Pressure-driven flows over a square cavity are studied numerically for Bingham plastics exhibiting a yield stress. The problem is encountered whenever pressure measurements are made by a drilled-hole based pressure transducer. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. Newtonian results are obtained for a wide range of Reynolds numbers $(0<Re\u22641000)$ for the cavity vortex position and intensity, and the excess pressure drop (entrance correction) in the system. To reduce the length of the computational domain for highly convective flows, an open boundary condition has been implemented at the outflow. For viscoplastic fluids the emphasis is on determining the extent and shape of yielded/unyielded regions along with the cavity vortex shape, size, and intensity for a wide range of Bingham numbers $(0\u2264Bn<\u221e)$. The entrance correction is found to be an increasing sigmoidal function of the Bn number, reaching asymptotically the value of zero. It is shown that for viscoplastic fluids not exhibiting normal stresses in shear flow (lack of viscoelasticity), the hole pressure is zero opposite the center of the hole. Thus, any nonzero pressure hole measured by this apparatus would signify the presence of a normal-stress difference in the fluid.