Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

ENUMERATION OF ROOTED VERTEX NON-SEPARABLE PLANAR MAPS

In a rooted planar map, the rooted vertex is said to be non-separable if the vertex onthe boundary of the outer face as an induced graph is not a cut-vertex. In this paper, the author derives a functional equation satisfied by the enumeratingfuuction of rooted vertex non-separable planar maps dependent on the edge number and thenumber of the edges on the outer face boundary, finds a parametric expression of itssolution, and obtains an explicit formula for the function. Particularly, the number of rooted vertex non-separable maps only replying on theedge number and that of rooted vertex non-separable tree-like maps defined in [4] accordingto the two indices, the edge number and the number of the edges on the outer face boundary,or only one index, the edge number, are also determined.     

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.     

Spanning Pre-disks in a Compression Body

A properly embedded essential planar surface P (not a disk) in a compression body V is called a spanning pre-disk with respect to J,if one boundary component of P is lying in (З)+V and all other boundary components of P are lying in (З)_ V and coplanar with J.In this paper,we show that the number of boundary components of spanning pre-disks in a compression body is unbounded.But the number of a maximal collection of spanning pre-disks is bounded.     

Horizontal convection: Effect of aspect ratio on Rayleigh number scaling and stability

Horizontal convection in a rectangular enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically using a spectral-element discretization for velocity and temperature fields. A Boussinesq approximation is employed to model buoyancy. The emphasis of this study is on the scaling of mean Nusselt number and boundary layer quantities with aspect ratio and Rayleigh number.     

Initial boundary value problem for a nonconservative system in elastodynamics

This article is concerned with the initial boundary value problem for a nonconservative system of hyperbolic equation appearing in elastodynamics in the space time domain x > 0, t > 0. The number of boundary conditions, to be prescribed at the boundary x = 0, depends on the number of characteristics entering the domain. Because our system is nonlinear, the characteristic speeds depends on the unknown and the direction of the characteristics curves are known apriori. As it is well known, the boundary condition has to be understood in a generalised way. One of the standard way is using vanishing viscosity method. We use this method to construct solution for a particular class of initial and boundary data, namely the initial and boundary datas that lie on the level sets of one of the Riemann invariants.     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

A 2D level set finite element grain coarsening study with heterogeneous grain boundary energies

In a previous article (Fausty et al., 2018) a new level-set finite element formulation for pure grain growth with heterogeneous grain boundary energies (i.e. one energy per grain interface) was developed and validated for simple configurations. In this work, the authors apply this new tool to the simulation of two dimensional grain growth of polycrystals using different disorientation dependent grain boundary energy functions. The results of these full-field calculations are assessed using the time dependent evolution of the following criteria: grain size, grain number, total interface energy, grain boundary disorientation distribution, grain boundary energy distribution and number of neighboring grains distribution. Of particular interest is the relationship between the grain boundary energy function and the evolution of the grain boundary network in the sense of both its morphology and its constitution. Some notable results are that the disorientation distribution evolution is inversely correlated to the grain boundary energy function itself and that the kinetics of grain growth are heavily effected by the heterogeneity of the system.     

Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes

This work presents nonsimilar boundary layer solutions for double-diffusion natural convection near a sphere with constant wall heat and mass fluxes in a micropolar fluid. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. Higher vortex viscosity tends to retard the flow, and thus decreases the local convection heat and mass transfer coefficients, raising the wall temperature and concentration. Moreover, the local convection heat and mass transfer coefficients near a sphere in Newtonian fluids are higher than those in micropolar fluids.     

Exact boundary controllability of partial nodal profile for network of strings

Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a complete theory on the exact boundary controllability of partial nodal profile.     

Third order boundary value problem with finite spectrum on time scales

The eigenvalues of a class of third order boundary value problem on time scales is investigated. It is shown that this kind of third order boundary value problem has finite number of eigenvalues, and the same results on time scales are previously known only for even order cases. It can be illustrated that the number of eigenvalues depend on the partition of the time scale and the order of the equation.     

Comparison between exact thermal boundary condition and harmonic mean conductivity condition at the solid–fluid interface for finite thickness shrouded non-isothermal fin array

A numerical study is made comparing the exact thermal boundary condition and a harmonic mean conductivity condition at the solid–fluid interface for a finite thickness shrouded non-isothermal fin array. Results highlight that there exists a significant deviation of pressure drop across the length of the fin for the exact thermal boundary condition, which is as high as 20% as compared to that obtained using the harmonic mean conductivity condition. The exact thermal boundary condition forecasts relatively more non-isothermal fin as compared to a harmonic mean conductivity condition. The greater the fin spacing the larger the non-isothermal behavior of the fin and it also depends upon Grashof number as well as inlet fluid velocity. The larger the Grashof number the greater is the non-isothermal behavior of fin. The greater the inlet velocity, the larger is the non-isothermal behavior of fin. Bulk fluid temperature is over predicted by as much as 13% by the harmonic mean conductivity condition for larger fin spacing with highest Grashof number coupled with larger velocity. This deviation is only 6% for smaller fin spacing. Overall Nusselt number is over predicted for the harmonic mean conductivity condition as compared to exact thermal boundary condition. This over prediction is limited to about 8%.     

Inflow Treatment for the Simulation of Subsonic Jets with Turbulent Nozzle Boundary Layers

The state of the boundary layer at the nozzle exit of a circular nozzle-jet configuration has an important influence on the development of the shear layer and the emitted sound. Of special interest is the acoustic near-field obtained when the nozzle exit boundary layer is fully turbulent. The turbulent inflow generation and the inflow boundary treatment are important issues to be addressed. We use the Synthetic Eddy Method (SEM) to generate a turbulent inflow which reproduces mean flow and Reynolds stress profiles of specified reference data. The spatially and temporally varying synthetic fluctuations are imposed in the simulation by a forcing term added to the governing equations which is active in a small region downstream of the inflow boundary. This forcing in combination with characteristic boundary conditions allows for passing of upstream-propagating acoustic waves and avoids an uncontrolled drift of mean-flow quantities. We employ this inflow boundary treatment for a subsonic nozzle-jet flow simulation at a Reynolds number of ∼ 9500 and Mach number of 0.9. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimizing singularities of generic plane disks with immersed boundaries

A cooriented circle immersion into the plane can be extended to a stable map of the disk which is an immersion in a neighborhood of the boundary and with outward normal vector field along the boundary equal to the given coorienting normal vector field. We express the minimal number of fold components of such a stable map as a function of its number of cusps and of the normal degree of its boundary. We also show that this minimum is attained for any cooriented circle immersion of normal degree not equal to one. The first author is a research fellow of the Royal Swedish Academy of Sciences sponsored by the Knut and Alice Wallenberg foundation.     

The porous medium equation with nonlinear absorption and moving boundaries

We consider a nonlinear parabolic equation containing the porous medium operator and a nonlinear absorption term, which causes the appearance of a moving boundary. Basic behavior and regularity results are obtained for the solution and the moving boundary under two different boundary conditions. Also, the behavior of the solution and the moving boundary as time goes to infinity is investigated. Supported in part by NSF grant number MCS 8104220.     

Minimal Morse flows on compact manifolds

In this paper we prove, using the Poincaré-Hopf inequalities, that a minimal number of non-degenerate singularities can be computed in terms only of abstract homological boundary information. Furthermore, this minimal number can be realized on some manifold with non-empty boundary satisfying the abstract homological boundary information. In fact, we present all possible indices and types (connecting or disconnecting) of singularities realizing this minimal number. The Euler characteristics of all manifolds realizing this minimal number are obtained and the associated Lyapunov graphs of Morse type are described and shown to have the lowest topological complexity.     

The effect of boundary conditions on the onset of chaos in Rayleigh–Bénard convection using energy-conserving Lorenz models

The influence of 16 boundary conditions on linear and nonlinear stability analyses of Rayleigh–Bénard system is reported. A Stuart–Landau amplitude equation for the Rayleigh–Bénard system between stress-free, isothermal boundary conditions is derived and the procedure used in this derivation serves as guidance for constructing an appropriate Fourier–Galerkin expansion for the other 15 boundary conditions to derive a generalized Lorenz model. The influence of the boundary conditions comes within the coefficients of the generalized Lorenz model. It is shown that the obtained generalized Lorenz model is energy conserving and for certain boundary conditions it retains features of the classical Lorenz model. Further, the principle of exchange of stabilities is shown to be valid for the present problem and hence it is the steady-state, linearized version of the generalized Lorenz model which yields an analytical expression for the Rayleigh number. On minimizing this expression with respect to wave number the critical Rayleigh number at which the onset of regular convective motion occurs in the form of rolls is determined for all 16 boundary conditions. It is found that these values are in good agreement with those of previous investigations leading to the conclusion that the chosen minimal Fourier–Galerkin expansion is a valid one. Exhibition of chaotic motion in the generalized Lorenz system at the Hopf Rayleigh number is studied. The phase-space plots which indicate a clear-cut visualization of the transition from regular convective motion to chaotic motion in the generalized Lorenz system are presented. Further, existence of a developing region for chaos (mildly chaotic motion) and windows of periodicity are captured using the bifurcation diagrams. It is concluded from the phase-space plots and the bifurcation diagrams that the generalized Lorenz model for certain sets of boundary conditions retains all the features of the classical Lorenz model. Such a conclusion cannot be made by a linear stability analysis and the need thus for a nonlinear analysis is highlighted in the paper.