Vanishing Shear Viscosity Research Articles

Vanishing shear viscosity limit for the compressible planar MHD system with boundary layer

This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp L∞ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.

Vanishing Shear Viscosity Limit for the Navier–Stokes Equations with Cylindrical Symmetry: Boundary Layer and Optimal Convergence Rate

We consider the initial boundary-value problem for the nonisentropic compressible Navier–Stokes equations with cylindrical symmetry and degenerate heat conductivity coefficient. The vanishing shear viscosity limit with strong boundary layer effect for arbitrarily large data is investigated. Compared with the existing result in [X. Qin et al. Arch. Rational Mech. Anal., 216 (2015) 1049–1086], where the optimal convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components, we derive the optimal convergence rate. The main challenges include that the presence of the strong boundary layer effect will prevent the convergence over the whole spatial domain, and that the quantities of a nonisentropic flow interact strongly with each other not only through the pressure term but also through the strongly nonlinear term in the temperature equation which would possibly slow down the convergence rate. Our main strategy is to find some new functions via asymptotic matching methods such that both the strong boundary layer effect and those quantities decaying at a lower speed can be canceled.

On the non-isentropic compressible Navier–Stokes equations with cylindrical symmetry: boundary layers and convergence rate at vanishing shear viscosity

On the non-isentropic compressible Navier–Stokes equations with cylindrical symmetry: boundary layers and convergence rate at vanishing shear viscosity

Optimal convergence rate of the vanishing shear viscosity limit for compressible Navier-Stokes equations with cylindrical symmetry

We consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry. The existence of boundary layers is well-known when the shear viscosity vanishes. In this paper, we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data.

On a simplified compressible Navier-Stokes equations with temperature-dependent viscosity

We consider a simplified compressible Navier-Stokes equations with cylindrical symmetry when viscosity coefficient λ and heat conductivity coefficient κ depend on temperature. We obtain global existence of strong solution and vanishing shear viscosity limit to the initial-boundary value problem in Eulerian coordinates. The analysis for the global existence is based on the assumption that μ=const.>0,1c˜θm≤λ(θ)≤c˜(1+θm),κ(θ)=θq, for m∈(0,1],q≥m. For the part of vanishing shear viscosity limit, we require in addition that 1c˜(1+θm)≤λ(θ)≤c˜(1+θm). In the paper, the acceleration effect in one direction is neglected, however, we do not need any smallness assumption for the initial data.

Vanishing shear viscosity limit and boundary layer study for the planar MHD system

We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity coefficient that depends on both the temperature and the density. Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the [Formula: see text] convergence rate is obtained together with the estimation on the thickness of the boundary layer.

The Behavior of Boundary Layer for the Compressible Planar MHD Equations at Small Shear Viscosity

The vanishing shear viscosity limit to an initial-boundary value problem of planar MHD equations for compressible viscous heat-conducting fluids is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at small shear viscosity, both the boundary-layer thickness and the boundary-layer corrector are discussed.

Odd surface waves in two-dimensional incompressible fluids

We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken. For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral and even exist in the absence of gravity and vanishing shear viscosity. In this limit, we derive effective nonlinear Hamiltonian equations for the surface dynamics, generalizing the linear solutions to the weakly nonlinear case. Within the small surface angle approximation, the equation of motion leads to a new class of non-linear chiral dynamics governed by what we dub the chiral Burgers equation. The chiral Burgers equation is identical to the complex Burgers equation with imaginary viscosity and an additional analyticity requirement that enforces chirality. We present several exact solutions of the chiral Burgers equation. For generic multiple pole initial conditions, the system evolves to the formation of singularities in a finite time similar to the case of an ideal fluid without odd viscosity. We also obtain a periodic solution to the chiral Burgers corresponding to the non-linear generalization of small amplitude linear waves.

Boundary-layer phenomena for the cylindrically symmetric Navier–Stokes equations of compressible heat-conducting fluids with large data at vanishing shear viscosity

This paper concerns the asymptotic behavior of the solution to an initial-boundary value problem of the cylindrically symmetric Navier–Stokes equations with large data for compressible heat-conducting ideal fluids, as the shear viscosity μ goes to zero. A suitable corrector function (the so-called boundary-layer type function) is constructed to eliminate the disparity of boundary values. As by-products, the convergence rates of the derivatives in L2 are obtained and the boundary-layer thickness (BL-thickness) of the value with is shown by an alternative method, compared with the results proved in Jiang and Zhang (2009 SIAM J. Math. Anal. 41 237–68) and Qin et al (2015 Arch. Ration. Mech. Anal. 216 1049–86).

On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit

In this paper, we consider an initial-boundary value problem for the one-dimensional equations of planar isentropic magnetohydrodynamics (MHD). The vanishing shear viscosity limit is justified. More important, both the thickness and the behavior of boundary layers are described. The proofs of these results are based on the full use of the material derivative, the “effective viscous flux”, and the mathematical structure of the equations.

Vanishing shear viscosity in the magnetohydrodynamic equations with temperature-dependent heat conductivity

We establish an initial-boundary value problem for the compressible magnetohydrodynamic equations in one space dimension with large initial data when the heat conductivity is some positive power of the temperature. We prove that as the shear viscosity vanishes, global weak solutions convergence to a solution of the original equations with zero shear viscosity.

A study on the boundary layer for the planar magnetohydrodynamics system

The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity μ. Under some conditions on the initial and boundary data, we show that the thickness is of the order μ|Inμ|. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.

Vanishing Shear Viscosity and Boundary Layer for the Navier–Stokes Equations with Cylindrical Symmetry

Both the global well-posedness for large data and the vanishing shear viscosity limit with a boundary layer to the compressible Navier–Stokes system with cylindrical symmetry are studied under a general condition on the heat conductivity coefficient that, in particular, includes the constant coefficient. The thickness of the boundary layer is proved to be almost optimal. Moreover, the optimal L1 convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components.

Boundary Layers for the Navier–Stokes Equations of Compressible Heat-Conducting Flows with Cylindrical Symmetry

We consider the Navier–Stokes equations of viscous compressible heat-conducting flows with cylindrical symmetry. Our main purpose is to study the boundary layer effect and the convergence rate as the shear viscosity $\mu$ goes to zero. We show that the boundary layer thickness and a convergence rate are of the orders $O(\mu^\alpha)$ with $0<\alpha<1/2$ and $O(\sqrt{\mu})$, respectively, thus extending the result in [H. Frid and V. V. Shelukhin, Commun. Math. Phys., 208 (1999), pp. 309–330] to the case of nonisentropic flows. As a byproduct, we also improve the convergence result in [H. Frid and V. V. Shelukhin, SIAM J. Math. Anal., 31 (2000), pp. 1144–1156] on the vanishing shear viscosity limit.

Vanishing Shear Viscosity Limit in the Magnetohydrodynamic Equations

We study an initial boundary value problem for the equations of plane magnetohydrodynamic compressible flows, and prove that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. As a by-product, this paper improves the related results obtained by Frid and Shelukhin for the case when the magnetic effect is neglected.

Vanishing Shear Viscosity in a Free-Boundary Problem for the Equations of Compressible Fluids

A free-boundary problem of describing a joint motion of two compressible fluids with different viscosities is considered. The passage to the limit is studied as the shear viscosity of one of the fluids vanishes.

Vanishing Shear Viscosity in the Equations of Compressible Fluids for the Flows with the Cylinder Symmetry

We analyze the question of the limit process when the shear viscosity goes to zero for global solutions to the Navier--Stokes equations for compressible heat conductive fluids for the flows which are invariant over cylindrical sheets.