Explore magnetic braking between parallel plates at fixed pressure gradient and zero transverse electric field. Adjust six inputs, compare velocity and flow, and reproduce the Ha=0, 1 and 10 examples.
Parameters
These σ and μ values are illustrative assumptions, not certified properties at a specified temperature. Switching presets does not change density ρ. For real materials, use properties for matching temperature and composition.
T
S/m
Pa·s
m
Pa/m
kg/m³
Density affects Re only, not velocity or Ha. The default 900 kg/m³ is also an illustrative assumption.
Model assumptions: Steady, incompressible, laminar flow between no-slip parallel plates; uniform transverse magnetic field; zero transverse electric field; low-Rm approximation. Externally driven electromagnetic pumping, finite-duct sidewalls, entrance effects and turbulence are not modeled.
Inputs and results remain interactive while the animation is paused.
Results
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Ha [-]
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u_max [m/s]
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u_avg [m/s]
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Re [-]
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Re_m [-]
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Flow ratio Q/Q₀
MHD flow (schematic)
Each profile in this schematic is normalized by its own maximum velocity. Particle speeds and arrow lengths are illustrative, not measurements of speed or force. Read absolute speeds in Results; use the chart below to compare profiles on a common velocity scale.
Hartmann profile (current)Parabola (B=0 reference)Magnetic field BLorentz force J×B
Velocity-profile comparison
The horizontal axis uses the zero-field maximum Gh²/(2μ) as its reference. B, σ, μ and h change the current curve; G and ρ do not change its normalized shape. At G=0 all velocities are zero; the chart shows the limiting shape as G→0.
Theory and equations
h is half of the full gap 2h. μ is dynamic viscosity (Pa·s); kinematic viscosity is ν=μ/ρ (m²/s). We assume G≥0 and zero transverse electric field E=0.
Ha = B h √(σ/μ)
u(y) = G h²/(μ Ha²) × [1 − cosh(Ha y/h)/cosh(Ha)]
ū = G h²/(μ Ha²) × [1 − tanh(Ha)/Ha]
Q/Q₀ = 3/Ha² × [1 − tanh(Ha)/Ha]
At B=0 the limit is u(y)=G(h²−y²)/(2μ), with mean velocity Gh²/(3μ) and Q/Q₀=1. At G=0 both Q and Q₀ vanish; the displayed ratio is the coefficient ratio in the limit G→0.
Re = ρ ū (2h)/μ, Rm = μ₀ σ ū (2h)
Re and Rm use the full gap 2h as their length scale. The model assumes Rm≪1 and laminar flow; Re alone is not used to predict transition. Changing the transverse electric field or current-closure conditions changes the analytical solution.
Understanding MHD and Hartmann flow
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Student: Can a magnetic field really change the flow of a conducting liquid? What does this simulator show?
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Professor: Yes, but the electrical conditions matter too. This tool calculates Hartmann flow between parallel plates with zero transverse electric field. Without a magnetic field, the velocity profile is parabolic. Increasing B at fixed pressure gradient produces an opposing Lorentz force and reduces mean velocity, while flattening the profile in the core.
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Student: Why does the illustrative sodium preset give such a strong magnetic effect?
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Professor: That preset assumes a large σ. Since Ha=Bh√(σ/μ), increasing σ increases Ha when B, h and μ are fixed. But the presets are not a temperature-specific property database. To assess a real material, enter σ, μ and ρ from sources covering the same temperature conditions.
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Student: Does a flatter profile mean more flow? How should I compare the numbers and figures?
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Professor: Not here. In the Ha=1 example, flow is about 71.52% of the zero-field value at the same pressure gradient. The schematic normalizes each profile by its own maximum, so its shape does not tell you whether the absolute speed is unchanged. Read mean velocity and Q/Q₀, then compare the common-scale chart. An electromagnetic pump uses externally applied current to drive fluid, unlike this braking model.
Frequently asked questions
At fixed pressure gradient and zero transverse electric field, increasing Ha flattens the core profile but decreases maximum and mean velocity. Velocity is zero at the walls. At large Ha, Hartmann layers of approximate thickness h/Ha form next to walls perpendicular to the magnetic field.
Yes. Ha=Bh√(σ/μ), so its dependence on σ is a square root. With G, h, μ and Ha fixed, this model gives the same velocity profile for different B–σ combinations. Rm also depends on σ, so the low-Rm approximation must be checked separately.
This analytical model helps explain magnetic braking and velocity profiles in conducting fluids. Real liquid-metal cooling systems require additional assessment of sidewalls, wall conductivity, current closure, entrance effects and temperature. It does not directly predict externally driven electromagnetic-pump or seawater-propulsion performance.
The displayed analytical solution is evaluated in floating-point arithmetic and results are shown to five significant digits. Small Ha uses a limiting series; large Ha uses expressions that avoid hyperbolic-function overflow. Agreement with the analytical solution is not a guarantee of accuracy for real fluids or equipment.
Applications and limitations
Magnetic effects in liquid-metal cooling:Currents and Lorentz forces change the pressure–flow relationship of conducting coolant in a magnetic field. In this fixed-pressure-gradient model, stronger B reduces flow. Maintaining the same flow requires a larger pressure gradient than in the zero-field case. A flatter profile alone does not imply reduced pressure loss.
Fusion blanket cooling:Strong fields confine plasma in magnetic-confinement fusion systems. Liquid-metal coolant concepts require assessment of MHD effects. Hartmann flow provides a useful idealized reference, not a complete blanket-design model.
Difference from an electromagnetic pump:A pump uses externally applied current and the J×B force to drive fluid. In this model, flow-induced current causes braking. Without specifying a power supply, electrodes and current closure, this page cannot calculate pump head or efficiency.
Ocean and geomagnetic context:Conducting seawater moving through the geomagnetic field can generate currents and magnetic signals. These phenomena involve MHD, but are not represented by the confined parallel-plate geometry used here.
Common misconceptions
Compare cases with matching fixed conditions. Q/Q₀ is relative to zero-field flow at the same G, h and μ. It is not directly comparable to constant-flow experiments or cases with different electrical conditions, such as insulating sidewalls or an open circuit. A nonzero transverse electric field changes the velocity equation itself.
μ is dynamic viscosity (Pa·s), not kinematic viscosity ν (m²/s). If only ν is known, use density to convert with μ=ρν. For material comparisons, match the temperature conditions and sources of σ, μ and ρ; do not use these illustrative presets as a property database.
This is an idealized educational model. Actual equipment may have different channel geometry, nonuniform fields and significant entrance or exit effects. After exploring the basic behavior, engineering assessment may require three-dimensional coupled CFD–MHD analysis with realistic boundary conditions. Do not use these results directly as a validated engineering design.
Start with the zero-field, Ha=1 and Ha=10 examples below to compare cases at the same pressure gradient.
Enter B (0–10 T), σ (0.1–10⁷ S/m) and μ (10⁻⁴–1 Pa·s) in the numeric fields. The σ and μ sliders use logarithmic scales.
Set half-gap h (0.005–0.2 m), G (0–10⁵ Pa/m) and ρ (1–20000 kg/m³). Results update automatically, including while the animation is paused.
Read mean velocity and flow ratio, and check the Rm warning. Use Share to copy a URL that saves your inputs.
Reproducible calculation examples
These are reference inputs for checking the analytical solution, not properties of a specific material: σ=1000 S/m, μ=0.1 Pa·s, h=0.05 m (full gap 0.1 m), G=100 Pa/m and ρ=1000 kg/m³. Only B changes.
B=0 T: Ha=0, maximum 1.25 m/s, mean 0.83333 m/s, Q/Q₀=1, Re=833.33.
B=0.2 T: Ha=1, maximum 0.87986 m/s, mean 0.59601 m/s, Q/Q₀=0.71522, Re=596.01.
B=2 T: Ha=10, maximum 0.024998 m/s, mean 0.022500 m/s, Q/Q₀=0.027000, Re=22.500.
At Ha=1, Q/Q₀=3[1−tanh(1)]≈0.71522. The pressure-gradient ratio needed to maintain the same flow is its reciprocal, about 1.3982. Profile flattening is not the same as reduced pressure loss.
Interpretation and validity limits
At large Ha, Hartmann layers next to plate walls perpendicular to B have approximate thickness h/Ha. Do not confuse them with finite-duct sidewall layers or assume the plotted points precisely resolve a very thin layer.
If Rm is not much smaller than one, neglecting the flow-induced magnetic field requires reassessment. A warning appears at Rm≥0.1; that threshold is not a guarantee of suitability for real equipment.
Re uses the full gap 2h as its reference length. Stability in a magnetic field also depends on electrical and wall conditions; an ordinary circular-pipe transition threshold is not applied here.
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