Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

Fang-Hua Lin, Jiajun Tong

Research output: Contribution to journalArticle

Abstract

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear nonlocal equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an H5/2-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parametrized circular configuration, then a global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as t→+∞. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - Jan 1 2018

Fingerprint

Immersed Boundary
Boundary Problem
Stokes
Solvability
Two Dimensions
Strings
Configuration
Lagrangian Coordinates
Nonlinear equations
Nonlocal Equations
Stokes Equations
Stokes Flow
Fundamental Solution
Three-dimension
Nonlinear Equations
Existence and Uniqueness
Converge
Closed
Motion
Formulation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Solvability of the Stokes Immersed Boundary Problem in Two Dimensions. / Lin, Fang-Hua; Tong, Jiajun.

In: Communications on Pure and Applied Mathematics, 01.01.2018.

Research output: Contribution to journalArticle

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