Why solve electromagnetics for arc welding?

Arc current creates its own magnetic field and feels an inward force from it (pinch force). Near the electrode current is concentrated, so pressure is high, and the difference from the wider downstream column blows plasma toward the workpiece. At 150 A pressure near the electrode is 2,865 Pa and the jet about 518 m/s. This flow carries heat to the work, so thermal calculations need the electromagnetic field too.

How much does arc pressure depress the weld pool?

At 150 A the arc pushes on the work with 179 Pa. Gravity alone would give a 2.6 mm depression, but in a 3 mm-radius pool surface tension contributes about 1,200 Pa, so the depression is much smaller. Pressure grows with current squared, so at high current the pool depresses significantly and penetration deepens.

What governs flow in the weld pool?

Mainly surface-tension differences with temperature (Marangoni effect) and electromagnetic force from the current. Here the surface-tension flow scale is about 1 m/s and the electromagnetic one about 0.1 m/s — roughly ten times smaller. Trace elements such as sulphur in steel can flip the sign of the surface-tension temperature coefficient, reversing flow and changing penetration depth dramatically.

Arc Welding Electromagnetic-Thermal Coupling

Category: Coupled Analysis – Electromagnetic-Thermal | Revised 2026-10-01

Theory: current squeezing itself

Overview

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Isn't arc-welding analysis just placing a heat source and solving temperature?

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For distortion estimates that is often enough. But to predict penetration depth and shape or bead profile, electromagnetic force must be considered. Current in the arc creates its own magnetic field and feels a force toward the axis. At the electrode tip current is concentrated, so the force is strong and pressure high; downstream the arc widens and pressure drops, and the difference drives plasma hard toward the work. That flow carries heat and presses the surface down. Current also flows through the weld pool, adding electromagnetic force that combines with surface-tension-driven flow to move molten metal. Solving these together is arc-welding electromagnetic-thermal coupling.

Pinch pressure and arc jet

$$ p(R) = \frac{\mu_0 I^2}{4\pi^2 R^2},\qquad \tfrac12\rho v^2 = p(R_1) - p(R_2) $$

I is current and R the radius it flows through; this is the axial pressure in a column of uniform current. The difference between near the electrode (R₁ = 0.5 mm) and the widened downstream column (R₂ ≫ R₁) accelerates the plasma. ρ is plasma density (0.02 kg/m³, illustrative).

Weld-pool flow scales

$$ u_{EM} \sim \sqrt{\frac{p(r)}{\rho_l}},\qquad u_{Ma} \sim \left(\frac{\tau^2 r}{\rho_l\,\mu}\right)^{1/3},\quad \tau = \frac{|d\gamma/dT|\,\Delta T}{r} $$

ρ_l is liquid steel density (7,000 kg/m³), μ viscosity (6×10⁻³ Pa·s), r pool radius (3 mm), dγ/dT the surface-tension temperature coefficient (4.3×10⁻⁴ N/mK) and ΔT the surface temperature difference (1,000 K). Both are order-of-magnitude scales.

Coffee Break Trivia

Same-grade steel, different penetration

In the early 1980s, a puzzling problem was that TIG welding stainless steel of the same grade under the same conditions gave very different penetration depending on the heat of material. Heiple and Roper explained that tiny amounts of sulphur or oxygen flip the sign of the surface-tension temperature coefficient, reversing surface flow in the pool from outward to inward and turning shallow, wide penetration into deep, narrow penetration. Differences of tens of ppm decide the weld's shape.

Worked examples

Example 1: current, pinch pressure and arc jet (fixed radii)

Current	Pressure near electrode	Plasma jet	Pressure on work (2 mm radius)
100 A	1,273 Pa	345 m/s	80 Pa
150 A	2,865 Pa	518 m/s	179 Pa
200 A	5,093 Pa	691 m/s	318 Pa

Pressure scales with current squared and jet speed with current. In practice the near-electrode radius also grows with current, so the increase is gentler. A 150 A, 12 V arc delivers 1,800 W; at 0.7 efficiency, 1,260 W enters the work.

Example 2: weld pool (150 A, 3 mm radius)

Item	Value
Arc pressure	179 Pa
Depression with gravity only	2.61 mm
Surface-tension pressure	1,200 Pa
Electromagnetic flow scale	0.107 m/s
Marangoni flow scale	1.14 m/s

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If Marangoni flow is ten times faster, can electromagnetic force be ignored?

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It depends. Marangoni flow acts along the pool surface, and whether surface flow runs outward or inward sets penetration shape. Electromagnetic force acts throughout the pool, driving flow downward toward the centre. When Marangoni flow is outward (low-sulphur steel), electromagnetic force works to dig the centre deeper, correcting the shape. As current rises, electromagnetic force grows with its square and arc pressure adds in, shifting the balance. The surface-tension coefficient can also change sign with temperature, so outward and inward flows may coexist in one pool. Real analyses therefore include both forces, solve coupled pool flow and temperature, and track free-surface deformation. The estimates are for first identifying which force leads.

Modelling workflow

Give plasma properties (conductivity, thermal conductivity, radiation) as functions of temperature.
Solve potential and magnetic field for current density and Lorentz force.
Solve plasma flow and temperature for heat and pressure on the work.
Solve pool flow and temperature with electromagnetic, Marangoni and buoyancy forces.
Compare predicted penetration with measured cross-section macrographs.

Coffee Break Trivia

“Welds that went shallow with a new material lot”

In automated TIG welding of piping, incomplete root penetration rose as soon as a new material lot was used, though welding conditions were unchanged. Composition checks showed slightly lower sulphur in the new lot, making pool surface flow outward and penetration shallow and wide. Coupled pool analysis was used to find pulsed-current conditions that periodically boost electromagnetic force and pressure, establishing settings that achieve full penetration regardless of lot.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Fixed heat-source distribution	Shape wrong when current changes	Solve arc or calibrate source
Constant surface-tension coefficient	Lot differences not reproduced	Function of composition and temperature
No Lorentz force in pool	High-current penetration wrong	Solve current distribution
Flat fixed pool surface	Arc-pressure effect missed	Track free surface
Constant plasma properties	Heat flow wrong	Temperature-dependent properties

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