Related Experiment Video
Updated: Jun 25, 2026

The Diffusion of Passive Tracers in Laminar Shear Flow
Published on: May 1, 2018
Analytical modeling of capillary flow in tubes of nonuniform cross section
William W Liou1, Yongqing Peng, Peter E Parker
1Department of Mechanical and Aeronautical Engineering, Western Michigan University, Kalamazoo, MI 49008, USA.
Abstract:
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.
Related Concept Videos
Steady, Laminar Flow in Circular Tubes
Applications of Integration to Find Blood Flow
Uniform Depth Channel Flow
Uniform Depth Channel Flow: Problem Solving
Typical Model Studies
Steady, Laminar Flow Between Parallel Plates

