Uniqueness and stability of slowly oscillating periodic solutions of delay equations with bounded nonlinearity

We study slowly oscillating periodic solutions of delay equations with small parameters. When the nonlinearity has finite and nonzero limits at infinities, the appearance of these solutions and their periods can be found though asymptotic analysis. Under further natural assumptions on the nonlinearity, we prove that slowly oscillating periodic solutions are unique and asymptotically stable when parameters are sufficiently small.     

Symmetry-breaking at non-positive solutions of semilinear elliptic equations

We consider symmetry-breaking bifurcations at non-positive, radially symmetric solutions of semilinear elliptic equations on a ball with Dirichlet boundary conditions. For nonlinearities which are asymptotically affine linear, we find solutions at which the symmetry breaks. The kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group O(n). For this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough. Thus we obtain, by the equivariant branching lemma, a large variety of isotropy subgroups of O(n) which occur as symmetries of the bifurcating solution branches.     

Solutions of the Jeffery-Hamel Problem Regularly Extendable in the Reynolds Number

The problem of steady viscous flow in a convergent channel is analyzed analytically and numerically for small, moderately large and asymptotically large Reynolds numbers over the entire range of allowed convergence angles. Attention is focused on regularly extendable problem solutions, for which purpose a high-accuracy hybrid numerical-analytical method of accelerated convergence and extension in a parameter is developed. For sufficiently large angles, the existence of tri-modal regimes symmetrical about the bisectrix and containing in- and outflow regions is established. The evolution of the velocity profiles with unbounded increase in the Reynolds number is investigated. Flow regimes for the critical convergence angle, which cannot be regularly extended in the parameter, are also studied. Several novel hydromechanical effects are noted.     

Asymptotic Behavior of Solutions of the Compressible Navier�CStokes Equation Around the Plane Couette Flow

Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L ² Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L ² norm as solutions of an n − 1 dimensional linear heat equation with a convective term.     

Separating Dissipative Pulses: The Exit Manifold

We prove that if a reaction-diffusion equation (in one space dimension) has asymptotically stable, exponentially localized traveling wave solutions then there are solutions of the system which are nearly the linear superposition of two such pulses moving in opposite directions away from one another. Moreover, such solutions are themselves asymptotically stable. This result is meant to complement analytic or numeric studies into interactions of such pulses over finite times which might result in the scenario treated here. Since the pulses are moving in opposite directions, it is not possible to put the problem into a moving reference frame which renders the linear problem autonomous. We overcome this difficulty by embedding the original system in a larger one wherein the linear part can be written as a time independent piece plus another piece which, even though it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.     

Pressure driven flow of dilute polymer solutions in a channel: Analytic results for very small and for very large channels

Kinetic theory of dilute macromolecular solutions is applied to pressure driven flow in a small channel where wall- (and interfacial) layers have to be reckoned with. The complete rheology is studied. It turns out that for very small channels both the shear stress and the normal stress are an order of magnitude larger than corresponding quantities in simple shear. On the other hand, when the channel is so wide that the wall layers are very thin in comparison, agreement with results appropriate for simple shear is found. The volume flow rate-pressure difference relation is derived and compared to the prediction which utilizes the slip velocity concept. For very small channels, this concept is five orders of magnitude off, but reproduces asymptotically correct results for very large channels.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

New Theory of Flight

We present a new mathematical theory explaining the fluid mechanics of subsonic flight, which is fundamentally different from the existing boundary layer-circulation theory by Prandtl–Kutta–Zhukovsky formed 100 year ago. The new theory is based on our new resolution of d’Alembert’s paradox showing that slightly viscous bluff body flow can be viewed as zero-drag/lift potential flow modified by 3d rotational slip separation arising from a specific separation instability of potential flow, into turbulent flow with nonzero drag/lift. For a wing this separation mechanism maintains the large lift of potential flow generated at the leading edge at the price of small drag, resulting in a lift to drag quotient of size 15–20 for a small propeller plane at cruising speed with Reynolds number \({Re\approx 10^{7}}\) and a jumbojet at take-off and landing with \({Re\approx 10^{8}}\) , which allows flight at affordable power. The new mathematical theory is supported by computed turbulent solutions of the Navier–Stokes equations with a slip boundary condition as a model of observed small skin friction of a turbulent boundary layer always arising for \({Re > 10^{6}}\) , in close accordance with experimental observations over the entire range of angle of attacks including stall using a few millions of mesh points for a full wing-body configuration.     

SIMULATION OF SEDIMENTATION OF TWO CIRCULAR PARTICLES WITH COLLISION CONSIDERED IN VERTICAL CHANNEL

D2Q9 model of lattice Boltzmann equation method was used to simulate the sedimentation of two circular particles in a bounded two dimension channel. The characteristics of the sedimentation shows some periodicity for the Reynolds number Re chosen, 0.1-20. The larger the Reynolds number, the stronger the interaction between the two particles and the larger the transversal displacements. For large Re, the two particles leading alternately; for small Re, the initially leading particle will keep its leading position and for moderate Re, the initially upper particle will get leading position and keep it. The influence of the initially relative position of the two particles on sedimentation is small. The width of the channel won＇t change the characteristics of the sedimentation as a whole, but will change the period of the sedimentation. The wider the channel, the longer the period will be.     

Existence, uniqueness and stability of steady flows of second and third grade fluids in an unbounded “pipe-like” domain

We study steady motions of viscous incompressible third-grade fluids in unbounded channels with arbitrary shape. Such flows exist for small fluxes, due to a pressure drop. We prove that they are asymptotically stable in time, provided the viscosity is sufficiently large, and the initial condition on the perturbation sufficiently small.     

Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer

The paper addresses a contact problem of the theory of elasticity,i.e.,the penetration of a circular indenter with a flat base into a soft functionally graded elastic layer.The elastic properties of a functionally graded layer arbitrarily vary with depth,and the foundation is assumed to be elastic,yet much harder than a layer.Approximated analytical solution is constructed,and it is shown that the solutions are asymptotically exact both for large and small values of characteristic dimensionless geometrical parameter of the problem.Numerical examples are analyzed for the cases of monotonic and nonmonotonic variations of elastic properties.Numerical results for the case of homogeneous layer are compared with the results for nondeformable foundation.     

Bistable diverter valve in microfluidics

Bistable diverter valves are useful for a large number of no-moving-part flow control applications, and there is a considerable interest in using them also in microfluidics, especially for handling small pressure-driven flows. However, with decreasing Reynolds number, the Coanda effect—on which the flow diverting effect depends—becomes less effective. Authors performed a study, involving flow visualisation, PIV experiments, measurements of the flow rates, and numerical flowfield computations, aimed at clarifying behaviour of a typical fluidic valve at low Reynolds numbers. A typical fluidic valve originally developed for high Re operation was demonstrated to be useful, though with progressively limited efficiency, down to surprisingly low Re values as small as Re = 800. Also observed was a previously not reported discontinuation in the otherwise monotonic decrease in performance at Re between 1,500 and 2,000.     

Artificial Boundary Conditions of Pressure Type for Viscous Flows in a System of Pipes

In a three-dimensional domain Ω with J cylindrical outlets to infinity the problem is treated how solutions to the stationary Stokes and Navier–Stokes system with pressure conditions at infinity can be approximated by solutions on bounded subdomains. The optimal artificial boundary conditions turn out to have singular coefficients. Existence, uniqueness and asymptotically precise estimates for the truncation error are proved for the linear problem and for the nonlinear problem with small data. The results include also estimates for the so called “do-nothing” condition.     

驱动长方形腔内流动非稳定性的数值模拟

本文对长宽比为２的驱动腔内流动进行了数值模拟．采用非均匀交错网格上的修正隐式Ｔｅｍａｍ格式，以及压力修正投影法，分别计算了Ｒｅ数为１００、４００、１０００、２０００、３０００、３５００、５０００、１００００的驱动长方形腔内流场。当Ｒｅ≤３０００时，流场收敛到定常状态；而Ｒｅ≥３５００时，只能得到渐近周期结果；其中应用了谱分析等方法说明数值是周期性变化，可见，Ｈｏｐｆ分叉点出现在Ｒｅ数３０００与３５００之间．     

Low Mach Number Limit of the Full Navier-Stokes Equations

The low Mach number limit for classical solutions of the full Navier-Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are taken into account. In particular, we consider general initial data. The equations lead to a singular problem, depending on a small scaling parameter, whose linearized system is not uniformly well-posed. Yet, it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the Mach number Ma ∈ (0,1], the Reynolds number Re ∈ [1,+∞] and the Péclet number Pe ∈ [1,+∞]. Based on uniform estimates in Sobolev spaces, and using a theorem of G. Métivier & S. Schochet [30], we next prove that the penalized terms converge strongly to zero. This allows us to rigorously justify, at least in the whole space case, the well-known computations given in the introduction of P.-L. Lions' book [26].     

On the Reynolds-number independence of mixed convection in a vertical channel subjected to asymmetric wall temperatures with and without flow reversal

The problem of mixed convection in a vertical channel with asymmetric wall temperatures including situations of flow reversal is studied numerically. The SIMPLER algorithm with a staggered grid system is employed to solve the corresponding numerical equations formulated by the finite-volume method. A second-order upwind scheme is used to model the convective term, and a suitable grid distribution is introduced. The ranges of the parameters studied are 0 r_t 1, 1 Re 1000, and 0 Gr/Re 500.

The numerical results, with the streamwise coordinate scaled by the Reynolds number (Re), show that solutions for the velocity and temperature fields are independent of the Reynolds number when Re 50, even in the presence of flow reversal. These solutions, however, are dependent on r_t and Gr/Re. Subsequently, correlations are proposed for the bulk temperature distribution and the local Nusselt numbers along the hot wall and the cold wall.     

考虑惯性效应的渗流方程及相关阻力标度律

本文基于体积平均法推导得到了多孔介质中考虑惯性效应时的局部线化宏观流动方程,由此可以递推得到较大雷诺数Re 时Navier-Stokes 方程的解,从而避免了直接求解Navier-Stokes 方程所带来的计算成本高和计算稳定性差的问题．针对正方形周期排列模型的算例表明,平均速度方向与宏观压力梯度方向并不总是一致,一般情况下,随着Re 增大,二者差异也会增大．当固定平均速度方向v ? 时,压差阻力在较大的Re 范围内存在标度律,标度指数约为3．该标度指数与弱惯性区域标度指数一致,但弱惯性区域Re 范围仅为0相似文献   

Resonance behaviour of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field

The oscillatory flow of a viscoelastic fluid in a circular pipe under the influence of a transversal magnetic field is studied. Exact solutions for the axial velocity and flow rate are presented. The velocity enhancement and the resonance behaviour are analysed both numerically and asymptotically in the case of small pipe radii. Approximations for the resonance frequencies and the achievable velocity enhancements are derived. Copyright © 2005 John Wiley & Sons, Ltd.     

Bifurcation of self-similar solutions describing a thermocapillary flow of a fluid in a thin layer

Thermocapillary flows of a fluid in a lamina with a rigid lower wall and a free upper surface, along which the temperature gradient is given in the radial direction, are investigated for large Marangoni numbers. Self-similar solutions which describe the axisymmetric flow regimes of a fluid without the circumferential velocity component are constructed numerically and asymptotically for a system of Prandtl equations. It is shown that a pair of new self-similar flow regimes of a fluid with rotation branches off from the regimes obtained. The new regimes ere calculated numerically and asymptotically. Rostov State University, Rostov-on-Don 344090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 137–142, May–June, 1999.