Zero-viscosity Limit Research Articles

On the steady Prandtl boundary layer expansions

In this paper, we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in (0,L) × ℝ+ with no-slip boundary conditions. By estimating the stream-function of the remainder, we justify the validity of the Prandtl boundary layer expansions. Specially, we show the global stability under the concavity condition of the Prandtl profile for an arbitrarily large constant L when the Euler flow is shear.

Asymptotic limit of the Navier-Stokes-Poisson-Korteweg system in the half-space

In this paper, we consider the quasi-neutral limit, zero-viscosity limit and vanishing capillarity limit for the compressible Navier-Stokes-Poisson system of Korteweg type in the half-space. The system is supplemented with the Neumann, Navier-slip and Dirichlet boundary conditions for density, velocity and electric potential, respectively. The stability of the approximation solutions involving the boundary layer is established by a conormal energy estimate, and then the convergence of solution of the Navier-Stokes-Poisson-Korteweg system to that of the compressible Euler equation is obtained with convergence rate.

The inviscid limit for the incompressible stationary magnetohydrodynamics equations in three dimensions

This paper is concerned with the zero-viscosity limit of the three-dimensional (3D) incompressible stationary magnetohydrodynamics (MHD) equations in the 3D unbounded domain [Formula: see text]. The main result of this paper establishes that the solution of 3D incompressible stationary MHD equations converges to the solution of the 3D incompressible stationary Euler equations as the viscosity coefficient goes to zero.

Zero viscosity and thermal diffusivity limit of the linearized compressible Navier–Stokes–Fourier equations in the half plane

We study the zero viscosity and thermal diffusivity limit of an initial boundary problem for the linearized Navier–Stokes–Fourier equations of a compressible viscous and heat conducting fluid in the half plane. We consider the case that the viscosity and thermal diffusivity converge to zero at the same order. The approximate solution of the linearized Navier–Stokes–Fourier equations with inner and boundary expansion terms is analyzed formally first by multiscale analysis. Then the pointwise estimates of the error terms of the approximate solution are obtained by energy methods. Thus establish the uniform stability for the linearized Navier–Stokes–Fourier equations in the zero viscosity and heat conductivity limit. This work is based on (Comm. Pure Appl. Math. 52 (1999), 479–541) and generalize their results from isentropic case to the general compressible fluid with thermal diffusive effect. Besides the viscous layer as in (Comm. Pure Appl. Math. 52 (1999), 479–541), the thermal layer appears and couples with the viscous layer linearly.

Linear stability and isotropy properties of athermal regularized lattice Boltzmann methods.

The present work proposes a general methodology to study stability and isotropy properties of lattice Boltzmann (LB) schemes. As a first investigation, such a methodology is applied to better understand these properties in the context of regularized approaches. To this extent, linear stability analyses of two-dimensional models are proposed: the standard Bhatnagar-Gross-Krook collision model, the original precollision regularization, and the recursive regularized model, where off-equilibrium distributions are partially computed thanks to a recursive formula. A systematic identification of the physical content carried by each LB mode is done by analyzing the eigenvectors of the linear systems. Stability results are then numerically confirmed by performing simulations of shear and acoustic waves. This work allows drawing fair conclusions on the stability properties of each model. In particular, among the aforementioned models, recursive regularization turns out to be the most stable one for the D2Q9 lattice, especially in the zero-viscosity limit. Two major properties shared by every regularized model are highlighted: (1) a mode filtering property and (2) an incorrect, and broadly anisotropic, dissipation rate of the modes carrying physical waves in under-resolved conditions. The first property is the main source of increased stability, especially for the recursive regularization. It is a direct consequence of the reconstruction of off-equilibrium populations before each collision process, decreasing the rank of the system of discrete equations. The second property seems to be related to numerical errors directly induced by the equilibration of high-order moments. In such a case, this property is likely to occur with any collision model that follows such a stabilization methodology.

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria.

A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on a numerical equilibrium that has been extensively used by discrete velocity methods since the late 1990s. Here, it is extended to reproduce an arbitrary number of moments of the Maxwell–Boltzmann distribution. These extensions to the standard 5-constraint (mass, momentum and energy) approach lead to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high Reynolds-number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the supersonic flow past a NACA0012 aerofoil confirms the excellent behaviour of this model in a low-viscosity and supersonic regime. The flow past a sphere is further simulated to investigate the 3D behaviour of our model in the low-viscosity supersonic regime. The proposed model is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures. It then opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach.This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Zero-Viscosity Limit of the Navier--Stokes Equations with the Navier Friction Boundary Condition

In this paper, we consider the zero-viscosity limit of the Navier-Stokes equations in a half space with the Navier friction boundary condition $$ (\beta u^{\varepsilon}-\varepsilon^{\gamma}\partial_y u^{\varepsilon})|_{y=0}=0, $$ where $\beta$ is a constant and $\gamma\in (0,1]$. In the case of $\gamma=1$, the convergence to the Euler equations and the Prandtl equation with the Robin boundary condition is justified for the analytic data. In the case of $\gamma\in (0,1)$, the convergence to the Euler equations and the linearized Prandtl equation is justified for the data in the Gevrey class $\frac 1 {\gamma}$.

Zero-Viscosity Limit of the Navier–Stokes Equations in a Simply-Connected Bounded Domain Under the Analytic Setting

In this paper, we consider the zero-viscosity limit of the 2-D Navier–Stokes equations in a simply-connected bounded domain with non-slip boundary condition. Based on the energy method in Wang et al. (Arch Ration Mech Anal 224(2):555–595, 2017), we justify the zero-viscosity limit under the analytic setting.

Zero-viscosity limit of the incompressible Navier-Stokes equations with sharp vorticity gradient

It is well-known that the 3D incompressible Euler equations admit some local-in-time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time-dependent hypersurface which evolves with the flow. In this paper we prove that such a solution can be obtained as zero-viscosity limit of strong solutions to the Navier-Stokes equations whose vorticity has sharp variations near the hypersurface associated with the inviscid limit. Indeed we exhibit some sequences of exact solutions to the Navier-Stokes equations with vanishing viscosity which are given by multi-scale asymptotic expansions involving some characteristic boundary layers given by some linear PDEs. The convergence and the validity of the expansion are guaranteed on the time interval associated with the solution to the Euler equations.

A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

ABSTRACTWe embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system and derive a relative entropy identity in the extended variables. Following the relative entropy formulation, we prove the convergence from thermoviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero. Also, convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. Finally, we establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.

Boundary Layers for the Navier–Stokes Equations Linearized Around a Stationary Euler Flow

We study the viscous boundary layer that forms at small viscosity near a rigid wall for the solution to the Navier–Stokes equations linearized around a smooth and stationary Euler flow (LNSE for short) in a smooth bounded domain $$\Omega \subset \mathbb {R}^3$$ under no-slip boundary conditions. LNSE is supplemented with smooth initial data and smooth external forcing, assumed ill-prepared, that is, not compatible with the no-slip boundary condition. We construct an approximate solution to LNSE on the time interval [0, T], $$0<T<\infty $$ , obtained via an asymptotic expansion in the viscosity parameter, such that the difference between the linearized Navier–Stokes solution and the proposed expansion vanishes as the viscosity tends to zero in $$L^2(\Omega )$$ uniformly in time, and remains bounded independently of viscosity in the space $$L^2([0,T];H^1(\Omega ))$$ . We make this construction both for a 3D channel domain and a smooth domain with a curved boundary. The zero-viscosity limit for LNSE, that is, the convergence of the LNSE solution to the solution of the linearized Euler equations around the same profile when viscosity vanishes, then naturally follows from the validity of this asymptotic expansion. This article generalizes and improves earlier works, such as Temam and Wang (Indiana Univ Math J 45(3):863–916, 1996), Xin and Yanagisawa (Commun Pure Appl Math 52(4):479–541, 1999), and Gie (Commun Math Sci 12(2):383–400, 2014).

Uniform regularity for free-boundary Navier–Stokes equations with surface tension

We study the zero-viscosity limit of free boundary Navier–Stokes equations with surface tension in unbounded domain thus extending the work of Masmoudi and Rousset [Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal. (2016), doi:10.1007/s00205-016-1036-5] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. Our main idea is to use Dirichlet–Neumann operator and time-derivatives.

Well-posedness of the Prandtl equation with monotonicity in Sobolev spaces

By using the paralinearization technique, we prove the well-posedness of the Prandtl equation for monotonic data in anisotropic Sobolev space with exponential weight and low regularity. The proof is very elementary, thus is expected to provide a new possible way for the zero-viscosity limit problem of the Navier–Stokes equations with the non-slip boundary condition.

Relative Entropy for Hyperbolic–Parabolic Systems and Application to the Constitutive Theory of Thermoviscoelasticity

We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.

On the zero-viscosity limit of the Navier–Stokes equations in [formula omitted] without analyticity

We consider the zero viscosity limit of the incompressible Navier–Stokes equations with non-slip boundary condition in R+3 for the initial vorticity located away from the boundary. Unlike 2-D case, this kind of data is not analytic in (x,y) for y close to the boundary in 3-D. Maekawa proved the local in time convergence of the Navier–Stokes equations in R+2 to the Euler equations outside a boundary layer and to the Prandtl equations in the boundary layer. His proof used Cauchy–Kowaleskaya theorem, where the vorticity formulation and Euler–Prandtl decomposition play an important role. In this paper, we generalize Maekawa's result to R+3 by using a direct energy method, which may be applicable for general physical domain.

Second-order transport, quasinormal modes and zero-viscosity limit in the Gauss-Bonnet holographic fluid

Gauss-Bonnet holographic fluid is a useful theoretical laboratory to study the effects of curvature-squared terms in the dual gravity action on transport coefficients, quasinormal spectra and the analytic structure of thermal correlators at strong coupling. To understand the behavior and possible pathologies of the Gauss-Bonnet fluid in 3 + 1 dimensions, we compute (analytically and non-perturbatively in the Gauss-Bonnet coupling) its second-order transport coefficients, the retarded two- and three-point correlation functions of the energy-momentum tensor in the hydrodynamic regime as well as the relevant quasinormal spectrum. The Haack-Yarom universal relation among the second-order transport coefficients is violated at second order in the Gauss-Bonnet coupling. In the zero-viscosity limit, the holographic fluid still produces entropy, while the momentum diffusion and the sound attenuation are suppressed at all orders in the hydrodynamic expansion. By adding higher-derivative electromagnetic field terms to the action, we also compute corrections to charge diffusion and identify the non-perturbative parameter regime in which the charge diffusion constant vanishes.

Zero-Viscosity Limit of the Navier–Stokes Equations in the Analytic Setting

In this paper, we consider the zero-viscosity limit of the Navier–Stokes equations in a half space with non-slip boundary condition. Based on the vorticity formulation and the use of conormal derivative, we develop an energy method to justify the zero-viscosity limit for the analytic data.

Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be \({O(\sqrt{\nu})}\). The main assumption is spatial analyticity of the initial datum.

Zero viscosity and diffusion vanishing limit of the incompressible magnetohydrodynamic system with perfectly conducting wall

This work investigates the incompressible viscous and diffusive MHD system with perfectly conducting wall boundary condition. For different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with different or same speeds, we establish the convergence to the ideal inviscid MHD system and the anisotropic inviscid MHD system by constructing the exact boundary layer functions.

Zero-viscosity limit in a holographic Gauss-Bonnet liquid

In recent papers, it was hypothesized that there exist dissipationless quantum liquids, i.e., liquids with zero or vanishingly small viscosity and zero entropy production, which nevertheless have nontrivial second-order transport coefficients. A natural candidate for a dissipationless liquid is the hypothetical conformal quantum liquid, whose holographically dual description in the infrared limit is given by the five-dimensional Gauss-Bonnet gravity. It is known that shear viscosity in that theory can be made arbitrarily small as the Gauss-Bonnet coupling parameter approaches a critical value. We evaluate the transport coefficients of a Gauss-Bonnet liquid (nonperturbatively in the coupling parameter; three of the six coefficients were previously unknown) and consider the zero-viscosity limit. We show that three of the five second-order coefficients are nonzero in this limit, but they do not satisfy the criterion of zero entropy production. Hence, the holographic Gauss-Bonnet liquid is not a dissipationless quantum liquid.