Dynamics of capillary flow in an undulated tube

From biology to engineering, while numerous applications are based on capillary phenomena in tubes having roughened surfaces, such as blood transport, paper-based rapid diagnostics, microfluidic fuel cells, and shale gas transport, the dynamics of such capillary flow remains poorly understood. We present a theoretical model for a circular undulated tube that has an idealized cosine-type inner wall characterized by two key morphological parameters: undulation amplitude and axial wave number. With the tube oriented at an arbitrary angle, we first characterize the apparent contact angle of the fluid as a function of local distortion angle and then establish a theoretical model involving inertia, viscosity, and gravity to describe the dynamics of capillary flow. A dimensionless number combining the three forces is introduced to quantify their influence. The model predictions reveal that, in an undulated tube with large wave numbers, the capillary height in equilibrium state is generally lower than that in a smooth tube of similar dimensions, whereas the reverse holds if the wave number becomes relatively small. When the viscosity of fluid is sufficiently small, capillary oscillation in an undulated tube is alleviated relative to that in a smooth tube, and hence stable capillary flow forms more easily in the former.