Adjust magnetic field B and fluid conductivity σ to change the Hartmann number Ha. Watch the velocity profile transform from parabolic Poiseuille flow to flat plug flow in real time.
What exactly is Hartmann Flow? I see it involves magnets and fluids, but how do they interact?
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Basically, it's the flow of an electrically conductive fluid, like liquid metal or saltwater, through a duct with a magnetic field applied across it. The moving fluid generates electric currents, which then interact with the magnetic field to create a braking force called the Lorentz force. In this simulator, you can see how that force reshapes the flow by adjusting the Magnetic Flux Density B slider.
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Wait, really? So the magnetic field can actually flatten the flow profile? Why does it change from a curved shape to a nearly flat one?
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Exactly! Without a magnetic field, you get classic parabolic Poiseuille flow—fast in the middle, slow at the walls. The magnetic braking force is strongest in the center where the fluid is fastest, and weaker near the walls. This evens out the velocity. Try it: set B to zero and see the parabola. Then crank it up high and watch it become "plug flow." The key parameter is the Hartmann number, which you control with B, σ, and μ.
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So the Hartmann number is just a combination of my slider settings? What does a high Ha physically mean for the fluid?
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Precisely. It's the dimensionless ratio of magnetic to viscous forces. A high Ha means magnetic forces dominate, leading to that flat profile with thin boundary layers at the walls. For instance, in a liquid-metal fusion reactor blanket, Ha can be in the thousands! In the simulator, use the Fluid Preset dropdown to "Liquid Sodium" and you'll instantly get a high Ha, showing you the extreme plug flow used in real engineering.
Physical Model & Key Equations
The core of this simulator is the Hartmann number, which predicts how the flow will behave. It combines all the physical properties you can adjust.
$$Ha = B \cdot h \sqrt{\frac{\sigma}{\mu}}$$
B is Magnetic Flux Density (Tesla), h is Duct Half-Height (m), σ is Electrical Conductivity (S/m), and μ is Dynamic Viscosity (Pa·s). A high Ha (>10) means magnetic effects are strong.
The velocity profile equation solves the balance between pressure, viscosity, and the electromagnetic Lorentz force. It shows how Ha directly shapes the flow.
u(y) is the velocity at position y from the centerline, and -dP/dx is the constant Pressure Gradient driving the flow. The cosh terms are hyperbolic cosines, which generate the flat profile when Ha is large.
Frequently Asked Questions
As Ha increases, the Lorentz force becomes stronger and the flow velocity in the central region becomes more uniform. The profile changes from a parabolic shape to a plug flow, and you can observe in real time that the velocity gradient near the wall becomes steeper.
Yes, both can be adjusted independently using sliders. Since Ha is proportional to the product of B and σ, even for the same Ha, different combinations of B and σ result in different contributions of the Lorentz force, leading to subtle differences in the velocity distribution.
It is applied to engineering fields that require control of conductive fluid flow in magnetic fields, such as the design of blankets for nuclear fusion reactors using liquid metal coolants, and the analysis of electromagnetic pumps and electromagnetic flowmeters.
Since this tool plots the analytical solution, it perfectly matches the theoretical values. However, in actual experiments, non-uniformities in fluid properties and end effects occur, so please use it as a reference for understanding qualitative trends.
Real-World Applications
Liquid-Metal Cooling for Nuclear Reactors: Advanced fission and fusion reactor designs use liquid metals like sodium or lead-bismuth to carry away intense heat. The magnetic fields from the reactor itself alter the flow, which must be accounted for in pump design and safety analysis to ensure proper cooling.
MHD Pumps & Flowmeters: These devices have no moving parts. By applying a magnetic field and running a current through the fluid, you can pump it silently—useful for corrosive liquid metals. Conversely, by measuring the voltage induced by a moving fluid in a magnetic field, you can measure its flow rate.
Fusion Reactor Blankets: In a tokamak, the blanket surrounding the plasma is often a flowing liquid lithium or molten salt. It must breed fuel (tritium) and withstand huge magnetic fields. Understanding Hartmann flow is critical for designing these blankets to ensure uniform flow and heat transfer.
Geophysics & Astrophysics: The Earth's outer core is a rotating, convecting liquid iron alloy, and its motion generates our planetary magnetic field. On a stellar scale, MHD principles govern the behavior of solar flares and the dynamics of interstellar plasma clouds.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few points that are easy to misunderstand. First, you might think "increasing the magnetic field increases the flow velocity," but that's incorrect. In this tool's settings, the "pressure gradient" is fixed, right? This simulates a condition where a pump applies a constant pushing pressure. Under that condition, strengthening the magnetic field B increases the braking force from the Lorentz force, so the average flow velocity actually decreases. Note that the "flattening profile" you see in the animation happens because the velocity near the wall increases while the central velocity is maintained, but the total flow rate (average velocity) is decreasing.
Next, regarding the material preset selection. Don't assume the electrical conductivity σ determines everything. Looking at the Hartmann number formula $Ha = B h \sqrt{\sigma / \mu}$, you see the kinematic viscosity μ is in the denominator. For example, mercury has a higher σ than sodium, but its μ is also larger, so their contributions to Ha cannot be compared simply. When evaluating new materials in practice, the golden rule is to use accurately measured values for both σ and μ.
Finally, remember this calculation is an "idealized academic model." In actual electromagnetic pump design, circular pipes are almost always used instead of two-dimensional parallel plates, the magnetic field is not uniform, and inlet/outlet effects cannot be ignored. After grasping the fundamental behavior with this tool, you need to move on to more realistic 3D coupled CFD-MHD analysis. It's risky to use this simulator's results directly for real-world design.
Set magnetic flux density (B) in Tesla and electrical conductivity (sigma) in S/m for your working fluid—liquid sodium at 600K typically uses B=0.5–2.0 T and sigma=7×10⁵ S/m
Adjust dynamic viscosity (Pa·s) and channel height (mm); for molten salt at 700°C, viscosity ≈0.002 Pa·s with 50mm ducts
Observe the Hartmann number (Ha) increase as magnetic field strengthens; at Ha>10, velocity profile transitions from parabolic to blunt plug flow, with peak velocity u_max shifting downward
Monitor Reynolds (Re) and magnetic Reynolds (Re_m) numbers; flow ratio Q/Q�0 indicates pressure-drop reduction compared to non-magnetic baseline
Worked Example
Liquid lithium in a 30mm channel with B=1.5 T, sigma=3.5×10⁶ S/m, viscosity=0.0015 Pa·s. Hartmann number Ha = B·H·√(sigma/rho·nu) ≈ 45. Maximum velocity drops from 2.8 m/s (no field) to 1.2 m/s (magnetic), while average velocity u_avg remains 0.95 m/s due to plug-like core formation. Magnetic Reynolds Re_m=U·H·mu₀·sigma ≈ 2100 indicates strong coupling. Flow ratio Q/Q₀≈0.82 shows 18% pressure-drop penalty from Lorentz braking.
Practical Notes
Aluminum smelter cells operate at Ha=200–500; extreme magnetic fields suppress turbulence and enable efficient DC current distribution while flattening velocity gradients
Liquid-metal blankets for fusion reactors (lithium, FLiBe) use Ha=50–150; plug-flow regime reduces thermal mixing and allows precise MHD pump design for tritium breeding
Increasing sigma or B beyond Ha=30 yields diminishing gains; focus instead on reducing duct height or exploiting Lorentz-force stabilization for flow control in high-temperature loops
Re_m <1 signals negligible magnetic diffusion; check that B·L·U product is sufficient for measurable MHD effects in your geometry