The diffuse interface approximation to fluid-structure interaction

Abstract

We consider a fluid-structure interaction problem in the Eulerian, phase-field formulation. The problem is described using the Navier--Stokes equations for a viscous, incompressible fluid, coupled with the incompressible hyperelasticity system, both written in the Eulerian coordinates. This allows the problem to be written in a unified formulation, using a single field for the fluid and structure velocities. To track the position of the domain, we use a phase-field approach, resulting in a coupled Cahn--Hilliard--Navier--Stokes-type of problem for the diffuse interface fluid-structure interaction. Under certain assumptions, we prove the convergence of the diffuse interface model to the sharp interface fluid-structure interaction problem. To solve the problem numerically, we propose a novel, strongly coupled, second-order partitioned computational method where the system is decoupled into the Cahn--Hilliard problem, the transport problem for the left Cauchy--Green deformation tensor, and the Navier--Stokes problem. The problems are solved iteratively until convergence at each time step. The performance of the method is illustrated on two computational examples.

Figures (7)

• Figure 1: An example of fluid and structure domains at time $t$.
• Figure 2: Relative errors computed at the final time.
• Figure 3: Norms of the solution used in Section \ref{['modelingerror']}.
• Figure 4: The phase-field function, $\phi$, indicating the position of the elastic structure in Case 1.
• Figure 5: The velocity streamlines colored by the velocity magnitude in Case 1.

Theorems & Definitions (11)

• Theorem 2.1
• proof
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• proof