A Semi-implicit Finite Volume Scheme for Incompressible Two-Phase Flows

This paper presents a mass and momentum conservative semi-implicit finite volume(FV)scheme for complex non-hydrostatic free surface flows,interacting with moving solid obstacles.A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase,a solid phase,and the surrounding void.According to the so-called diffuse interface approach,the different phases and consequently the void are described by means of a scalar volume fraction function for each phase.In our numerical scheme,the dynamics of the liquid phase and the motion of the solid are decoupled.The solid is assumed to be a moving rigid body,whose motion is prescribed.Only after the advection of the solid volume fraction,the dynamics of the liquid phase is considered.As usual in semi-implicit schemes,we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly,while the pressure terms are treated implicitly.The non-conservative products arising in the transport equation for the solid volume fraction are treated by a path-conservative approach.The resulting semi-implicit FV discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure which can be efficiently solved with a nested Newton-type technique.The time step size is only limited by the velocities of the two phases contained in the domain,and not by the gravity wave speed nor by the stiff algebraic relaxation source term,which requires an implicit discretization.The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and moving solid phase.

Local existence and uniqueness of the incompressible Navier-Stokes-Landau-Lifshitz equations with the Dzyaloshinskii-Moriya interaction and V-flow term

In this paper,we prove that there exists a unique local solution for the Cauchy problem of a system of the incompressible Navier-Stokes-Landau-Lifshitz equations with the Dzyaloshinskii-Moriya interaction and V-flow term inR^(2) and R^(3).Our methods rely upon approximating the system with a perturbed parabolic system and parallel transport.

CONVEXITY OF THE FREE BOUNDARY FOR AN AXISYMMETRIC INCOMPRESSIBLE IMPINGING JET

This paper is devoted to the study of the shape of the free boundary for a threedimensional axisymmetric incompressible impinging jet.To be more precise,we will show that the free boundary is convex to the fluid,provided that the uneven ground is concave to the fluid.

INCOMPRESSIBLE LIMIT OF IDEAL MAGNETOHYDRODYNAMICS IN A DOMAIN WITH BOUNDARIES

We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary.The boundary condition is characteristic and the initial data is general.We first establish the uniform existence of classical solutions with respect to the Mach number.Then,we prove that the solutions converge to the solution of the incompressible MHD system.In particular,we obtain a stronger convergence result by using the dispersion of acoustic waves in the half space.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discretizations,we show that the simple bound-preserving limiter in Li et al.(SIAM J Numer Anal 56:3308–3345,2018)can enforce the strict bounds of the vorticity,if the velocity field satisfies a discrete divergence free constraint.For reducing oscillations,a modified TVB limiter adapted from Cockburn and Shu(SIAM J Numer Anal 31:607–627,1994)is constructed without affecting the bound-preserving property.This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.

Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows

Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.

New algorithm for solving 3D incompressible viscous equations based on projection method

A new algorithm based on the projection method with the implicit finite difference technique was established to calculate the velocity fields and pressure.The calculation region can be divided into different regions according to Reynolds number.In the far-wall region,the thermal melt flow was calculated as Newtonian flow.In the near-wall region,the thermal melt flow was calculated as non-Newtonian flow.It was proved that the new algorithm based on the projection method with the implicit technique was correct through nonparametric statistics method and experiment.The simulation results show that the new algorithm based on the projection method with the implicit technique calculates more quickly than the solution algorithm-volume of fluid method using the explicit difference method.

Incompressible Pairwise Incompressible Surfaces in Knot Exteriors

We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.

ZERO KINEMATIC VISCOSITY-MAGNETIC DIFFUSION LIMIT OF THE INCOMPRESSIBLE VISCOUS MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS

We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω⊂R^(3).It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient.The solution is uniformly bounded in a conormal Sobolev space and W^(1,∞)(Ω)which allows us to take the zero kinematic viscosity-magnetic diffusion limit.Moreover,we also get the rates of convergence in L^(∞)(0,T;L^(2)),L^(∞)(0,T;W^(1,p))(2≤p<∞),and L^(∞)((0,T)×Ω)for some T>0.

INCOMPRESSIBLE PAIRWISE INCOMPRESSIBLE SURFACES IN LINK COMPLEMENTS

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S3 - L. We discuss the properties that the surface F intersects with 2-spheres in S3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves （R-move and S2-move）, and define the characteristic number of the topological graph for F∩S2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating （or almost alternating） link complements. We prove that the genus of the surface equals zero if the component number of F∩S2＋（or F∩S2-） is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #（F） ≤8.

UNIFORM ATTRACTOR FOR NONAUTONOMOUS INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH A NEW CLASS OF EXTERNAL FORCES

This paper is joint with [27]. The authors prove in this article the existence and reveal its structure of uniform attractor for a two-dimensional nonautonomous incompressible non-Newtonian fluid with a new class of external forces.

Finite elastic torsional instability of incompressible thermo-hyperelastic cylinder

Torsional instability of an incompressible thermo-hyperelastic cylindrical rod, subjected to axial stretching and large torsions, is examined within the framework of finite elasticity. When the cylinder is stretched and twisted by a sufficiently large degree, a knot may form suddenly at one point. This inherent elastic instability is analyzed with the minimum potential energy principle and the critical values of torsion are obtained. The distribution of stresses as well as the tensile force and the torque are studied. Effect of tem- perature change is specifically discussed.

EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION TO THE INCOMPRESSIBLE NAVIER-STOKES-LANDAU-LIFSHITZ MODEL IN 2-DIMENSION

In this paper, we prove the existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations in two-dimension with finite energy.The main techniques is the Faedo-Galerkin approximation and weak compactness theory.

QUALITATIVE ANALYSIS OF DYNAMICAL BEHAVIOR FOR AN IMPERFECT INCOMPRESSIBLE NEO-HOOKEAN SPHERICAL SHELL

The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carded out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carded out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.

BOUNDARY LAYER ASYMPTOTIC BEHAVIOR OF INCOMPRESSIBLE NAVIER-STOKES EQUATION IN A CYLINDER WITH SMALL VISCOSITY

The object of this article is to study the boundary layer appearing at large Reynolds number （small viscosity ε） incompressible Navier Stokes Equation in a cylinder in space dimension three. These are Navier-Stokes equations linearized around a fixed velocity flow： the authors study the convergence as ε →0 to the inviscid type equations, the authors define the correctors needed to resolve the boundary layer and obtain convergence results valid up to the boundary and the authors also study the behavior of the boundary layer when, simultaneously, time and the Reynolds number tend to infinity, in which case the boundary layer tends to pervade the whole domain.

ENERGY DISSIPATION FOR WEAK SOLUTIONS OF INCOMPRESSIBLE MHD EQUATIONS

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D（u, B） that steins from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.

CAVITY FORMATION AND ITS VIBRATION FOR A CLASS OF GENERALIZED INCOMPRESSIBLE HYPER-ELASTIC MATERIALS

The problem of radial symmetric motion for a solid sphere composed of a class of generalized incompressible neo-Hookean materials, subjected to a suddenly applied surface tensile dead load, is examined.The analytic solutions for this problem and the motion equation of cavity that describes cavity formation and growth with time are obtained. The e?ect of radial perturbation of the materials on cavity formation and its motion is discussed. The plane of the perturbation parameters of the materials is divided into four regions. The existential conditions and qualitative properties of solutions of the motion equation of the cavity are studied in di?erent parameters’ regions in detail. It is proved that the cavity motion with time is a nonlinear periodic vibration. The vibration center is then determined.

QUALITATIVE ANALYSIS OF SPHERICAL CAVITY NUCLEATION AND GROWTH FOR INCOMPRESSIBLE GENERALIZED VALANIS-LANDEL HYPERELASTIC MATERIALS

A cavitated bifurcation problem is examined for a sphere composed of a class of generalized Valanis-Landel materials subjected to a uniform radial tensile dead-load. A cavitated bifurcation equation is obtained. An explicit formula for the critical value associated with the vari- ation of the imperfection parameters is presented. The distinguishing between the left-bifurcation and right-bifurcation of the nontrivial solution of the cavitated bifurcation equation at the critical point is made. It is proved that there exists a secondary turning bifurcation point on the nontrivial solution branch, which bifurcates locally to the left. It is shown that the dimensionless cavitated bifurcation equation is equivalent to normal forms with single-sided constraint conditions at the critical point by using the singularity theory. The stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed.

Cavitation for Incompressible Anisotropic Hyper-Elastic Materials

The cavitation problem in a solid sphere composed of an incompressible anisotropic hyper elastic material under a uniform radial tensile dead load was examined. A new analytical solution was obtained. The stress contributions were given and the jumping and concentration of stresses were discussed. The stability of solutions and the effect of the degree of anisotropy of the material were analyzed.

On the pressure and stress singularities induced by steady flows of incompressible viscous fluids

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, to date there has been relatively little explicit identification of stress singularities caused by fluid flows. In this study, stress and pressure singularities induced by steady flows of viscous incompressible fluids are asymptotically identified. This is done by taking advantage of an earlier result that the Navier-Stokes equations are locally governed by Stokes flow in angular corners. Findings for power singularities are confirmed by developing and using an analogy with solid mechanics. This analogy also facilitates the identification of flow-induced log singularities. Both types of singularity are further confirmed for two global configurations by applying convergence-divergence checks to numerical results. Even though these flow-induced stress singularities are analogous to singularities in solid mechanics, they nonetheless render a number of structural configurations singular that were not previously appreciated as such from identifications within solid mechanics alone.