Regularity Results for Stationary Electro-Rheological Fluids

We prove regularity results for weak solutions to systems modelling electro-rheological fluids in the stationary case, as proposed in [27, 31]; a particular case of the system we consider is where ?(u) is the symmetric part of the gradient Du and the variable growth exponent p(x) is a Hölder continuous function larger than 3n/(n+2).     

Vortex Motion in Two-Dimensional Viscous Fluid Flows

The diffusion and annihilation of vortices in axisymmetric and plane incompressible viscous fluid flows are considered. A formula relating the pressure with the velocity of the vortices in the viscous fluid is obtained.     

Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress

We prove global regularity of solutions of Oldroyd-B equations in two spatial dimensions with spatial diffusion of the polymeric stresses.     

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

Vortex Formations in Plane Flows of Viscous and Ideal Fluids

An analytic representation of individual vortex formations with different cross-sections is obtained for plane-parallel flows of viscous and ideal fluids. A flow in which one of the streamlines has cusp is also found. In these flows the vorticity is constant; therefore, the solutions obtained for the equations describing the kinematic variables in ideal fluid flows must simultaneously satisfy the Navier-Stokes equations.     

Some Global Stability Results for Shear Flows of Viscoelastic Fluids

We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases. This research was supported by the National Science Foundation under Grant DMS-0405810.     

Existence and Regularity of Steady Flows for Shear-Thinning Liquids in Exterior Two-Dimensional

We show that the two-dimensional exterior boundary-value problem (flow past a cylinder) associated with a class of shear-thinning liquid models possesses at least one solution for data of arbitrary “size”. This result must be contrasted with its counterpart for the Navier–Stokes model, where a similar result is known to hold, to date, only if the size of the data is sufficiently restricted.     

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.     

Shear Thinning Viscous Fluids in Cylindrical Domains. Regularity up to the Boundary

We consider the motion of a non-Newtonian fluid with shear dependent viscosity between two cylinders. We prove regularity results for the second derivatives of the velocity and the first derivatives of the pressure up to the boundary. A similar problem is studied in reference [2] in the case of a flat boundary. Here we extend the techniques applied in [2] to cylindrical coordinates.      

On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier–Stokes/Cahn–Hilliard system, which is capable of describing the evolution of droplet formation and collision during the flow. We prove the existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures—in contrast to the usual case of a smooth free energy density—that the concentration stays in the physical reasonable interval. Furthermore, we find that unique “strong” solutions exist in two dimensions globally in time and in three dimensions locally in time. Moreover, we show that for any weak solution the concentration is uniformly continuous in space and time. Because of this regularity, we are able to show that any weak solution becomes regular for large times and converges as t → ∞ to a solution of the stationary system. These results are based on a regularity theory for the Cahn–Hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a Stokes system with variable viscosity and forces in L ²(0, ∞; H ^s (Ω)), ${s \in [0, \frac12)}$ , which are new themselves.     

Erratum to: Existence and Regularity of Steady Flows for Shear-Thinning Liquids in Exterior Two-Dimensional Domains

We show that the two-dimensional exterior boundary-value problem (flow past a cylinder) associated with a class of shear-thinning liquid models possesses at least one solution for data of arbitrary “size”. This result must be contrasted with its counterpart for the Navier–Stokes model, where a similar result is known to hold, to date, only if the size of the data is sufficiently restricted.     

Regularity of Non-Newtonian Fluids

In this paper, we consider a non-Newtonian fluids with shear dependent viscosity in a bounded domain ${\Omega \subset \mathbb{R}^n, n = 2, 3}$ . For the power-law model with the viscosity as in (1.4), we show the global in time existence of a weak solution for ${q \geq \frac{11}{5}}$ when n = 3 (see Theorem 1.1), and the local in time existence of a weak solution for ${2 > q > \frac{3n}{n+2}}$ , when n = 2,3 (see Theorem 1.2).     

Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations

We study the initial boundary value problem of two-dimensional viscous Boussinesq equations over a bounded domain with smooth boundary. We show that the equations have a unique classical solution for H ³ initial data and the no-slip boundary condition. In addition, we show that the kinetic energy is uniformly bounded in time.     

类升力体外形高超声速粘性绕流拓扑结构分析

采用交替方向隐式分解的隐式NND格式求解全N-S方程模拟“类升力体”外形在高超声速下的大攻角流动,给出了“类升力体”外形表面极限流线随攻角变化的拓扑结构及40°攻角下垂直于体轴的横截面流线拓扑结构。结果表明:类升力体外形三维流场结构十分复杂,攻角从0°～40°变化时,背风面表面极限流线依次由不分离、开式分离向起始于鞍、结点组合的高阶奇点的分离方式转化,翼面横向分离亦随攻角增大而增大;垂直于体轴的横截面流动拓扑结构与文献[2]给出的理论分析一致。     

Boundary Regularity of Shear Thickening Flows

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W^{1,(np+2-p)/(n-2)}(W){W^{1,(np+2-p)/(n-2)}(\Omega)} . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.     

以压力为基本求解变量数值模拟粘性超、跨音速流动

应用以压力为基本求解变量的SIMPLE方法 ,对一双喉喷管中的层流超音速流动和一扩压器中的紊流跨音速流动进行了数值计算。计算结果显示 ,本文的计算结果与文献数据及实验结果相符很好。表明本文方法对可压缩流动有很高的模拟精度。进而表明经过可压缩推广的SIMPLE方法适用于任何马赫数的流动计算     

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.     

Hyperbolic Approximation of the Navier-Stokes Equations for Viscous Mixed Flows

Simplified two-dimensional Navier-Stokes equations of the hyperbolic type are derived for viscous mixed (with transition through the sonic velocity) internal and external flows as a result of a special splitting of the pressure gradient in the predominant flow direction into hyperbolic and elliptic components. The application of these equations is illustrated with reference to the calculation of Laval nozzle flows and the problem of supersonic flow past blunt bodies. The hyperbolic approximation obtained adequately describes the interaction between the stream and surfaces for internal and external flows and can be used over a wide Mach number range at moderate and high Reynolds numbers. Examples of the calculation of viscous mixed flows in a Laval nozzle with large longitudinal throat curvature and in a shock layer in the neighborhood of a sphere and a large-aspect-ratio hemisphere-cylinder are given. The problem of determining the drag coefficient of cold and hot spheres is solved in a new formulation for supersonic air flow over a wide range of Reynolds numbers. In the case of low and moderate Reynolds numbers a drag reduction effect is detected when the surface of the sphere is cooled.     

Stability of Creeping Flows of Maxwell Fluids

We derive rigorous criteria for the linear stability of viscoelastic flows under periodic boundary conditions. The results are based on recent work of R. Shvydkoy (Commun Math Phys 265:507–545, 2006).     

Incompressible Viscous Flows in Borderline Besov Spaces

We establish two new estimates for a transport-diffusion equation. As an application we treat the problem of global persistence of the Besov regularity with , for the two-dimensional Navier–Stokes equations with uniform bounds on the viscosity. We provide also an inviscid global result.