COMSOL 6.4 - Viscoelastic Flow Through a Channel with a Flexible Wall

Viscoelastic Flow Through a Channel with a Flexible Wall

The correct description of blood circulation requires an understanding of its complex rheological behavior and its interaction with the blood vessel walls. Depending on the size of the vessel, we may consider microcirculation as the flow in arterioles, venules, and capillaries, and macrocirculation as the flow in larger arteries. In macrocirculation, inertia effects tend to be dominant and the non-Newtonian behavior of the blood plays a lesser role. In microcirculation, inertia effects are smaller and we need to include the rheological behavior of blood.

In Ref. 1 and Ref. 2, a benchmark geometry for fluid-structure interaction with a Newtonian liquid in a thin two-dimensional channel, partially bounded by an elastic wall, was extended to viscoelastic flow to serve as a preliminary study of microcirculation in vessels where capturing the elastic nature of the wall and the viscoelastic character of blood is important. This models reproduces their results using the Viscoelastic Flow interface and the Fluid-Structure Interaction multiphysics coupling.

This application requires the Polymer Flow Module and the Structural Mechanics Module.

Model Definition

The geometry of the model is depicted in Figure 1 and Table 1. The flow is from the left to the right, and it is affected by an elastic section of the wall which is loaded with a uniform external pressure Pext = 1.755 Pa. The upper wall has a thickness b = 0.1 mm, and is made of a solid material obeying Hooke’s law with a Young modulus of 35.9 kPa and a Poison ratio of 0.45. The flow enters from the left with an average velocity Uin = 0.03 m/s, and reaches an outlet pressure Pout = 0 Pa at the right-hand side of the channel. A no-slip boundary condition is enforced at the walls.

Figure 1: Channel geometry.

Table 1: Dimensions of the geometry.
Parameter	Value
D	0.01[m]
b	0.01*D
Ld	5*D
Lw	5*D
Lu	30*D

The fluid is assumed to be an Oldroyd-B fluid with density 1000 kg/m3. The solvent viscosity ratio is β = μs/μ0 = 0.5, with μp = μs = 0.5 mPa⋅s. These parameters correspond to an inlet Reynolds number of

Studies of viscoelastic flows use the Weissenberg number to compare the magnitude of the elastic forces to the viscous forces. In this case, it can be defined as

where λ represents the relaxation time of the polymer. In this model, three different relaxation times are used — 0.05 s, 0.1 s, and 0.2 s — corresponding to Wi = 0.15, Wi = 0.3, and Wi = 0.6, respectively.

Results and Discussion

The deformation of the plate and the boundary loads on the wall are plotted in Figure 2 for a Weissenberg number of 0.6. Arrows pointing upward represent the loads due to the interaction with the fluid, while the downward arrows represent the external pressure on the elastic plate. As commented in Ref. 1 and formally proven in Ref. 3, the normal component of the viscous and elastic stresses are negligible, and the normal forces on the wall are mainly due to the pressure on the elastic plate, which is plotted for all studied Weissenberg numbers in Figure 3. The pressure from the fluid, pointing upward, acts against the external pressure on the plate, but has smaller magnitude. This results in a downward deformation of the plate, as plotted in Figure 4 for all Weissenberg numbers. Even though the normal component of the viscoelastic stresses does not have a direct effect on the plate, different relaxation times result in different distributions of pressure and plate displacement. Specifically, increasing the Weissenberg number results in an increased pressure downstream of the narrowest gap in the channel and a slightly smaller plate displacement. Narrowing the channel also affects the velocity distribution as seen in Figure 5.

Figure 2: Boundary loads applied to the wall for Wi = 0.6. The upper arrows represent the fluid forces exerted on the wall. The downward pointing arrows represent the uniform pressure on the flexible wall.

Figure 3: Distribution of fluid pressure on the elastic wall for the different Weissenberg numbers.

Figure 4: Elastic wall displacement for the different Weissenberg numbers.

Figure 5: Velocity magnitude in the deformed channel for Wi = 0.6.

Notes About the COMSOL Implementation

The present application solves a coupled fluid-structure interaction problem with a stationary study. A robust way to solve this model is to provide a good initial solution for the nonlinear problem by first solving the fluid and solid parts separately. First, solve the viscoelastic flow in the deformed channel. Then, solve for the structural displacements using the fluid loads for an undeformed channel. Finally, solve the coupled problem for the different Weissenberg numbers using the previous solution fields as the initial values.

References

1. E. Turkoz, Numerical Modeling of Fluid-Structure Interaction with Rheologically Complex Fluids, PhD thesis, Department of Mechanical Engineering, Technische Universität, 2014.

2. D. Chakraborty and others, “Viscoelastic Flow in a Two-Dimensional Collapsible Channel,” J. Non-Newtonian Fluid Mech., vol. 165, pp. 1204–1218, 2010.

3. N.A. Patankar and others, “Normal Stresses on the Surface of a Rigid Body in an Oldroyd-B Fluid,” J. Fluid Eng., vol. 124, pp. 279–280, 2002.

Polymer_Flow_Module/Verification_Examples/viscoelastic_fsi_flexible_wall

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