On the spectral instability of parallel shear flows

Emmanuel Grenier; Yan Guo; Toan T. Nguyen

Displaying similar documents to “On the spectral instability of parallel shear flows”

Stability with respect to domain of the low Mach number limit of compressible heat-conducting viscous fluid

Aneta Wróblewska-Kamińska (2023)

Archivum Mathematicum

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We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $ε \to 0$ , the Froude number proportional to $\sqrt{ε}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $ε \to 0$ . The limit velocity field is solenoidal and satisfies the incompressible Oberbeck–Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we...

Numerical comparison of unsteady compressible viscous flow in convergent channel

Pořízková, Petra, Kozel, Karel, Horáček, Jaromír

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This study deals with a numerical solution of a 2D flows of a compressible viscous fluids in a convergent channel for low inlet airflow velocity. Three governing systems – Full system, Adiabatic system, Iso-energetic system $b a s e d o n t h e N a v i e r - S t o k e s e q u a t i o n s f o r l a m i n a r f l o w a r e t e s t e d . T h e n u m e r i c a l s o l u t i o n i s r e a l i z e d b y f i n i t e v o l u m e m e t h o d a n d t h e p r e d i c t o r - c o r r e c t o r M a c C o r m a c k s c h e m e w i t h J a m e s o n a r t i f i c i a l v i s c o s i t y u s i n g a g r i d o f q u a d r i l a t e r a l c e l l s . T h e u n s t e a d y g r i d o f q u a d r i l a t e r a l c e l l s i s c o n s i d e r e d i n t h e f o r m o f c o n s e r v a t i o n l a w s u s i n g A r b i t r a r y L a g r a n g i a n - E u l e r i a n m e t h o d . T h e n u m e r i c a l r e s u l t s, a c q u i r e d f r o m a d e v e l o p e d p r o g r a m, a r e p r e s e n t e d f o r i n l e t v e l o c i t y$ u=4.12 ms-1 $a n d R e y n o l d s n u m b e r R e =$ 4 103 $.$

THE Navier-stokes flow around a rotating obstacle with time-dependent body force

Toshiaki Hishida (2009)

Banach Center Publications

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We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with $F \in B U C (ℝ; L_{3 / 2, \infty} (D))$ , we consider this problem in D × ℝ and prove that there exists a unique solution $u \in B U C (ℝ; L_{3, \infty} (D))$ when F and |ω| are sufficiently small. If, in particular, the external...

Global existence of smooth solutions for the compressible viscous fluid flow with radiation in $ℝ^{3}$

Hyejong O, Hakho Hong, Jongsung Kim (2023)

Applications of Mathematics

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This paper is concerned with the 3-D Cauchy problem for the compressible viscous fluid flow taking into account the radiation effect. For more general gases including ideal polytropic gas, we prove that there exists a unique smooth solutions in $[0, \infty)$ , provided that the initial perturbations are small. Moreover, the time decay rates of the global solutions are obtained for higher-order spatial derivatives of density, velocity, temperature, and the radiative heat flux.

Global regular solutions to the Navier-Stokes equations in a cylinder

Wojciech M. Zajączkowski (2006)

Banach Center Publications

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The existence and uniqueness of solutions to the Navier-Stokes equations in a cylinder Ω and with boundary slip conditions is proved. Assuming that the azimuthal derivative of cylindrical coordinates and azimuthal coordinate of the initial velocity and the external force are sufficiently small we prove long time existence of regular solutions such that the velocity belongs to $W_{5 / 2}^{2, 1} (Ω \times (0, T))$ and the gradient of the pressure to $L_{5 / 2} (Ω \times (0, T))$ . We prove the existence of solutions without any restrictions on the...

The Leray problem for 2D inhomogeneous fluids

Farid Ammar-Khodja, Marcelo M. Santos (2005)

Banach Center Publications

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We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an $L^{\infty}$ function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.

Consistent streamline residual-based artificial viscosity stabilization for numerical simulation of incompressible turbulent flow by isogeometric analysis

Bohumír Bastl, Marek Brandner, Kristýna Slabá, Eva Turnerová (2022)

Applications of Mathematics

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In this paper, we propose a new stabilization technique for numerical simulation of incompressible turbulent flow by solving Reynolds-averaged Navier-Stokes equations closed by the SST $k$ - $ω$ turbulence model. The stabilization scheme is constructed such that it is consistent in the sense used in the finite element method, artificial diffusion is added only in the direction of convection and it is based on a purely nonlinear approach. We present numerical results obtained by our in-house...

A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition

H. Morimoto, H. Fujita (2001)

Mathematica Bohemica

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We consider the steady Navier-Stokes equations in a 2-dimensional unbounded multiply connected domain $Ω$ under the general outflow condition. Let $T$ be a 2-dimensional straight channel $ℝ \times (- 1, 1)$ . We suppose that $Ω \cap {x_{1} < 0}$ is bounded and that $Ω \cap {x_{1} > - 1} = T \cap {x_{1} > - 1}$ . Let $V$ be a Poiseuille flow in $T$ and $μ$ the flux of $V$ . We look for a solution which tends to $V$ as $x_{1} \to \infty$ . Assuming that the domain and the boundary data are symmetric with respect to the $x_{1}$ -axis, and that the axis intersects every component of the boundary, we have shown...

On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem

Hugo Beirão da Veiga (2009)

Journal of the European Mathematical Society

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We establish regularity results up to the boundary for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case $p \geq 2$ . Actually, we are interested in proving regularity results in $L^{q} (Ω)$ spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous...

Steady compressible Oseen flow with slip boundary conditions

Tomasz Piasecki (2009)

Banach Center Publications

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We prove the existence of solution in the class H²(Ω) × H¹(Ω) to the steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with boundary of class $H^{5 / 2}$ . The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to the problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under an additional assumption on the geometry of...

Stability of oscillating boundary layers in rotating fluids

Nader Masmoudi, Frédéric Rousset (2008)

Annales scientifiques de l'École Normale Supérieure

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We prove the linear and non-linear stability of oscillating Ekman boundary layers for rotating fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to $ε$ . This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated.

Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions

W. M. Zajączkowski (2005)

Studia Mathematica

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Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $I = \sum_{i = 1}^{2} (| | \partial_{x ₃}^{i} {v (0) | |}_{L ₂ (Ω)} + | | \partial_{x ₃}^{i} {f | |}_{L ₂ (Ω \times (0, T))})$ is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by...

Stability of Constant Solutions to the Navier-Stokes System in ℝ³

Piotr Bogusław Mucha (2001)

Applicationes Mathematicae

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The paper examines the initial value problem for the Navier-Stokes system of viscous incompressible fluids in the three-dimensional space. We prove stability of regular solutions which tend to constant flows sufficiently fast. We show that a perturbation of a regular solution is bounded in $W_{r}^{2, 1} (ℝ ³ \times [k, k + 1])$ for k ∈ ℕ. The result is obtained under the assumption of smallness of the L₂-norm of the perturbing initial data. We do not assume smallness of the $W_{r}^{2 - 2 / r} (ℝ ³)$ -norm of the perturbing initial data or smallness...

On an estimate for the linearized compressible Navier-Stokes equations in the $L_{p}$ -framework

Piotr Bogusław Mucha (2001)

Colloquium Mathematicae

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An $L_{p}$ -estimate with a constant independent of time for solutions of the linearized compressible Navier-Stokes system in the whole space (under the assumption that solutions have compact supports in space) is obtained.

Global existence of axially symmetric solutions to Navier-Stokes equations with large angular component of velocity

Wojciech M. Zajączkowski (2004)

Colloquium Mathematicae

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Global existence of axially symmetric solutions to the Navier-Stokes equations in a cylinder with the axis of symmetry removed is proved. The solutions satisfy the ideal slip conditions on the boundary. We underline that there is no restriction on the angular component of velocity. We obtain two kinds of existence results. First, under assumptions necessary for the existence of weak solutions, we prove that the velocity belongs to $W_{4 / 3}^{2, 1} (Ω \times (0, T))$ , so it satisfies the Serrin condition. Next, increasing...

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Weighted L² and $L^{q}$ approaches to fluid flow past a rotating body

R. Farwig, S. Kračmar, M. Krbec, Š. Nečasová, P. Penel (2009)

Banach Center Publications

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Consider the flow of a viscous, incompressible fluid past a rotating obstacle with velocity at infinity parallel to the axis of rotation. After a coordinate transform in order to reduce the problem to a Navier-Stokes system on a fixed exterior domain and a subsequent linearization we are led to a modified Oseen system with two additional terms one of which is not subordinate to the Laplacean. In this paper we describe two different approaches to this problem in the whole space case....

Global existence of solutions for incompressible magnetohydrodynamic equations

Wisam Alame, W. M. Zajączkowski (2004)

Applicationes Mathematicae

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Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_{p}^{2, 1} (Ω \times (0, T))$ and the pressure q satisfies $\nabla q \in L_{p} (Ω \times (0, T))$ for p ≥ 7/3.

On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids

Piotr Boguslaw Mucha, Wojciech Zajączkowski (2002)

Annales Polonici Mathematici

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The local existence of solutions for the compressible Navier-Stokes equations with the Dirichlet boundary conditions in the $L_{p}$ -framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made in Lagrangian coordinates. The result is sharp in the $L_{p}$ -approach, because the velocity belongs to $W_{r}^{2, 1}$ with r > 3.

Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component

Zujin Zhang (2018)

Czechoslovak Mathematical Journal

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We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of $u_{3}$ and $ω_{3}$ , which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M. Pokorný (2004).

Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

Zujin Zhang, Chenxuan Tong (2022)

Applications of Mathematics

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We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $| ω^{r} (x, t) | + | ω^{z} (r, t) | \leq \frac{C}{r^{10}}, 0 < r \leq \frac{1}{2} .$ By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing $ω^{r}$ , $ω^{z}$ and $ω^{θ} / r$ on different hollow cylinders, we are able to improve it and obtain $| ω^{r} (x, t) | + | ω^{z} (r, t) | \leq \frac{C | \ln r |}{r^{17 / 2}}, 0 < r \leq \frac{1}{2} .$