Density-dependent Viscosity Research Articles

Analytical solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure

We study some particular solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure, in radial symmetry. With an extension of the previous known blow-up solutions for the Euler-Poisson equations with pressureless Navier-Stokes-Poisson density-dependent viscosity, we constructed the corresponding self-similar blow-up solutions for the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure. Our solutions can provide concrete examples for testing the validation and stabilities of numerical methods for the systems.

Global C∞-solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity and free boundaries. The initial density ρ0∈W1,2n is bounded below away from zero and the initial velocity u0∈L2n. The viscosity coefficient µ is proportional to ρθ with 0<θ⩽1, where ρis the density. The existence and uniqueness of global solutions in Hi([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50:023101; Meth. Appl. Anal. 2005; 12:239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7:373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity and free boundaries when the initial data ρ0 and u0 are smooth. Copyright © 2011 John Wiley & Sons, Ltd.

Global solutions of compressible barotropic Navier–Stokes equations with a density-dependent viscosity coefficient

In this paper, we consider the global existence and uniqueness of the classical (weak) solution for the 2D or 3D compressible Navier–Stokes equations with a density-dependent viscosity coefficient (λ = λ(ρ)). Initial data and solutions are only small in the energy-norm. We also give a description of the large time behavior of the solution. Then, we study the propagation of singularities in solutions. We obtain that if there is a vacuum domain initially, then the vacuum domain will exist for all time, and vanishes as time goes to infinity.

On the dynamics of Navier–Stokes equations for a shallow water model

In this paper, we study a free boundary problem of one-dimensional compressible Navier–Stokes equations with a density-dependent viscosity, which include, in particular, a shallow water model. Under some suitable assumptions on the initial data, we obtain the global existence, uniqueness and the large time behavior of weak solutions. In particular, it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations.

Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.

Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities

In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.

Regularity and asymptotic behavior of 1D compressible Navier–Stokes–Poisson equations with free boundary

In this paper, firstly, we consider the regularity of solutions in H i ( [ 0 , 1 ] ) ( i = 2 , 4 ) to the 1D Navier–Stokes–Poisson equations with density-dependent viscosity and the initial density that is connected to vacuum with discontinuities, and the viscosity coefficient is proportional to ρ θ with 0 < θ < 1 . Furthermore, we get the asymptotic behavior of the solutions when the viscosity coefficient is a constant. This is a continuation of [S.J. Ding, H.Y. Wen, L. Yao, C.J. Zhu, Global solutions to one-dimensional compressible Navier–Stokes–Poisson equations with density-dependent viscosity, J. Math. Phys. 50 (2009) 023101], where the existence and uniqueness of global weak solutions in H 1 ( [ 0 , 1 ] ) for both cases: μ ( ρ ) = ρ θ , 0 < θ < 1 and μ = constant have been established.

CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46.We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

A two-dimensional snow creep model for alpine terrain

The paper is focused on the investigation of snow creep and settlement of alpine snowpacks. A detailed investigation of the constitutive behavior of snow for long-term creeping under low-strain rates is provided. The snowpack is considered as a compressible, viscous fluid. An assumption for a temperature and density-dependent viscosity and viscous Poisson ratio is provided in this paper, based on observations in nature and laboratory. The provided model is particularly applicable for settlement calculation of seasonal alpine snowpacks and computation of reaction forces on obstacles. The model is evaluated by numerous laboratory and field experiments. Different case studies are provided in this paper.

Global weak solutions and asymptotic behavior to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier–Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρ θ with 1 2 < θ < max { 3 − γ , 3 2 } , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ = 1 , γ = 2 ).

Asymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosity

Asymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosity

Global Weak Solutions to Compressible Navier–Stokes Equations for Quantum Fluids

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier–Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Méhats [Derivation of viscous correction terms for the isothermal quantum Euler model, 2009, submitted] from a Wigner equation using a moment method and a Chapman–Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier–Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak solutions to the viscous quantum Euler model is shown by using the Faedo–Galerkin method and weak compactness techniques. As a consequence, we deduce the existence of solutions to the quantum Navier–Stokes system if the viscosity constant is smaller than the scaled Planck constant.

Compressible Flows with a Density-Dependent Viscosity Coefficient

We prove the global existence of weak solutions for the two-dimensional compressible Navier–Stokes equations with a density-dependent viscosity coefficient ($\lambda=\lambda(\rho)$). Initial data and solutions are small in energy-norm with nonnegative density having arbitrarily large sup-norm. Then, we show that if there is a vacuum domain at the initial time, then the vacuum domain will retain for all time and vanishes as time goes to infinity. At last, we show that the condition of $\mu=\text{constant}$ will induce some singularities of the system at vacuum. Thus, the viscosity coefficient $\mu$ plays a key role in the Navier–Stokes equations.

Two-component viscous flows with density-dependent transition layer

We present a model describing flows which undergo a phase change caused by chemical reactions in the material flow. In mathematical terms, we consider a free boundary problem for the onedimensional, compressible Navier–Stokes system coupled with an Allen–Cahn equation. One free boundary arises at the boundary of the flow to a vacuum state with a jump in the density. Another approximate free boundary arises due to the Allen–Cahn equation modeling the phase transition of the flow. We consider a density-dependent viscosity µ = ρ^θ and a density-dependent transition layer of thickness δ = \tilde δρ^{−1} . We establish the existence of a unique weak global solution, provided that θ &lt; 1/3 .

Dynamical behaviors for 1D compressible Navier–Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier–Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T ∗ > 0 , which is compressed at an algebraic rate and vanishes at the time T ∗ , meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.

On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities

We consider the Cauchy problem for the equations of selfgravitating motions of a barotropic gas with density-dependent viscosities μ(ρ), and λ(ρ) satisfying the Bresch–Desjardins condition, when the pressure P(ρ) is not necessarily a monotone function of the density. We prove that this problem admits a global weak solution provided that the adiabatic exponent γ associated with P(ρ) satisfies \({\gamma > \frac{4}{3}}\).

Remarks on one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity and vacuum

The Navier-Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper. When the viscosity coefficient µ is proportional to ρ θ with 0 0 is also given.

Asymptotic Behavior of Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

In this paper, we study the one-dimensional Navier-Stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient μ(ρ) is proportional to ρ θ with θ > 0, where ρ is the density, we give the asymptotic behavior and the decay rate of the density function ρ(x, t). Furthermore, the behavior of the density function ρ(x, t) near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied. The analysis is based on some new mathematical techniques and some new useful estimates. This fills a final gap on studying Navier-Stokes equations with the viscosity coefficient μ(ρ) dependent on the density ρ.

Analytical solutions to the Navier–Stokes equations with density-dependent viscosity and with pressure

This article is the continued version of the analytical solutions for the pressureless Navier–Stokes equations with density-dependent viscosity [Yuen, M. W., “Analytical solutions to the Navier-Stokes equations,” J. Math. Phys. 49, 113102 (2008)]. We are able to extend the similar solution structure to the case with pressure under some restriction to the constants γ and θ.

Analytical blowup solutions to the pressureless Navier–Stokes–Poisson equations with density-dependent viscosity in RN

We study the pressureless Navier–Stokes–Poisson equations with density-dependent viscosity. With the extension of the blowup solutions for the Euler–Poisson equations, the analytical blowup solutions, in radial symmetry, in RN (N ⩾ 2) are constructed.