The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

Abstract

Abstract. The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

“…One way to see this is to smooth the initial domain by a convolution with the parameter κ to form Ω κ . Then, by the properties of the convolution, κ Ω κ Similarly as in Section 9 of [8], this provides us with a time of existence T κ = T 1 independent of κ and an estimate on (0, T 1 ) independent of κ of the typẽ E(t) 2 ≤ N 0 (u 0 ), as long as the conditions (19.2) hold. Now, since η(t) 3 ≤ Id 3 + t 0 ṽ 3 , we see that condition (19.2b) will be satisfied for t ≤ 1 N 0 (u 0 ) .…”

“…The addition of the artificial viscosity term allows us to prove that E κ (t) is continuous; thus, following the development in [8], there exists a sufficiently small time T , which is independent of κ, such that sup t∈[0,T ] E κ (t) <M 0 forM 0 > M 0 . We then find κ-independent nonlinear estimates for the σ = 0 case for the energy function (20.1).…”

“…Simultaneously, we introduce a new type of parabolic-type artificial viscosity boundary operator on Γ (of the same order in space as the surface tension operator). Note that unlike the case of interface motion in the fluid-structure interaction problem that we studied in [8], there is not a unique choice of the artificial viscosity term; in particular, other choices of artificial viscosity are possible for the asymptotic limit as the artificial viscosity is taken to zero.…”

On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations

“…Regarding the forcing term of the structure, thanks to the continuity of the stresses on Γ 0 w given in (8), and the continuity of the velocities on Γ t w (the first of (7)), we have:…”

“…However, for this problem recent well posedness results for strong solutions have been obtained in [8].…”