The ductus arteriosus is one of several shunts in the cardiovascular system. It is a small vessel connecting the aortic arch and pulmonary artery that allows blood to bypass the pulmonary circulation. It is open during foetal development because the foetal lungs cannot function and oxygenation of the blood occurs by exchange with the maternal blood in the placenta. Normally it closes a few days after birth; however, in a small number of people closure does not occur, leading to a condition known as patent ductus arteriosus. In this paper our aim is to investigate the resulting cardiovascular effects. We develop a mathematical model of the haemodynamics in three different idealised geometries by assuming that the entry flow is irrotational and remains so in the core until at least the shunt position. We argue that separation or diffusion of vorticity into the core flow is delayed due to the high frequency associated with the pulsatile component of the flow profile. The analysis uses complex potential theory, Schwarz-Christoffel transformations, conformal mappings and Fourier series. The main results are based on the assumption that the flow in patients with patent ductus arteriosus is similar to the flow in healthy adults, and we apply this assumption using boundary conditions that are representative of physiological values in healthy adults. The model suggests that the pressures in the aorta and pulmonary artery are likely to equalise, that the shear stress increases near the edges of the shunt and that backflow of large volumes may occur from the pulmonary artery into the aorta or towards the ventricles due to the presence of the patent shunt. Our results strongly suggest that an abnormal compensatory physiology develops in patients with patent ductus arteriosus.