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Bifurcating periodic solution from a blood flow with variable body force: via Crandall-Rabinowitz bifurcation theorem

Yuchao He, Yongli Song, Yonghui Xia

TL;DR

This work addresses the existence of time-periodic traveling-wave solutions in a two-dimensional, viscous incompressible blood flow with a free boundary and a variable body force. The authors reformulate the free-boundary problem as a fixed-domain quasilinear elliptic problem via a height-function transformation, first identifying a laminar baseline and then analyzing a linearized eigenproblem that leads to a critical parameter where bifurcation occurs. By establishing a one-dimensional kernel and a transversality condition through a Sturm–Liouville framework and the Rayleigh quotient $\mu(\xi)$, they apply the Crandall–Rabinowitz theorem to prove the existence of a local $C^1$ curve of small-amplitude periodic solutions emanating from the laminar flow, under a key inequality $Q>2c_3+2\sqrt{3}c_2$. The analysis extends prior studies by treating harmonic vorticity and non-constant body forces in a free-boundary setting, with implications for rigorous understanding and numerical modeling of cardiovascular flows.

Abstract

This paper employs the Crandall-Rabinowitz bifurcation theorem to investigate the existence of periodic solutions in blood flow propagating through vessels with free boundary conditions. It is rigorously proved that a local $C^1$-curve of small-amplitude periodic solutions is bifurcated. In contrast to previous studies on periodic flows that primarily focus on constant vorticity, our work emphasizes the bifurcation analysis of periodic solutions in blood flow with harmonic vorticity distribution and external body forces.

Bifurcating periodic solution from a blood flow with variable body force: via Crandall-Rabinowitz bifurcation theorem

TL;DR

This work addresses the existence of time-periodic traveling-wave solutions in a two-dimensional, viscous incompressible blood flow with a free boundary and a variable body force. The authors reformulate the free-boundary problem as a fixed-domain quasilinear elliptic problem via a height-function transformation, first identifying a laminar baseline and then analyzing a linearized eigenproblem that leads to a critical parameter where bifurcation occurs. By establishing a one-dimensional kernel and a transversality condition through a Sturm–Liouville framework and the Rayleigh quotient , they apply the Crandall–Rabinowitz theorem to prove the existence of a local curve of small-amplitude periodic solutions emanating from the laminar flow, under a key inequality . The analysis extends prior studies by treating harmonic vorticity and non-constant body forces in a free-boundary setting, with implications for rigorous understanding and numerical modeling of cardiovascular flows.

Abstract

This paper employs the Crandall-Rabinowitz bifurcation theorem to investigate the existence of periodic solutions in blood flow propagating through vessels with free boundary conditions. It is rigorously proved that a local -curve of small-amplitude periodic solutions is bifurcated. In contrast to previous studies on periodic flows that primarily focus on constant vorticity, our work emphasizes the bifurcation analysis of periodic solutions in blood flow with harmonic vorticity distribution and external body forces.
Paper Structure (8 sections, 4 theorems, 76 equations, 2 figures)

This paper contains 8 sections, 4 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Equivalent reformulation of blood flow model
  3. Preliminaries and assumptions
  4. Equivalent reformulations
  5. Main result
  6. Laminar solution
  7. Linearised problem
  8. Local bifurcation

Key Result

Lemma 3.1

If there exists $\beta\in (3,\infty]$ such that where and then there exists $2\Gamma_{\max}+\frac{c_3}{2}>\xi>2\Gamma_{\max}$ such that $\mu(\xi)\leq-1$.

Figures (2)

  • Figure 1: Cross section of blood flow in blood vessels.
  • Figure 2: Sketch of the blood flow region transformation.

Theorems & Definitions (7)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4