We consider the problem of capillary imbibition into an axisymmetric tube for which the tube radius decreases in the direction of increasing imbibition. For tubes with constant radius, imbibition is described by Washburn’s law (referred to here as the ‘BCLW law’ to recognize the contributions of Bell, Cameron and Lucas that pre-date Washburn). We show that imbibition into tubes with a power-law relationship between radius and length generally occurs more quickly than imbibition into a constant radius tube. By a suitable choice of the shape exponent it is possible to decrease the time taken for the liquid to imbibe from one position to another by a factor of two compared to the BCLW law. We then show that a further, small, decrease in the imbibition time may be obtained by using a tube consisting of a cylinder joined to a cone of three times the cylinder length. For a given inlet radius, this composite shape attains the minimum imbibition time possible. We confirm our theoretical results with experiments on the tips of micropipettes and discuss the possible significance of these results for the control of liquid motion in microfluidic devices.
