Highly filled rubber compounds exhibit a unique rheological behavior, which is affected by its filler–filler and filler–matrix interactions leading to pronounced nonlinear viscoelasticity. The necessity to consider these characteristics in rheological testing and modeling, adds further complexity providing universally valid numerical descriptions. In the present study, the pressure driven contraction and capillary flow of a carbon black filled hydrogenated acrylonitrile–butadiene rubber compound is studied both experimentally and numerically. Rheological testing indicates no pronounced slippage at the wall but a shear sensitive plug flow at the centerline. The viscoelastic Kaye-Bernstein–Kearsley–Zapas/Wagner, the viscoplastic Herschel–Bulkley and the viscous power-law models are used in computational fluid dynamic simulations aiming to predict measured pressure drops in an orifice and various capillary dies. Viscoelastic modeling is found of particular importance describing contraction flow dominated areas, whereas viscous models are able to predict pressure drops of capillary flows well.