Density-dependent Viscosity Research Articles

Discontinuous solutions for the Navier–Stokes equations with density-dependent viscosity

We prove the existence of a unique global-in-time weak solution of the Navier–Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space H^{1}(\mathbb{R}^{2}) , and the initial fluid density is \alpha -Hölder continuous on both sides of a \mathscr{C}^{1+\alpha} -regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one. Our result generalizes a previous known result of Hoff [Comm. Pure Appl. Math. 55 (2002), 1365–140] and Hoff and Santos [Arch. Ration. Mech. Anal. 188 (2008), 509–543] concerning the propagation of regularity for discontinuity surfaces by allowing a more general non-linear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli, and Paicu [Anal. PDE 13 (2020), 275–316] with global-in-time well-posedness, even for density-dependent viscosity, and we achieve uniqueness in a large space.

Global strong solutions to the radially symmetric compressible MHD equations in 2D solid balls with arbitrary large initial data

In this paper, we prove the global existence of strong solutions to the two-dimensional compressible MHD equations with density dependent viscosity coefficients (known as Kazhikhov-Vaigant model) on 2D solid balls with arbitrary large initial smooth data where shear viscosity μ being constant and the bulk viscosity λ be a polynomial of density up to power β. The global existence of the radially symmetric strong solutions was established under Dirichlet boundary conditions for β > 1. Moreover, as long as β>max{1,γ+24}, the density is shown to be uniformly bounded with respect to time. This generalizes the previous result [Fan et al., Arch. Ration. Mech. Anal. 245, 239–278 (2022)] of the compressible Navier-Stokes equations on 2D bounded domains where they require β > 4/3 and also improves the result [Chen et al., SIAM J. Math. Anal. 54(3), 3817–3847 (2022)] of of compressible MHD equations on 2D solid balls.

Global existence and exponential decay of strong solutions to the 3D nonhomogeneous nematic liquid crystal flows with density-dependent viscosity

Global existence and exponential decay of strong solutions to the 3D nonhomogeneous nematic liquid crystal flows with density-dependent viscosity

Mathematical analysis of the motion of a piston in a fluid with density dependent viscosity

We study a free boundary value problem modelling the motion of a piston in a viscous compressible fluid. The fluid is modelled by 1D compressible Navier–Stokes equations with possibly degenerate viscosity coefficient, and the motion of the piston is described by Newton’s second law. We show that the initial boundary value problem has a unique global in time solution, and we also determine the large time behaviour of the system. Finally, we show how our methodology may be adapted to the motion of several pistons.

H(div)-conforming and discontinuous Galerkin approach for Herschel–Bulkley flow with density-dependent viscosity and yield stress

This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.

Global Well-Posedness and Exponential Decay of Strong Solution to the Three-Dimensional Nonhomogeneous Bénard System with Density-Dependent Viscosity and Vacuum

Global Well-Posedness and Exponential Decay of Strong Solution to the Three-Dimensional Nonhomogeneous Bénard System with Density-Dependent Viscosity and Vacuum

Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries

Abstract In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ(ρ) = ρ θ , λ(ρ) = (θ − 1)ρ θ in R N ${\mathbb{R}}^{N}$ . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ., vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for θ = 2 3 $\theta =\frac{2}{3}$ , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, γ = 5 3 $\gamma =\frac{5}{3}$ , respectively.

Relative entropy inequality for capillary fluids with density dependent viscosity and applications

We derive a relative entropy inequality for capillary compressible fluids with density dependent viscosity. Applications in the context of weak–strong uniqueness analysis, pressureless fluids and high-Mach number flows are presented.

Inviscid limit for the compressible Navier-Stokes equations with density dependent viscosity

We consider the compressible Navier-Stokes system describing the motion of a barotropic fluid with density dependent viscosity confined in a three-dimensional bounded domain Ω. We show the convergence of the weak solution to the compressible Navier-Stokes system to the strong solution to the compressible Euler system when the viscosity and the damping coefficients tend to zero.

Global well-posedness for 2D nonhomogeneous asymmetric fluids with magnetic field and density-dependent viscosity

Global well-posedness for 2D nonhomogeneous asymmetric fluids with magnetic field and density-dependent viscosity

Initial-Boundary Value Problems for One-Dimensional pth Power Viscous Reactive Gas with Density-Dependent Viscosity

Initial-Boundary Value Problems for One-Dimensional pth Power Viscous Reactive Gas with Density-Dependent Viscosity

Analytical Solutions to the Cylindrically Symmetric Compressible Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum Free Boundary

Analytical Solutions to the Cylindrically Symmetric Compressible Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum Free Boundary

On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

<p>In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space $ \mathbb{R}^3 $ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.</p>

Exact spherically symmetric solutions of pressureless Navier-Stokes equations with density-dependent viscosity

Exact spherically symmetric solutions of pressureless Navier-Stokes equations with density-dependent viscosity

Global existence and large-time behavior of solutions to the 1D compressible Navier-Stokes equations with density-dependent viscosity in the half space

Global existence and large-time behavior of solutions to the 1D compressible Navier-Stokes equations with density-dependent viscosity in the half space

A blow-up criterion for the strong solutions to the two-dimensional non-isothermal inhomogeneous liquid crystal flow with density-dependent viscosity

A blow-up criterion for the strong solutions to the two-dimensional non-isothermal inhomogeneous liquid crystal flow with density-dependent viscosity

Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity

Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity

Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity

Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity

Global solutions to 1D compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and free-boundary

Global solutions to 1D compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and free-boundary

Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum

Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum