Numerical study of shear-dependent non-Newtonian fluids in compliant vessels

On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

“…Thus, we have proven that the global iterative method with respect to the domain deformation, used in our numerical approximation of the problem (1)-(8), see [1,2,28], converges at least for sufficiently small time. In this construction of the solution the contact of the moving wall with the fixed bottom boundary is avoided.…”

“…1.1. The fluid problem (1) and the structure equation (2) are coupled through the natural kinematic condition describing the continuity of the velocities on the moving interface Γ w (t), v x 1 , R 0 (x 1 ) + η(x 1 , t), t = 0, ∂ t η(x 1 , t) .…”

“…Here P in , P out are some given pressures and 0 < x 2 < R 0 (x 1 ) + η(x 1 , t) with x 1 = 0, L. On the bottom boundary Γ c the flow symmetry is considered, v 2 (x 1 , 0, t) = 0 , µ ∂v 1 ∂x 2 (x 1 , 0, t) = 0, 0 < x 1 < L, 0 < t < T. (6) We equip equation (2) with the following clamped boundary conditions, η(0, t) = η(L, t) = 0, η x 1 (0, t) = η x 1 (L, t) = 0.…”

“…Let us note that in [3] the convergence of the iterative process described above and the uniqueness of the fixed point have been obtained only for a semi-pervious and pseudo-compressible approximation of the original fluidstructure interaction problem. The right-hand side of structure equation (2) has been replaced by κ(v 2 −η t ) for a fixed κ ∈ R. This implies more regularity for η than in the original case (2). The continuous dependence on data and consequently the contractivity of the fixed point mapping has been shown in [3] only for non-solenoidal solutions and fixed κ.…”

“…We prove the convergence of an iterative process with respect to the computational domain geometry. In our previous works on numerical approximation of similar problems we refer this approach as the global iterative method [1,2]. This iterative approach can be understood as a linearization of the socalled geometric nonlinearity of the underlying model.…”