Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.     

Viscous compressible barotropic symmetric flows with free boundary under general mass force Part I: Uniform‐in‐time bounds and stabilization

We consider symmetric flows of a viscous compressible barotropic fluid with a free boundary, under a general mass force depending both on the Eulerian and Lagrangian co‐ordinates, with arbitrarily large initial data. For a general non‐monotone state function p, we prove uniform‐in‐time energy bound and the uniform bounds for the density ρ, together with the stabilization as t → ∞ of the kinetic and potential energies. We also obtain H¹‐stabilization of the velocity v to zero provided that the second viscosity is zero. For either increasing or non‐decreasing p, we study the L^λ‐stabilization of ρ and the stabilization of the free boundary together with the corresponding ω‐limit set in the general case of non‐unique stationary solution possibly with zones of vacuum. In the case of increasing p and stationary densities ρ_S separated from zero, we establish the uniform‐in‐time H¹‐bounds and the uniform stabilization for ρ and v. All these results are stated and mainly proved in the Eulerian co‐ordinates. They are supplemented with the corresponding stabilization results in the Lagrangian co‐ordinates in the case of ρ_S separated from zero. Copyright © 2005 John Wiley & Sons, Ltd.     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

One time‐step finite element discretization of the equation of motion of two‐fluid flows

We discretize in space the equations obtained at each time step when discretizing in time a Navier‐Stokes system modelling the two‐dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions at the boundary and at the interface between the two fluids. We discretize this system with the “mini‐element” and establish error estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A two‐grid stabilization method for solving the steady‐state Navier‐Stokes equations

We formulate a subgrid eddy viscosity method for solving the steady‐state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On stationary two‐dimensional flows around a fast rotating disk

We study the two‐dimensional stationary Navier–Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described precisely by exhibiting a boundary layer structure and an axisymmetrization of the flow.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulations for two‐dimensional stochastic incompressible Navier‐Stokes equations

This article mainly concerns modeling the stochastic input and its propagation in incompressible Navier‐Stokes(N‐S) flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to derive the equations in the weak form. The resulting set of deterministic equations is then solved with standard methods to obtain the mean solution. In this article, the main method employs the Hermite polynomial as the basis in random space. Numerical examples are given and the error analysis is demonstrated for a model problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

An optimization‐based domain decomposition method for numerical simulation of the incompressible Navier‐Stokes flows

This article is concerned about an optimization‐based domain decomposition method for numerical simulation of the incompressible Navier‐Stokes flows. Using the method, an classical domain decomposition problem is transformed into a constrained minimization problem for which the objective functional is chosen to measure the jump in the dependent variables across the common interfaces between subdomains. The Lagrange multiplier rule is used to transform the constrained optimization problem into an unconstrained one and that rule is applied to derive an optimality system from which optimal solutions may be obtained. The optimality system is also derived using “sensitivity” derivatives instead of the Lagrange multiplier rule. We consider a gradient‐type approach to the solution of domain decomposition problem. The results of some numerical experiments are presented to demonstrate the feasibility and applicability of the algorithm developed in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Strong solutions to 1D compressible Navier‐Stokes/Allen‐Cahn system with free boundary

This paper is concerned with a diffuse interface model for two‐phase flow of compressible fluids with a type of free boundary. We establish the existence and uniqueness of global strong solutions of a coupled Navier‐Stokes/Allen‐Cahn system in 1D.     

On a finite element method for three‐dimensional unsteady compressible viscous flows

In this article we analyze a finite element method for three‐dimensional unsteady compressible Navier‐Stokes equations. We prove the existence and uniqueness of the numerical solution, and obtain a priori error estimates uniform in time. Numerical computations are carried out to test the orders of accuracy in the error estimates. Blend function interpolations are applied in the calculation of numerical integrations. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 432–449, 2004.     

Discretization of the Navier‐Stokes equations with slip boundary condition

We propose and analyze a two‐level method of discretizing the nonlinear Navier‐Stokes equations with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries, flows past chemically reacting walls, and other examples where the usual no‐slip condition u = 0 is not valid. The two‐level algorithm consists of solving a small nonlinear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. The two‐level method exploits the quadratic nonlinearity in the Navier‐Stokes equations. Our error estimates show that it has optimal order accuracy, provided that the best approximation to the true solution in the velocity and pressure spaces is bounded above by the data. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 26–42, 2001     

Navier边界条件下湖方程的边界层问题

考虑光滑区域上二维粘性湖方程在Navier边界条件下的无粘极限问题,证明了具有Navier边界条件粘性湖方程的边界层在Sobolev空间中是非线性稳定的,验证了具有较弱强度的边界层的渐近展开的合理性.     

Homogenization of a two‐phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two‐phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid‐fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two‐phase fluid in the free domain, a coupled Darcy law connecting two‐phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.