## 2D Fluid-Structure Interaction in a Micro Channel ###### Micro Channel Flow with FSI Investigation for an Obstacle

This model demonstrates how to set up a fluid-structure interaction problem in COMSOL Multiphysics. It illustrates how fluid flow can deform solid structures and how to solve for the flow in a continuously deforming geometry.

The Fluid-Structure Interaction (FSI) Multiphysics interface combines fluid flow with solid mechanics to capture the interaction between the fluid and the solid structure. A Solid Mechanics interface and a Single-Phase Flow interface model the solid and the fluid, respectively. The FSI couplings appear on the boundaries between the fluid and the solid. The Fluid-Structure Interaction interface uses an arbitrary Lagrangian-Eulerian (ALE) method to combine the fluid flow formulated using an Eulerian description and a spatial frame with solid mechanics formulated using a Lagrangian description and a material (reference) frame.

### Introduction

The following example demonstrates techniques for modeling fluid-structure interactions in COMSOL Multiphysics. It illustrates how fluid flow can deform structures and how to solve for the flow in a continuously deforming geometry using the arbitrary Lagrangian-Eulerian (ALE) technique. The model geometry consists of a horizontal flow channel in the middle of which is an obstacle, a narrow vertical structure (Figure 1). The fluid flows from left to right, except where the obstacle forces it into a narrow path in the upper part of the channel, and it imposes a force on the structure’s walls resulting from the viscous drag and fluid pressure. The structure, being made of a deformable material, bends under the applied load. Consequently, the fluid flow also follows a new path, so solving the flow in the original geometry would generate incorrect results. Figure 1. Fluid flows into this horizontal flow channel from the left, and it enters with a parabolic velocity profile. A narrow vertical structure in the channel (the straight vertical structure) forces the flow into a narrow path. Due to fluid pressure and viscous drag, the originally vertical structure bends. This simulation models the fluid flow in a deformed, moving mesh that follows the movement of the bending structure.

### Model Definition

In this example the flow channel is 100 μm high and 300 μm long. The vertical structure — 5 μm wide, 50 μm high, and with a semicircular top — sits 100 μm away from the channel’s left boundary. Assume that the structure is long in the direction perpendicular to the image.

The fluid is a water-like substance with a density ρ = 1000 kg/m3 and dynamic viscosity η = 0.001 Pa·s. To demonstrate the desired techniques, assume the structure consists of a flexible material with a density ρ = 7850 kg/m3 and Young’s modulus E = 200 kPa.

### Fluid Flow

The fluid flow in the channel is described by the incompressible Navier-Stokes equations for the velocity field, u (u, v), and the pressure, p, in the spatial (deformed) moving coordinate system:

$\displaystyle \rho \frac{{\partial \vec{u}}}{{\partial t}}-\nabla .\left[ {-p{\mathrm I}+\eta \left( {\nabla \vec{u}+{{{\left( {\nabla \vec{u}} \right)}}^{T}}} \right)} \right]+\rho \left( {\left( {\vec{u}-{{{\vec{u}}}_{m}}} \right).\nabla } \right)\vec{u}=\vec{F}$

$\displaystyle -\nabla .\vec{u}=0$

In these equations, I denotes the unit diagonal matrix and F is the volume force affecting the fluid. Assume that no gravitation or other volume forces affect the fluid, so that F=0. The coordinate system velocity is um (um, vm).

At the channel entrance on the left, the flow has fully developed laminar characteristics with a parabolic velocity profile but its amplitude changes with time. At first flow increases rapidly, reaching its peak value at 0.215 s; thereafter it gradually decreases to a steady-state value of 5 cm/s. The centerline velocity in the x direction, uin , with the steady-state amplitude U comes from the equation

$\displaystyle {{u}_{{in}}}=\frac{{U.{{t}^{2}}}}{{\sqrt{{{{{\left( {0.04-{{t}^{2}}} \right)}}^{2}}+{{{\left( {0.1t} \right)}}^{2}}}}}}$

Where t must be expressed in seconds.

At the outflow (right-hand boundary), the condition is p = 0. On the solid (non-deforming) walls, no slip conditions are imposed, u = 0, v = 0, while on the deforming interface the velocities equal the deformation rate, u0 = ut and v0 = vt (the default condition; note that u and v on the right-hand sides refer to the displacement components).

### Structural Mechanics

The structural deformations are solved for using an elastic formulation and a nonlinear geometry formulation to allow large deformations. The obstacle is fixed to the bottom of the fluid channel. All other object boundaries experience a load from the fluid, given by

$\displaystyle {{\vec{F}}_{T}}=-\vec{n}.\left( {-p{\mathrm I}+\eta \left( {\nabla \vec{u}+{{{\left( {\nabla \vec{u}} \right)}}^{T}}} \right)} \right)$

Where n is the normal vector to the boundary. This load represents a sum of pressure and viscous forces.

### Moving Mesh

The Navier-Stokes equations are solved on a freely moving deformed mesh, which constitutes the fluid domain. The deformation of this mesh relative to the initial shape of the domain is computed using Hyperelastic smoothing. At the exterior boundaries of the flow domain, the deformation is set to zero in all directions.

### Results and Discussion

Figure 2 shows the geometry deformation and flow at t = 4 s when the system is close to its steady state. Due to the channel’s small dimensions, the Reynolds number of the flow is small (Re << 100), and the flow stays laminar in most of the area. The swirls are restricted to a small area behind the structure. The amount of deformation as well as the size and location of the swirls depend on the magnitude of the inflow velocity. Figure 2. Flow velocity and geometry deformation at t = 4 s. The streamlines indicate the flow direction and the color indicates flow-velocity magnitude.

Figure 3 shows the mesh velocity at t = 0.15 s. The boundaries of the narrow structure are the only moving boundaries of the flow channel. Therefore the mesh velocity also has its largest values near the structure. Depending on the current state of the deformation — whether it is increasing, decreasing or stationary — the mesh velocity can have a very different distribution. Figure 3. Mesh velocity (arrows) and mesh and geometry deformation at t = 0.15 s.

Figure 4 further illustrates this point; it compares the average inflow velocity to the horizontal mesh velocity and the horizontal mesh displacement just beside the top of the structure. Most of the time the deformation follows the inflow velocity quite closely. Whenever the inflow velocity starts to decrease, the deformation also decreases, which you can observe as the negative values on the horizontal mesh velocity. Toward the end of the simulation, when inflow and structure deformation approach their steady-state values, the mesh velocity also decreases to zero. Figure 4. Inflow velocity, horizontal mesh velocity, and mesh deformation. The blue curve shows the average x direction velocity at the inflow boundary (m/s); the green shows 104×mesh displacement in the x direction (dx_ale; m) at the geometry point (1.05·10-4, 0.5·10-4); and the red curve shows 103×mesh velocity in the x direction (xt; m/s), also at the point (1.05·10-4, 0.5·10-4).

Figure 5 compares the meshes at different times. The first image shows the initial mesh, which you generate prior to solving the model. This mesh is equally distributed around the top of the structure. The second image shows the mesh in its deformed form. Because the structure deforms more in the horizontal direction, the mesh also changes more in this direction: On the left, the mesh elements are stretched; on the right, they are compressed in the x direction. Figure 5. Geometry and mesh near the top of the structure at t = 0 s and 2 s.

### Notes about the COMSOL Implementation

This example implements the model using Fluid-Structure Interaction interface. By default the Fluid-Structure Interaction interface treats all domains as fluid. Activate solid material model node in the area of the narrow structure. To get a more accurate computation of the large strains, large deformation analysis is the default setting. The interface automatically identifies the fluid-solid interaction boundaries and assigns the boundary condition to those boundaries.

#### – Simulation Details:

Physics: Fluid-Structure Interaction > Fluid-Solid Interaction (Laminar Flow + Solid Mechanics)

Study: Time Dependent

– The Moving (Deformable) Mesh is used for Fluid Domain.

Materials for the fluid flow:

Materials for the solid mechanics:

Source: COMSOL Blogs, Models, Cyclopedia
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