Minimodule: Modeling Blood Flow in Vessels

Abstract

We introduce cardiovascular fluid mechanics and explore a model of blood flow through a cylindrical vessel. Initially, we arrive at the Poiseuille-Hagen flow equation, and then we incorporate the Casson equation to account for the non-Newtonian properties of blood. The resulting second-order linear ordinary differential equations (ODEs) can be solved analytically and/or numerically. This Minimodule can be either a challenging group project for an introduction to differential equations course or a problem set for an undergraduate fluid mechanics course. This project could also be adapted for a graduate-level fluid mechanics course by beginning with the Navier-Stokes equations and requiring students to determine the applicable assumptions for the flow themselves.

Note: The information below was created with the assistance of AI.

Level of Mathematics

Overall Level:
Upper Undergraduate (Primary) - Intro Graduate (Adaptable)

Evidence:

• Explicitly designed for:
• “an introduction to differential equations course”
• “an undergraduate fluid mechanics course”
• “could also be adapted for a graduate-level course”

Mathematical sophistication:

• Second-order linear ODEs
• Multivariable calculus (gradients, divergence)
• Navier–Stokes equations (simplified)
• Analytical + numerical solution methods

Interpretation:

• Core level: 2nd–3rd year undergraduate
• Advanced extension: graduate fluid mechanics

Subject Matter
Core Mathematical Topics:

• Differential Equations
• Second-order ODEs
• Boundary value problems
• Calculus (Multivariable)
• Derivatives, gradients, cylindrical coordinates
• Fluid Mechanics Mathematics
• Navier–Stokes equations
• Poiseuille flow
• Nonlinear Modeling
• Casson equation (non-Newtonian fluids)

Supporting Topics:

• Mathematical modeling
• Piecewise-defined functions (plug flow region)
• Integration and physical interpretation

Application Areas
Primary Application:

• Biomedical Engineering / Cardiovascular Modeling
• Blood flow in arteries, veins, capillaries

Broader Applications:

• Fluid mechanics
• Pipe flow, viscous flow systems
• Medical diagnostics
• Modeling non-invasive blood flow
• Biomechanics
• Computational fluid dynamics (CFD)

Real-world relevance:

• Modeling replaces invasive measurement in medicine
• Used to understand:
• Aortic flow
• Vessel behavior
• Effects of blood properties

Prerequisites
Required Background:

Mathematics:

• Calculus I–III:
• Derivatives and integrals
• Multivariable calculus basics
• Differential equations:
• Solving second-order ODEs
• Algebraic manipulation

Physics / Engineering:

• Basic mechanics (Newton’s laws)
• Intro fluid mechanics concepts:
• Pressure, viscosity, shear stress

Recommended:

• Linear algebra (basic familiarity)
• Numerical methods (optional extension)

Advanced (for full depth):

• Partial differential equations
• Vector calculus (∇, divergence, Laplacian)

Correlation to Mathematics Standards

US Common Core (High School)
Limited direct alignment (above standard HS level), but connects to:

• HSS-MD: Modeling with mathematics
• HSA-CED: Creating equations
• HSA-REI: Solving equations

Mainly through modeling and interpretation, not technical depth

AP Courses
AP Calculus BC

• Differential equations (intro level)
• Integration and modeling
• Applications of derivatives

AP Physics (C: Mechanics / Fluids concepts)

• Fluid flow intuition (not formal Navier–Stokes)

Undergraduate Standards
Strong alignment with:

• Differential Equations courses
• Fluid Mechanics (Engineering)
• Mathematical Modeling courses
• Applied Mathematics / Biomechanics

Mathematical Practices (Process Standards)
This module strongly emphasizes:

• MP4: Model with mathematics
• Real-world system → equations → solution
• MP2: Reason quantitatively
• Physical meaning of variables and parameters
• MP1: Problem solving
• Multi-step derivations and interpretation
• MP7: Structure
• Recognizing simplifications (assumptions → equations)