Hydromagnetische Stabilität laminarer Grenzschicht an einer längsangeströmten ebenen Platte

Summary The purpose of the paper is to consider the stability for wavelike disturbances in the steady, twodimensional, laminar boundary layer of a magnetic field, which is applied uniformly normal to the flat plate. The results show that the critical Reynolds number (R _c ^* ) increases remarkably with the characteristic parameter (). The increase of the critical Reynolds number depends not only on the shape parameter of the velocity distribution in the boundary layer but also on the peculiarity of the velocity profile. It is also found that the boundary layer holds itself laminar all over the flat plate, when the magnetic parameterN is greater than 1.25×10^–7, then a reduction of the skin-frictin drag might be expeced.     

The Trailing Edge Problem For Mixed Convection Flow Past a Horizontal Plate

The influence of buoyancy onto the boundary‐layer flow past a horizontal plate aligned parallel to a uniform free stream is characterized by the buoyancy parameter K = Gr/Re^5/2 where Gr and Re are the Grashof and Reynolds number, respectively. An asymptotiy analysis of the complete flow field including potential flow, boundary layer, wake and interaction region is given for small buoyancy parameters and large Reynolds numbers in the distinguished limit KRe^1/4 = O(1). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Small Vorticity Nonlinear Critical Layer for Kelvin Modes on a Vortex

We consider in this paper the propagation of neutral modes along a vortex with velocity profile being the radial coordinate. In the linear stability theory governing such flows, the boundary in parameter space separating stable and unstable regions is usually comprised of modes that are singular at some value of r denoted r_c , the critical point. The singularity can be dealt with by adding viscous and/or nonlinear effects within a thin critical layer centered on the critical point. At high Reynolds numbers, the case of most interest in applications, nonlinearity is essential, but it develops that viscosity, treated here as a small perturbation, still plays a subtle role. After first presenting the scaling for the general case, we formulate a nonlinear critical layer theory valid when the critical point occurs far enough from the center of the vortex so that the vorticity there is small. Solutions are found having no phase change across the critical layer thus permitting the existence of modes not possible in a linear theory. It is found that both the axial and azimuthal mean vorticity are different on either side of the critical layer as a result of the wave–mean flow interaction. A long wave analysis with O (1) vorticity leads to similar conclusions.     

Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC ²-function at both isolated critical point and infinity.     

Viscosity vanishing limit of the nonlinear pipe magnetohydrodynamic flow with diffusion

We establish viscosity vanishing limit of the nonlinear pipe magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed diffusion. The convergence is verified under various Sobolev norms, including the L^∞(H¹) norm.     

A new approach to the nonlinear stability of viscous flow in a coplanar magnetic field

We present a new Lyapunov function for laminar flow, in the x‐direction, between two parallel planes in the presence of a coplanar magnetic field for three‐dimensional perturbations with stress‐free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(R_e) and magnetic Reynolds numbers(R_m) below π²/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.     

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L ² least-squares scheme or a weighted L ² least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L ² least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L ² least-squares scheme seems to be better than the iterative L ² least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.     

Nonlinear Stability of Nonstationary Cross-Flow Vortices in Compressible Boundary Layers

The nonlinear evolution of long-wavelength non stationary cross-flow vortices in a compressible boundary layer is investigated; the work extends that of Gajjar [1] to flows involving multiple critical layers. The basic flow profile considered in this paper is that appropriate for a fully three-dimensional boundary layer with O(1) Mach number and with wall heating or cooling. The governing equations for the evolution of the cross-flow vortex are obtained, and some special cases are discussed. One special case includes linear theory, where exact analytic expressions for the growth rate of the vortices are obtained. Another special case is a generalization of the Bassom and Gajjar [2] results for neutral waves to compressible flows. The viscous correction to the growth rate is derived, and it is shown how the unsteady nonlinear critical layer structure merges with that for a Haberman type of viscous critical layer.     

Turbulence as a nonequilibrium phase transition

The transition from laminar to turbulent flow is studied on the basis of an exact equation for the averaged velocity and an approximate nonlinear equation for the Reynolds stress . The stationary state can be determined from the condition of minimum of a functional that is analogous to the Landau functional in the theory of phase transitions. The Reynolds stress plays the role of a parameter. It is shown that a nontrivial solution for corresponding to a steady turbulent regime exists only for Reynolds numbersR that exceed a certain critical valueR _cr. The results of a numerical calculation of the profile of the averaged velocity, the friction coefficient, and the Reynolds stress in a wide range of values ofR agree well with experimental data for channel flow.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 2, pp. 293–311, August, 1992.     

A Suggested Mechanism for Nonlinear Wall Roughness Effects on High Reynolds Number Flow Stability

The interactions between an uneven wall and free stream unsteadiness and their resultant nonlinear influence on flow stability are considered by means of a related model problem concerning the nonlinear stability of streaming flow past a moving wavy wall. The particular streaming flows studied are plane Poiseuille flow and attached boundary-layer flow, and the theory is presented for the high Reynolds number regime in each case. That regime can permit inter alia much more analytical and physical understanding to be obtained than the finite Reynolds number regime; this may be at the expense of some loss of real application, but not necessarily so, as the present study shows. The fundamental differences found between the forced nonlinear stability properties of the two cases are influenced to a large extent by the surprising contrasts existing even in the unforced situations. For the high Reynolds number effects of nonlinearity alone are destabilizing for plane Poiseuille flow, in contrast with both the initial suggestion of earlier numerical work (our prediction is shown to be consistent with these results nevertheless) and the corresponding high Reynolds number effects in boundary-layer stability. A small amplitude of unevenness at the wall can still have a significant impact on the bifurcation of disturbances to finite-amplitude periodic solutions, however, producing a destabilizing influence on plane Poiseuille flow but a stabilizing influence on boundary-layer flow.     

Large Mass Global Solutions for a Class of L 1-Critical Nonlocal Aggregation Equations and Parabolic-Elliptic Patlak-Keller-Segel Models

We consider a class of L ¹ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the L ¹ scaling symmetry and we prove that all ‘sufficiently spread out’ initial data, even with supercritical mass, results in global, decaying solutions. In particular, we produce decaying solutions with arbitrary mass in cases for which finite time blow up solutions or non-decaying solutions are also known to exist for sufficiently large mass. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with supercritical mass blow up in finite time. The results with linear diffusion are proved using properties of the Fokker-Planck semi-group whereas the results with nonlinear diffusion are proved using a more interesting bootstrap argument coupling the entropy-entropy dissipation methods of the porous media equation together with higher L ^p estimates similar to those used in small-data and local theory for PKS-type equations.     

Large-Amplitude Long-Wave Instability of a Supersonic Shear Layer

For sufficiently high Mach numbers, small disturbances on a supersonic vortex sheet are known to grow in amplitude because of slow nonlinear wave steepening. Under the same external conditions, linear theory predicts slow growth of long-wave disturbances to a thin supersonic shear layer. An asymptotic formulation that adds nonzero shear-layer thickness to the weakly nonlinear formulation for a vortex sheet is given here. Spatial evolution is considered for a spatially periodic disturbance having amplitude of the same order, in Reynolds number, as the shear-layer thickness. A quasi-equilibrium inviscid nonlinear critical layer is found, with effects of diffusion and slow growth appearing through a nonsecularity condition. Other limiting cases are also considered, in an attempt to determine a relationship between the vortex-sheet limit and the long-wave limit for a thin shear layer; there appear to be three special limits, corresponding to disturbances of different amplitudes at different locations along the shear layer.     

Strong stability of stationary solutions and Karush-Kuhn-Tucker points in nonlinear optimization

The concepts of strongly stable stationary solutions (in Kojima's sense) and of strongly regular Karush-Kuhn-Tucker points (in Robinson's sense) for optimization problems with twice differentiable data are crucial in theory and applications of nonlinear optimization with data perturbations. In this paper we give interconnections between both concepts and extend some ideas to standard nonlinear programs withC ¹ data (underC ¹ perturbations). The main purpose of this paper is to survey several equivalent characterizations of strong stability in the classical case of programs withC ² data underC ² perturbations. The unified approach proposed here is essentially based on arguments from the analysis of Lipschitzian mappings.Sponsored by the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria.     

On the stability and transition for the Navier-Stokes-alpha model

The main aim of this paper is to investigate the stability and transition of the Navier-Stokes-alpha model. By using the continued-fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non-dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio-temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid-axis of the channel.     

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H¹₀ versus C¹₀ minimizers of a C¹-functional.     

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M₃, solute gradient, S₁, and Taylor number, T_A1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.     

Controllability and Hedgibility of Black-Scholes Equations with <Emphasis Type="Italic">N</Emphasis> Stocks

This paper is to investigate the controllability and observability properties of linear and certain nonlinear Black-Scholes (B-S) type equations consisting of N stocks in an appropriate bounded domain I of ℝ₊ ^N . In this model both the stock volatility and interest rate are influenced by the stock prices and the control which is related to the hedging ratio in option pricing of finance is distributed over a subdomain of I. The proof of the controllability result for the linear B-S equations relies on the suitable observability inequality for the associated adjoint problem, and for the nonlinear model, fixed point technique is applied. Our result leads to that the dynamic hedgibility in finance is proved in the context of controllability theory.     

Long Nonlinear Surface Waves in the Presence of a Variable Current

We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a current beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [ 1 ] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range of ). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes. In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profile being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner–Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.     

Upper-Branch Instability of Flow in Pipes of Large Aspect Ratio with Three-Dimensional Nonlinear Viscous Critical Layers

We consider the upper-branch neutral stability of flow in pipesof large aspect ratio, basically extending the work of F. T.Smith to the nonlinear regime. The inclusion of weak nonlinearityleads to an eigenproblem whose solution depends on the propertiesof three-dimensional nonlinear critical layers. Two specialcases are considered. The first is for very small amplitude perturbations, where R is a Reynolds numberbased on the height of the tube and which is assumed large.Then a fully analytical solution of the three-dimensional criticallayers is possible, from which the linear results of Smith maybe deduced. The second case studied is that of flow in a rectangularpipe, where a solution of the nonlinear critical layer problemcan be obtained. Further analysis of neutral modes in this lattercase suggests the possible existence, inter alia, of neutralmodes for finite aspect ratio tubes. These modes depend on thescaled amplitude and have O(1) wavespeeds.