Analytical modeling of capillary flow in tubes of nonuniform cross section

The interface rise for the flow in a capillary with a nonuniform cross section distribution along a straight center axis is investigated analytically in this paper. Starting from the Navier–Stokes equations, we derive a model equation for the time-dependent rise of the capillary interface by using an approximated three-dimensional flow velocity profiles. The derived nonlinear, second-order differential equation can be solved numerically using the Runge–Kutta method. The nonuniformity effect is included in the inertial and viscous terms of the proposed model. The present model is validated by comparing the solutions for a circular cylindrical tube, rectangular cylindrical microchannels, and convergent–divergent and divergent–convergent capillaries. The validated model has been applied to capillaries with parabolic varying wall, sinusoidal wall, and divergent sinusoidal wall. The inertial and viscous effects on the dynamic capillary rise and the equilibrium height are investigated in detail.

Graphical abstractA new analytical model for the time-dependent rise of the capillary interface in nonuniform cross-sectional tubes.Figure optionsDownload full-size imageDownload as PowerPoint slide