In-Flow and Out-Flow Problem for the Stokes System
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Journal of Mathematical Fluid Mechanics
In-Flow and Out-Flow Problem for the Stokes System Bernard Nowakowski
and Gerhard Str¨ ohmer
Communicated by Y. Giga
Abstract. We investigate the existence and regularity of solutions to the stationary Stokes system and non-stationary Navier– Stokes equations in three dimensional bounded domains with in- and out-lets. We assume that on the in- and out-flow parts of the boundary the pressure is prescribed and the tangential component of the velocity field is zero, whereas on the lateral part of the boundary the fluid is at rest. Mathematics Subject Classification. Primary 35B65, 35Q30, 76D03; Secondary 76D05. Keywords. In- and out-flow, Stokes system, Dynamic boundary conditions.
1. Introduction Let us consider the Stokes system −νΔu + ∇Π = f div u = g
in Ω, in Ω,
(1)
where Ω ⊂ R3 is a bounded solid with piecewise C 2 -boundary Γ := ∂Ω. Functions u (the velocity) and Π (the pressure) are unknown. Given are f (the external force), g and a positive viscosity coefficient ν. We are interested in the existence and regularity of solutions (u, Π) to (1) when Ω has cylindrical in- and outlets. More specifically, we assume that Γ is the union of Γin,out and Γlat , where Γin,out = 1≤k≤N Γkin,out . We also assume that for all 1 ≤ k ≤ N the surfaces Γkin,out are flat and Γkin,out is orthogonal to Γlat (see Fig. 1). The latter assumption ensures that we do not need to work in the framework of weighted Sobolev spaces and can rely on the reflection principle. We need to supplement (1) with proper boundary conditions. On Γlat we assume that the fluid is at rest. The choice of the boundary conditions on Γkin,out depends on the problem we would like to model. For some examples of such problems and further motivations originating from real-life situations we refer the reader to the Introduction in e.g. [1,2] and [3, Sect. 3]. The most frequently used boundary conditions are: • prescribed pressure drops (see e.g. [3,4]) 1 Π dS = Πk (t), Γk Γk in,out in,out where Πk (t) are given functions and Γkin,out denotes the surface area of Γkin,out , • prescribed net fluxes (see e.g. [3,5,6]) u · n dS = Fk (t), Γk in,out
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B. Nowakowski and G. Str¨ ohmer
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Γ2in,out
Γlat
Γ1in,out
Γ3in,out Fig. 1. An example of Ω when N = 3
where Fk (t) are given functions, • prescribed pressure and tangential components of the velocity (see e.g. [1,2]) Π|Γkin,out = Πk , v × n = ak × n,
(2)
where Πk and ak are given functions. Other boundary conditions that appear in the literature include the so-called artificial boundary conditions (ABCs, see e.g. [7,8]), prescribed normal components of the velocity field on Γkin,out (see e.g. [9,10]), modified do-nothing boundary conditions (see e.g. [11,12]) and other (see e.g. [13–15]). This list, however, is far from being complete but forms a good starting point for further research. In our work we consider (2) with ak = 0. It can be rewritten as utan = 0 Π = Πkin,out u · n|Γlat = 0
on Γ, on Γkin,out , on Γlat ,
(3)
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