[Paper Review] The Flow of Newtonian Fluids in Axisymmetric Corrugated Tubes

Taha Sochi|arXiv (Cornell University)|Jun 8, 2010

Rheology and Fluid Dynamics Studies20 references5 citations

TL;DR

This paper presents an analytical method to derive the pressure drop–flow rate relationship for Newtonian fluids in axisymmetric corrugated tubes with varying cross-sections. By generalizing the Hagen-Poiseuille law through integration of 1/r⁴ over axial variation, it derives exact analytical expressions for conical, parabolic, hyperbolic cosine, and sinusoidal tube geometries, enabling precise flow characterization in non-uniform capillaries with applications in fluid dynamics and porous media modeling.

ABSTRACT

This article deals with the flow of Newtonian fluids through axially-symmetric corrugated tubes. An analytical method to derive the relation between volumetric flow rate and pressure drop in laminar flow regimes is presented and applied to a number of simple tube geometries of converging-diverging nature. The method is general in terms of fluid and tube shape within the previous restrictions. Moreover, it can be used as a basis for numerical integration where analytical relations cannot be obtained due to mathematical difficulties.

Motivation & Objective

• To develop a general analytical framework for predicting pressure drop in Newtonian fluid flow through axisymmetric, non-uniform tubes with varying radii.
• To extend the Hagen-Poiseuille law to tubes with converging-diverging profiles by integrating the inverse fourth power of the radius along the axial coordinate.
• To provide exact analytical solutions for pressure-flow relationships in specific geometries (conical, parabolic, hyperbolic cosine, sinusoidal) to enable precise modeling without numerical approximation.
• To establish a foundation that can be adapted for non-Newtonian flows and extended to non-axially symmetric geometries.
• To validate the analytical results through dimensional consistency and numerical verification, ensuring reliability for practical applications.

Proposed method

• The method generalizes the Hagen-Poiseuille equation by expressing the infinitesimal pressure drop as dP = (8μQ / π) · dx / r⁴ for a differential tube element.
• The total pressure drop is computed via integration: P = (8μQ / π) ∫₀ᴸ dx / r⁴(x), where r(x) defines the axial radius variation.
• For each tube geometry (conical, parabolic, etc.), the radius function r(x) is analytically defined over the tube length L, and the integral is evaluated symbolically.
• The integration yields closed-form analytical expressions for pressure drop P as a function of flow rate Q, fluid viscosity μ, and geometric parameters (R_max, R_min, L).
• The method is validated by checking dimensional consistency and cross-verified with numerical integration for accuracy.
• The approach is extendable to non-Newtonian fluids and other regular geometries, serving as a basis for numerical integration when analytical solutions are intractable.

Experimental results

Research questions

• RQ1 What is the analytical relationship between pressure drop and volumetric flow rate in a Newtonian fluid flowing through a conically shaped corrugated tube?
• RQ2 How can the Hagen-Poiseuille law be generalized to tubes with parabolic radius variation along the axis?
• RQ3 What is the exact pressure-flow relationship for a tube with a hyperbolic cosine radius profile?
• RQ4 Can an analytical solution be derived for a sinusoidal corrugated tube, and how does it compare to other geometries?
• RQ5 To what extent can this method be generalized to non-Newtonian fluids or non-axially symmetric geometries?

Key findings

• For a conical tube, the pressure drop is P = (8LQμ / 3π) · [1/R_min³ - 1/R_max³] / (R_max - R_min), derived from 1/r⁴ integration.
• For a parabolic tube, the pressure drop is P = (4LQμ / π) · [1/(3R_min R_max³) + 5/(12R_min² R_max²) + 5/(8R_min³ R_max) + (5/8) · arctan(√(R_max² - R_min²)/R_min) / (R_min³ √(R_max² - R_min²))].
• For a hyperbolic cosine tube, the pressure drop is P = (8LQμ / 3π R_min⁴) · [tanh(acosh(R_max/R_min)) · (sech²(acosh(R_max/R_min)) + 2)] / acosh(R_max/R_min), incorporating hyperbolic functions.
• For a sinusoidal tube spanning one wavelength, the pressure drop is P = (LQμ / 2π) · [2(R_max + R_min)³ + 3(R_max + R_min)(R_max - R_min)²] / (R_max R_min)⁷ᐟ², derived using trigonometric integral identities.
• All derived expressions are dimensionally consistent and verified via numerical integration, confirming their analytical validity.
• The method provides exact analytical solutions for multiple geometries, offering a foundation for modeling complex flow in porous media, viscoelastic fluids, and non-Newtonian systems.