Abstract
Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate.
Highlights
Boundary Conditions: AsymptoticThe asymptotic behavior of solutions of partial differential equations in thin domains is extensively studied in the vast mathematical literature
This asymptotic expansion was used to justify the method of asymptotic partial decomposition of the domain firstly introduced for the stationary Navier–Stokes equations in thin tube structures in [4]
This method allowed reducing the computational costs that the Navier–Stokes equations posed in thin tube structures
Summary
The asymptotic behavior of solutions of partial differential equations in thin domains is extensively studied in the vast mathematical literature. We will construct an asymptotic expansion of a weak solution of the stationary Navier–Stokes equations in the whole thin tube structure with the Bernoulli boundary conditions for the inflows and outflows. This construction uses the stabilization theorem of the Stokes equations in a cylinder with the no-slip conditions on the lateral boundary, with the proof provided in Appendix A.
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