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A 2D stochastic fluid-structure interaction problem in compliant arteries with non-zero longitudinal displacement

Krutika Tawri

TL;DR

This paper considers the case where the structure is allowed to have non-zero longitudinal displacement, and applies the stochastic force to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary.

Abstract

In this paper, we study a nonlinear fluid-structure interaction problem driven by a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic wall whose elastodynamics is described by membrane equations. The stochastic force is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known a priori. In particular, we consider the case where the structure is allowed to have non-zero longitudinal displacement.

A 2D stochastic fluid-structure interaction problem in compliant arteries with non-zero longitudinal displacement

TL;DR

This paper considers the case where the structure is allowed to have non-zero longitudinal displacement, and applies the stochastic force to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary.

Abstract

In this paper, we study a nonlinear fluid-structure interaction problem driven by a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic wall whose elastodynamics is described by membrane equations. The stochastic force is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known a priori. In particular, we consider the case where the structure is allowed to have non-zero longitudinal displacement.
Paper Structure (13 sections, 20 theorems, 166 equations, 1 figure)

This paper contains 13 sections, 20 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setup
  3. The deterministic model and a weak formulation
  4. The stochastic framework and definition of martingale solutions
  5. ALE mappings
  6. Definition of martingale solutions
  7. Operator splitting scheme
  8. Definition of the splitting scheme
  9. Approximate solutions
  10. Passing $N\rightarrow\infty$
  11. Tightness results
  12. Almost sure convergence
  13. Passing to the limit $\varepsilon\to 0$

Key Result

Lemma 3.1

(Existence for the structure sub-problem.) Assume that ${\boldsymbol{\eta}}^n$ and ${\bf v}^{n}$ are ${\bf H}^2_0(0,L)$ and ${\bf L}^2(0,L)$ valued $\mathcal{F}_{t^n}$-measurable random variables, respectively. Then there exist ${\bf H}^2_0(0,L)$- valued $\mathcal{F}_{t^n}$-measurable random variabl where corresponds to numerical dissipation.

Figures (1)

  • Figure 1: A realization of the domain

Theorems & Definitions (34)

  • Definition 1
  • Remark 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 1
  • proof
  • Lemma 3.3
  • proof
  • ...and 24 more