We have studied the flow of superfluid 3He-B forced through small apertures. There are unexpectedly large dissipative currents, which can be described by two independent processes. One process involves the creation of quasiparticles within the aperture and their subsequent acceleration in the ambient pressure gradient. The second process involves the dissipative precession of a texture in a geometry-induced anisotropic order parameter. For both mechanisms we make a simple estimate of the relevant effect and find these agree well with the data.