sees the Shercliff layers; the diagonal cut helps compare the fully two-dimensional cases.
Model summary. The cross-section solve keeps the low-Rm quasi-static system with imposed field B0 || +ŷ and flow u || ẑ:
∇⊥2u - Ha2(u + ∂xφ) + 1 = 0, ∇⊥2φ = -∂xu,
with Jx = -(∂xφ + u) and Jy = -∂yφ. No-slip gives u = 0 at every wall. Conducting walls set φ = 0, while insulating walls impose J·n = 0, which reduces to homogeneous Neumann here because the wall speed vanishes. The dimensional pressure gradient is then inferred from the unit-forcing solve and the chosen bulk speed.
Afterward the explorer reconstructs only the induced axial field from Ampère's law by solving Jx = ∂yβ, Jy = -∂xβ, equivalently ∇⊥2β = ∂yJx - ∂xJy, where β ∝ bz and the additive constant is fixed by a zero-mean gauge. No separate transverse or poloidal magnetic field is evolved in this reduced model.