The axial valley along a slow spreading mid-ocean ridge may be explained by a vertical pressure gradient due to viscous flow of asthenosphere upwelling in a relatively narrow conduit beneath the ridge axis. Along a ridge segment, the axial valley floor deepens by as much as 2--3 km over distances of several tens of kilometers approaching a ridge-transform intersection. This deepening may be explained, in part, by a pressure gradient associated with horizontal flow in the ridge axis conduit. Vanishing vertical velocity on a conduit end wall at a ridge-transform intersection results in reduced vertical flow within about one conduit width of the intersection. Horizontal flow along the conduit, toward the intersection, must occur to form lithosphere at a uniform rate along the ridge axis. A simple model with a conduit of uniform width that terminates with vertical planar endwalls at ridge-transform intersections is considered. The model is based on an analytical solution for flow in a narrow conduit and boundary layer approximations for the flow structure near the vertical end walls. This model indicates that the induced horizontal flow and pressure gradient extend for a distance along the ridge axis of several lithosphere thicknesses from an intersection. For reasonable values of the conduit width and asthenosphere viscosity the model predicts that induced horizontal flow can contribute significantly to the deepening of an axial valley approaching an intersection.