The flow of liquid helium II has been investigated under gradients of pressure and temperature in slits of 1$\mu $ diameter. Besides the flow rate, the heat current and the pressure difference at the ends of the slit, the pressure at an intermediate point within the slit has been determined. It was found that for superflow the entire drop in pressure and temperature occurs at the narrowest place in the slit. In the remainder of the slit mass flow takes place under effectively zero gradient of pressure or temperature. The experiments also indicate the existence of a critical flow rate beyond which frictional dissipation makes its appearance. The critical rate was determined by four different criteria which yielded consistent results. The temperature dependence of the critical rate is similar to that observed in the helium II film. With flow under a temperature gradient and for higher flow rates the hydrostatic pressure within the slit was found to drop below that at the ends and an explanation for this effect has been suggested. Some experiments with wider slits have shown that in these even for small velocities the transport is a mixture of superflow and normal flow which renders the phenomena very complex.