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Capillary Rise in Pipes with Variable Cross Section

Isidora Rapajić, Srboljub Simić

TL;DR

The paper addresses capillary rise in pipes with a variable cross section by deriving a generalized Washburn equation from first principles, preserving inertia and gravity while assuming a slowly varying radius $R(z)$. The authors perform local and global balance analyses, introduce a mean cross-section velocity, and obtain a dimensionally consistent ODE that reduces to the classical Washburn model when inertia is neglected. The key contributions are the generalized Washburn equation, its dimensionless formulation with a single parameter $\omega$, and a rigorous comparison to lubrication theory, showing equivalence in the inertia-free limit $\omega\to0$ and distinct inertial dynamics otherwise. Numerical results reveal monotone or oscillatory approaches to equilibrium depending on $\omega$, validating the model against the lubrication limit and illustrating the importance of inertia for tapered geometries. This framework provides a rigorous basis for further analytical study of existence, uniqueness, and stability, and it sets the stage for exploring more complex cross-sectional variations.

Abstract

This work proposes an ODE model for a capillary rise in pipes with variable cross section and compares it to the lubrication theory model. Two key assumptions are made: (1) radius of the pipe varies with axial coordinate, and (2) pipe's convergence angle is small. The model reduction process involves the identification of critical parameters and simplifies the governing equations by neglecting higher-order terms. Under appropriate scaling, it is shown that generalized Washburn's equation for capillary rise in pipes with variable cross section reduces to the known lubrication theory model.

Capillary Rise in Pipes with Variable Cross Section

TL;DR

The paper addresses capillary rise in pipes with a variable cross section by deriving a generalized Washburn equation from first principles, preserving inertia and gravity while assuming a slowly varying radius . The authors perform local and global balance analyses, introduce a mean cross-section velocity, and obtain a dimensionally consistent ODE that reduces to the classical Washburn model when inertia is neglected. The key contributions are the generalized Washburn equation, its dimensionless formulation with a single parameter , and a rigorous comparison to lubrication theory, showing equivalence in the inertia-free limit and distinct inertial dynamics otherwise. Numerical results reveal monotone or oscillatory approaches to equilibrium depending on , validating the model against the lubrication limit and illustrating the importance of inertia for tapered geometries. This framework provides a rigorous basis for further analytical study of existence, uniqueness, and stability, and it sets the stage for exploring more complex cross-sectional variations.

Abstract

This work proposes an ODE model for a capillary rise in pipes with variable cross section and compares it to the lubrication theory model. Two key assumptions are made: (1) radius of the pipe varies with axial coordinate, and (2) pipe's convergence angle is small. The model reduction process involves the identification of critical parameters and simplifies the governing equations by neglecting higher-order terms. Under appropriate scaling, it is shown that generalized Washburn's equation for capillary rise in pipes with variable cross section reduces to the known lubrication theory model.

Paper Structure

This paper contains 26 sections, 61 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Assumptions and local analysis
  3. Assumptions
  4. Local analysis
  5. Mass balance
  6. Momentum balance
  7. Global analysis and the generalized Washburn equation
  8. Global order of magnitude estimates
  9. Global mass balance and mean velocity of the meniscus
  10. Global momentum balance
  11. The generalized Washburn equation
  12. Remark about the initial conditions
  13. Lubrication theory model and its approximation
  14. Equilibrium height
  15. Dimensionless models
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Poiseuille flow in a capillary pipe of variable radius, with meniscus height $h(t)$ at time $t$.
  • Figure 2: Solution $H(T)$ of the generalized Washburn equation \ref{['eq::DLessGeneralisedWashburn']} for different values of critical parameter; $L = 20h_\textrm{e}, R_0 = 1$. Left: $\omega = 0.01$; right: $\omega = 10$.
  • Figure 3: Comparison of the solution $H_{\mathrm{Wash}}(T)$ of the generalized Washburn equation \ref{['eq::DLessGeneralisedWashburn']} to the solution $H_{\mathrm{LubApp}}(T)$\ref{['eq::DLessLubrication-Approx']} and $H_{\mathrm{Lub}}(T)$\ref{['eq::DLessLubrication']}, $L = 20 h_\textrm{e}$, $\omega = 0.01$.
  • Figure 4: Comparison of the solution $H_{\mathrm{Wash}}(T)$ of the generalized Washburn equation \ref{['eq::DLessGeneralisedWashburn']} to the solution $H_{\mathrm{LubApp}}(T)$\ref{['eq::DLessLubrication-Approx']} and $H_{\mathrm{Lub}}(T)$\ref{['eq::DLessLubrication']}, $L = 20 h_\textrm{e}$, $\omega = 10$.