Weak Solutions for An Incompressible, Generalized Newtonian Fluid Interacting with a Linearly Elastic Koiter Type Shell

Abstract

In this paper we analyze the interaction of an incompressible, generalized Newtonian fluid with a linearly elastic Koiter shell whose motion is restricted to transverse displacements. The middle surface of the shell constitutes the mathematical boundary of the three-dimensional fluid domain. We show that weak solutions exist as long as the magnitude of the displacement stays below some (possibly large) bound which is determined by the geometry of the undeformed shell.

Compressible Fluids Interacting with a Linear-Elastic Shell

“…Such a Korn-type inequality is well-known for Lipschitz domains, see [43]. In our context of domains with less regularity, a Korntype inequality for symmetric gradients is shown in [31,Prop. 2.9] following ideas of [1].…”

“…A solution exists provided the magnitude of the displacement stays below some bound (depending only on the geometry of the reference domain) which excludes self-intersections. The results from [32] have been extended to some incompressible non-Newtonian cases in [31]; see also [24]. Results for incompressible fluids in cylindrical domains have been shown in [38,39] and [6].…”

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

“…Existence of a weak solution for a FSI problem between a 3D incompressible, viscous fluid and a 2D viscoelastic plate was considered by Chambolle et al in [9], while Grandmont improved this result in [28] to hold for a 2D elastic plate. These results were extended to a more general geometry in [37], and then to the case of generalized Newtonian fluids in [36], and to a non-Newtonian shear dependent fluid in [40]. In these works existence of a weak solution was proved for as long as the elastic boundary does not touch "the bottom" (rigid) portion of the fluid domain boundary.…”

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

“…In the case of a twodimensional fluid flow and one-dimensional structure the authors obtained in [5,6] the existence of a weak solution for the same regularity of the domain deformation as in [3] using the Schauder fixed point argument, without giving any details on the construction of the domain deformation and without providing the uniqueness of the solution. Using a similar fixed point procedure for the geometry the existence of weak solution for a fluid-structure interaction problem for a generalized power-law shear-dependent fluid including the Newtonian case has been shown in our recent joint works [7,8], see also [9].…”

“…Herec Ko is the coercivity constant of the viscous form coming from the Korn inequality, α, K are given by (9), µ, ρ, E, a, b, c are given by the physical model and C 7 is a constant depending on α, α −1 , K,c −1 Ko and on the norms h i L ∞ (0,T ;H 2 (0,L)) , u i L ∞ (0,T ;L 2 (D)) , i = 1, 2. The matrix R in the above estimate arises due to the transformation of weak solution from Ω(h 1 ) to Ω(h 2 ).…”