Exact discharge matching can conceal velocity errors in regularized Herschel–Bulkley duct flow

Abstract A regularized yield-stress model can reproduce an ideal discharge through compensation between constitutive approximations. We characterize this compensation at finite parameters for fully developed Herschel–Bulkley pipe and plane-channel flow. For every finite exponential regularization parameter, positive subcritical yield ratio 0 < Y < 1, and shear thickening index n > 1, there is a unique shear-rate smoothing parameter that matches the ideal discharge exactly. At that match the centerline velocity remains too large, and the velocity error changes sign once inside the section. Shear-thinning and Bingham indices do not admit this cancellation for the stated law. A deterministic matrix of 96 discharge matches, 576 pressure-transfer evaluations, and 128 control tests the result. Despite the discharge agreement to numerical precision, 43 matches exceed a 1% centerline-error threshold; within the subset with dimensionless smoothing, at most 0.01, 6 of 58 do so. A pipe case with n = 1.5, Y = 0.99, M = 104, and smoothing E ≃ 0.02522 has a 14.1% centerline error. Holding dimensional regularization parameters fixed while changing pressure exposes additional discharge discrepancies. The previously derived near-yield limit explains the shrinking pressure interval in which exponential core flow competes with the ideal yielded shell. These exact-solution benchmarks test verification based on integral discharge; they do not establish accuracy in general geometries or experimental validity.