This is a cross-section view of the annular gap the fluid is actually pushed through inside the damper — the microscopic mechanism the lumped Bingham-plastic force law abstracts away. Between two flat walls separated by gap 2B, a pressure gradient G drives flow. The local shear stress grows linearly from zero at the centerline to G·B at the wall:
τ(y) = G·y (force balance on a fluid slab)
Bingham plastic constitutive law:
du/dy = 0 where |τ(y)| ≤ τ_y (rigid plug)
du/dy = -(G·y - τ_y)/η where |τ(y)| > τ_y (yielded, shearing)
Plug half-width: y_p = τ_y / G (fluid stays rigid inside this band)
Integrating that velocity gradient from each wall (u=0, no-slip) inward gives a parabola-like profile in the yielded region that flattens into a flat-topped plug in the middle — the flatter and wider the plug, the more of the channel is moving as a rigid block rather than shearing. The panel solves this the way a real damper does in reverse: you set a piston speed, the sim computes the flow rate that must pass through the gap (continuity, Q = vpiston·Apiston), then numerically inverts the flow-rate integral for the pressure gradient G that produces exactly that flow against the current yield stress τ_y(I) — the same saturating field response as the 3D damper's lumped model.
- y_p — plug half-width. At I=0, τ_y=0 so y_p=0 (pure Newtonian parabola, no plug). Raising the field grows the plug until it can span the whole gap, at which point the fluid locks solid and no flow is possible at that pressure.
- Centerline velocity — how fast the rigid plug core moves; it equals the flow-weighted "top speed" of the whole profile.
- Force — total damper force is the pressure drop across the gap's flow length, ΔP = G·L, times the piston area: F = G·L·Ap. This is the same physical force the 3D lumped model reports, arrived at from the actual flow field instead of an assumed force-velocity curve.
Real-world relevance: this exact slit Poiseuille-flow-with-a-yielded-core analysis (Bird, Stewart & Lightfoot) is how MR/ER damper valve gaps are sized in practice — engineers pick the gap B and length L so the target force range at max current stays within the coil's saturation limit and the fluid's real yield stress (tens of kPa for a fluid like MRF-132DG).