This paper supplements an earlier study (Singh 1974) of the steady case of ‘Entry flow in a curved pipe’. Here we consider an entrance profile of the form \[ w = Q(t)/(1+\delta r\cos\alpha), \] which is physiologically more relevant for blood being pumped from the left ventricle into the ascending aorta. A boundary-layer analysis is applied to determine the effects of curvature and an adverse pressure gradient (associated with the primary flow) on the wall shear. The study shows how the negative wall shear and backflow near the wall develop during the decelerating phase of the cycle as the boundary layer grows. The analysis shows how the increasing effect of the secondary flow due to curvature draws off slower moving fluid azimuthally from the outer bend to the inner bend; this induces a cross-flow of faster moving fluid from the inner bend to the outer bend which results in a thinning of the boundary layer at the outer bend and a thickening at the inner bend. This implies an increased wall shear at the outer bend compared with that at the inner bend as the flow develops further downstream; this is in contrast with the initial stages of the motion near the entrance where the higher wall shear occurs at the inner bend owing to the external flow and to geometric factors. The analysis shows that the displacement effect of the boundary layer on the core is not significant because the boundary layer remains thin, about one-tenth of the tube diameter.