Navier‐Stokes equations with imposed pressure and velocity fluxes

Abstract

SUMMARYBoundary value problems for Stokes and Navier-Stokes equations with non-standard boundary conditions are studied. Included is the case where the pressure or its normal derivative is given on some part of the boundary or the pressure is given up to a constant but given velocity flux. First, a variational formulation is introduced which is shown to be equivalent to the Stokes equations with the non-standard boundary conditions under consideration. The existence and uniqueness of the solution of the variational problem are studied. Secondly, most of the results obtained for the Stokes equations are extended to the case of the Navier-Stokes equations. The final section is devoted to numerical experiments, flows in pipes and physiological flows.KEY WORDS velocity-pressure formulation; pressure boundary condition; single and branched pipes; steady and periodic flows NTRODUCTIONThis paper is concerned with the stationary Stokes and Navier-Stokes equations with non-standard boundary conditions. Specifically, the case where the pressure is given on some part of the boundary will be considered.To be precise, the flow of a viscous incompressible fluid which occupies a bounded domain R of R3 is studied. The velocity u and pressure p are assumed to satisfy in this domain R either the stationary Stokes equationsor the stationary Navier-Stokes equations

Geometric multiscale modeling of the cardiovascular system, between theory and practice

“…In this case, the do-nothing approach corresponds to the boundary condition p = P on Γ, the only assumption being that the pressure is constant on Γ, without any further request on the smallness of the viscous boundary term (see [41,194,195]). Remark 1.…”

“…This approach relies on the introduction of a set of functions that represents the lifting of the flow rate data inside the domain, in a way similar to what is done for standard Dirichlet conditions. See also [41] for a curl-curl formulation. However, the lifting functions, called flux-carriers, are not easy to construct in general.…”

Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations

“…It is interesting that boundary conditions (2.17) are quite often used for the analysis of the (Navier-)Stokes equations in a bounded domain, see e.g. [3,27] and references therein, and are physical motivated [8].…”

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

“…This type of mixed boundary conditions appears in a large number of physical situations, the simplest one being a tank closed by a membrane on a part of its boundary (the index "m" in Γ m means membrane). An other example is the flow in a bifurcating pipe [8].…”