Stationary Navier Stokes Flows Research Articles

Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel

Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel

Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data

We analyze the forced incompressible stationary Navier-Stokes flow in R+n, n>2. Existence of a unique solution satisfying a global integrability property measured in a scale of tent spaces is established for small data in homogeneous Sobolev space with s=−12 degree of smoothness. The velocity field is shown to be locally Hölder continuous while the pressure belongs to Llocp for any p∈(1,∞). Our approach is based on the analysis of the inhomogeneous Stokes system for which we derive a new solvability result involving Dirichlet data in Triebel-Lizorkin classes with negative amount of smoothness and is of independent interest.

$$\varvec{L}^{\varvec{r}}$$-results of the stationary Navier–Stokes flows around a rotating obstacle

$$\varvec{L}^{\varvec{r}}$$-results of the stationary Navier–Stokes flows around a rotating obstacle

Existence of the stationary Navier-Stokes flow in [formula omitted] around a radial flow

We consider the stationary Navier-Stokes equations on the whole plane R2. We show that for a given small and smooth external force around a radial flow, there exists a classical solution decaying like |x|−1. In our result, it is not necessary to impose any symmetric conditions on external forces.

Existence and Uniqueness for Plane Stationary Navier–Stokes Flows with Compactly Supported Force

We study the stationary Navier--Stokes equations in the whole plane with a compactly supported force term and with a prescribed constant spatial limit. Prior to this work, existence of solutions to this problem was only known under special symmetry and smallness assumptions. In the paper we solve the key difficulties in applying Leray's {\it invading domains method} and, as a consequence, prove the existence of $D$-solutions in the whole plane for arbitrary compactly supported force. The boundary condition at infinity are verified in two different scenarios: (I) the~limiting velocity is sufficient large with respect to the external force, (II) both the total integral of force and the~limiting velocity vanish. Hence, our method produces large class of new solutions with prescribed spatial limits. Moreover, we show the uniqueness of $D$-solutions to this problem in a perturbative regime. The main tools here are two new estimates for general Navier-Stokes solutions, which have rather simple forms. They control the difference between mean values of the velocity over two concentric circles in terms of the Dirichlet integral in the annulus between them.

Asymptotic stability of stationary Navier–Stokes flow in Besov spaces

We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.

Attainability of a Stationary Navier-Stokes Flow around a Rigid Body Rotating from Rest

We consider the large time behavior of the three-dimensional Navier-Stokes flow around a rotating rigid body. Assume that the angular velocity of the body gradually increases until it reaches a small terminal one at a certain finite time and it is fixed afterwards. We then show that the fluid motion converges to a steady solution as time $t\rightarrow\infty$.

A Connection Between Symmetry Breaking for Sobolev Minimizers and Stationary Navier–Stokes Flows Past a Circular Obstacle

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier–Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.

Leray's plane stationary solutions at small Reynolds numbers

In the celebrated 1933 paper [22] by J. Leray, the invading domains method was proposed to construct D-solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D-solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, as well as the ideas developed in the recent works [19,20].

Uniqueness of Plane Stationary Navier–Stokes Flow Past an Obstacle

We study the exterior problem for stationary Navier–Stokes equations in two dimensions describing a viscous incompressible fluid flowing past an obstacle. It is shown that, at small Reynolds numbers, the classical solutions constructed by Finn and Smith are unique in the class of D-solutions (that is, solutions with finite Dirichlet integral). No additional symmetry or decay assumptions are required. This result answers a long-standing open problem. In the proofs, we developed the ideas of the classical Ch. Amick paper (Acta Math. 1988).

Existence of a Stationary Navier–Stokes Flow Past a Rigid Body, with Application to Starting Problem in Higher Dimensions

We consider the large time behavior of the Navier-Stokes flow past a rigid body in $\mathbb{R}^n$ with $n\geq 3$. We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time $t\rightarrow\infty$ in $L^q$ with $q\in[n,\infty]$. This is called Finn's starting problem and the three-dimensional case was affirmatively solved by Galdi, Heywood and Shibata $(1997).$ The present paper extends their result to the case of higher dimensions. Even for the three-dimensional case, our theorem provides new convergence rate, that is determined by the summability of the stationary solution at infinity and seems to be sharp.

On Stationary Navier–Stokes Flows Around a Rotating Obstacle in Two-Dimensions

We study the two-dimensional stationary Navier-Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of the obstacle and the given exterior force are sufficiently small.

Green tensor of the Stokes system and asymptotics of stationary Navier–Stokes flows in the half space

We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier–Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.

Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow

We apply a phase field approach for a general shape optimization problem of a stationary Navier-Stokes flow. To be precise we add a multiple of the Ginzburg{Landau energy as a regularization to the objective functional and relax the non-permeability of the medium outside the fluid region. The resulting diffuse interface problem can be shown to be well-posed and optimality conditions are derived. We state suitable assumptions on the problem in order to derive a sharp interface limit for the minimizers and the optimality conditions. Additionally, we can derive a necessary optimality system for the sharp interface problem by geometric variations without stating additional regularity assumptions on the minimizing set.

A remark on Leray’s problem on stationary Navier–Stokes flows with large fluxes in infinite cylindrical domains

We consider Leray’s problem on stationary Navier–Stokes flows with arbitrary large fluxes in an unbounded cylinder with several exits to infinity. For a stationary Navier–Stokes flow with large fluxes in the unbounded cylinder, we prove that, if the difference between the pressure of the main flow and the pressure of the Poiseuille flow with the same flux in a branch of the cylinder remains bounded at |x| → ∞, then the flow behaves at infinity of the branch like the Poiseuille flow.

Artificial boundary conditions for stationary Navier–Stokes flows past bodies in the half-plane

We discuss artificial boundary conditions for stationary Navier–Stokes flows past bodies in the half-plane, for a range of low Reynolds numbers. When truncating the half-plane to a finite domain for numerical purposes, artificial boundaries appear. We present an explicit Dirichlet condition for the velocity at these boundaries in terms of an asymptotic expansion for the solution to the problem. We show a substantial increase in accuracy of the computed values for drag and lift when compared with results for traditional boundary conditions. We also analyze the qualitative behavior of the solutions in terms of the streamlines of the flow. The new boundary conditions are universal in the sense that they depend on a given body only through one constant, which can be determined in a feed-back loop as part of the solution process.

On the stationary Navier-Stokes flow with isotropic streamlines in all latitudes on a sphere or a 2D hyperbolic space

In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with isotropic streamlines. When the sphere is replaced by a 2-dimensional hyperbolic space, we also give the analog existence result for stationary parallel laminar Navier-Stokes flows along a circular-arc boundary portion of some compact obstacle in the 2-D hyperbolic space. The existence of stationary parallel laminar Navier-Stokes flows along a straight boundary of some obstacle in the 2-D hyperbolic space is also studied. In any one of these cases, we show that a parallel laminar flow with a Poiseuille's flow profile ceases to be a stationary Navier-Stokes flow, due to the curvature of the background manifold.

Spatial stability of 3D exterior stationary Navier–Stokes flows

In this paper, we study the stability of stationary solutions w for the Navier–Stokes flows in an exterior domain with zero velocity at infinity. With suitable assumptions of w, by the works of Chen (1993), Kozono–Ogawa (1994) and Borchers–Miyakawa (1995), if u0−w∈Lr(Ω)∩L3(Ω) then one can obtain‖u(t)−w‖p=O(t−32(1r−1p))for 1<r<p<∞,‖∇(u(t)−w)‖p=O(t−32(1r−1p)−12)for 1<r<p<3, where u(x,t) is a solution of the Navier–Stokes equations with the initial condition u0. In this paper, we will prove that for any 0<α<3 if |x|α(u0−w) belongs to Lr(Ω) then one has‖|x|α(u(t)−w)‖Lp=O(t−32(1r−1p)+α2)for p>3r3−rα.

Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows

In this article, we study the long time numerical stability and asymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global $H^2-$stability result for the fully discrete finite element solution. Secondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd problem converges to the stationary Navier-Stokes flows if the body force $f(x,t)$ approaches to a steady-state $f_\infty(x)$ as $t\rightarrow\infty$. Finally, some numerical experiments are given to verify the theoretical predictions.

On the stationary Navier–Stokes flows around a rotating body

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body \( \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega}\) which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier–Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying \({\nabla u, p \in L_{3/2, \infty} (\Omega)}\) and \({u \in L_3, \infty (\Omega)}\) under the smallness condition on \({|U| + |\omega| + ||F||_{L_{3/2, \infty} (\Omega)}}\) . Then the uniqueness is shown for solutions (u, p) satisfying \({\nabla u, p \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}\) and \({u \in L_{3, \infty} (\Omega) \cap L_{q*, r} (\Omega)}\) provided that 3/2 < q < 3 and \({{F \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}}\) . Here L q,r (Ω) denotes the well-known Lorentz space and q* = 3q /(3 − q) is the Sobolev exponent to q.