Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.     

The effect of buoyancy on upper-branch Tollmien-Schlichting waves

We investigate the effect of buoyancy on the upper-branch linearstability characteristics of an accelerating boundary-layerflow. The presence of a large thermal buoyancy force significantlyalters the stability structure. As the factor G (which is relatedto the Grashof number of the flow, and defined in Section 2)becomes large and positive, the flow structure becomes two layeredand disturbances are governed by the Taylor-Goldstein equation.The resulting inviscid modes are unstable for a large componentof the wavenumber spectrum, with the result that buoyancy isstrongly destabilizing. Restabilization is encountered at sufficientlylarge wavenumbers. For G large and negative the flow structureis again two layered Disturbances to the basic flow are nowgoverned by the steady Taylor—Goldstein equation in themajority of the boundary layer, coupled with a viscous walllayer. The resulting eigenvalue problem is identical to thatfound for the corresponding case of lower-branch Tollmien—Schlichtingwaves, thus suggesting that the neutral curve eventually becomesclosed in this limit.     

Amplitude-dependent Stability of Boundary-layer Flow with a Strongly Non-linear Critical Layer

The amplitude-dependent neutral stability properties, mainlyof an accelerating boundary-layer flow, are studied theoreticallyfor large Reynolds numbers when the disturbance size is sufficientlylarge to provoke a strongly non-linear critical layer withinthe flow field. The theory has a rational basis aimed at a detailedunderstanding of the delicate physical balances controllingstability. It shows that when the fundamental disturbance size rises to O(R^-1/3, where R is the Reynolds number based on theboundary-layer thickness, the neutral wavelength shortens andthe wavespeed increases in such a way that they become comparablewith the typical thickness and speed, respectively, of the basicflow. In this Rayleigh-like situation a new (previously negligible)feature emerges, that of a substantial pressure variation acrossthe critical layer, which strongly affects the jump conditionson the Rayleigh solutions holding outside the critical layer.As a result of the strong non-linearity the total velocity jumpis affected non-linearly by the critical layer vorticity, whilein contrast the phase shift remains linearly dependent on thevorticity. Furthermore, it is shown that the phase shift, notthe total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbanceis greater than the basic flow. The link between the viscouseffects in the wall layers and in the critical layer fixes theamplitude-dependence of the neutral modes throughout. As thedisturbance amplitude increases the critical layer with vorticitytrapped within it moves toward the edge of the boundary layerand is forced to leave the boundary layer when exceeds O(R^-1/3,if neutral stability is to be maintained. This departure israther abrupt, involving a dependence on (scaled amplitude)^–12.A study of the more practical application to temporally growingdisturbances should be interesting.     

Logarithmic Orbifold Euler Numbers of Surfaces with Applications

We introduce orbifold Euler numbers for normal surfaces withboundary Q-divisors. These numbers behave multiplicatively underfinite maps and in the log canonical case we prove that theysatisfy the Bogomolov–Miyaoka–Yau type inequality.Existence of such a generalization was earlier conjectured byG. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241–282].Most of the paper is devoted to properties of local orbifoldEuler numbers and to their computation. As a first application we show that our results imply a generalizedversion of R. Holzapfel's ‘proportionality theorem’[Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg,Braunschweig, 1998)]. Then we show a simple proof of a necessarycondition for the logarithmic comparison theorem which recoversan earlier result by F. Calderón-Moreno, F. Castro-Jiménez,D. Mond and L. Narváez-Macarro [Comment. Math. Helv.77 (2002) 24–38]. Then we prove effective versions of Bogomolov's result on boundednessof rational curves in some surfaces of general type (conjecturedby G. Tian [Springer Lecture Notes in Mathematics 1646 (1996)143–185)]. Finally, we give some applications to singularitiesof plane curves; for example, we improve F. Hirzebruch's boundon the maximal number of cusps of a plane curve. 2000 MathematicalSubject Classification: 14J17, 14J29, 14C17.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

Stability of flames in an exothermic-endothermic system

The propagation of a premixed laminar flame supported by anexothermic chemical reaction under adiabatic conditions butsubject to inhibition through a parallel endothermic chemicalprocess is considered. The temporal stability to longitudinalperturbations of any resulting flames is investigated. The heatloss through the endothermic reaction, represented by the dimensionlessparameter , has a strong quenching effect on wave propagation.The wave speed–cooling parameter (, c) curves are determinedfor a range of values of the other parameters. These curvescan be monotone decreasing or S-shaped, depending on the valuesof the parameters ß, representing the rate at whichinhibitor is consumed relative to the consumption of fuel, µ,the ratio of the activation energies of the reactants and theLewis numbers. This gives the possibility of having either one,two or three different flame velocities for the same value ofthe cooling parameter . For Lewis numbers close to unity, whenthere are three solutions, two of them are stable and one isunstable, with two saddle-node bifurcation points on the (,c) curve. For larger values of the Lewis numbers there is aHopf bifurcation point on the curve, dividing it into a stableand an unstable branch. The saddle-node and Hopf bifurcationcurves are also determined. The two curves have a common, Takens–Bogdanovbifurcation point.     

On the asymptotic behaviour for an electromagnetic system with a dissipative boundary condition

In this work we study some properties of solutions for the systemdescribing a three-dimensional non-homogeneous non-conductingdielectric with a general boundary condition with memory. Wefirst show the existence of the inverse of this boundary condition,which allows us to introduce a boundary free energy, similarto the one considered by Fabrizio & Morro (1996, Arch. Rat.Mech. Anal., 136, 359–381). Then, we prove existence anduniqueness theorems for weak and strong solutions of the evolutiveproblem in a finite time interval. Moreover, following Rivera& Olivera (1997, Boll. U.M.I., 11-A, 115–127), weexamine some dissipative properties of the boundary conditionand of its inverse and we give a useful energy estimate. Finally,when there is no memory in the boundary condition the exponentialdecay of the solution is proved.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

The Rotation of a Gravitating Sphere in a Monatomic Gas,II

The purpose of this work is to study the fluid motion caused by the high speed rotation of a gravitating sphere in a monatomic gas. It has been possible to find a stable steady solution only for very small Prandtl number, which can be interpreted to mean an optically thick gas. The flow is characterized by a flat radial jet in the equatorial plane and a viscous boundary layer on the spherical surface which, in some cases, lies beneath a thermal boundary layer. That the outer region must be hydrostatic puts very stringent constraints on the associated velocity field which necessitate still another boundary layer on the sphere. This last layer is shown to be unstable to small disturbances in certain temperature ranges. Finally, a similar solution that exists for order one Prandtl number must be disregarded because this last boundary layer is always unstable.     

An Analytic Approach for Calculating Absolutely Unstable Inviscid Modes of the Boundary Layer on a Rotating Disk

An analytical treatment of inviscidly absolutely unstable modes is pursued using the long-wavelength asymptotic approach. It is shown using the inviscid Rayleigh scalings in conjunction with the linear critical layer theory that the rotating-disk boundary layer flow undergoes a region of absolute instability for some small azimuthal wave numbers. The analytically calculated branch points for the absolute instability are found to be in good agreement with those obtained via a numerical solution of the inviscid Rayleigh equation.     

The Effect of Sidewalls on Overstable Convection in a Rotating Fluid

A simple two-dimensional model is used to demonstrate some interestingeffects which arise when Chandrasekhar's (1962) theory of overstableconvection in an infinite rotating fluid layer is modified totake account of lateral walls. The aim of the investigationis to determine how sidewalls aligned with the convective rollsaffect the critical Rayleigh number and frequency of oscillationand also how the overstable eigensolutions are related to thepreviously determined stationary solutions of the equations(Daniels, 1977). For containers of large aspect ratio, L, thecritical Rayleigh number for overstability is R_o+O(L^–1)(where R_o is the value for the infinite layer) and in the neighbourhoodof this single perturbed value it is found that there is aninfinite spectrum of overstable eigenvalues with frequencieswhich differ by O(L^–1). The O(L^–1) correction toR_o is determined analytically for the case of small Prandtlnumber and rapid rotation.     

Long-wavelength thermal convection in a weak shear flow

The problem of thermal convection in an imposed shear flow isexamined for a horizontal layer of fluid between poorly conductingboundaries. The horizontal scale H of the convective motionnear its onset is much greater than the depth h of the fluidlayer, with h/H being proportional to the one-fourth power ofa Biot number appearing in the condition applied to the temperatureat the horizontal boundaries. It is known that an asymptoticexpansion in powers of h/H yields a nonlinear long-wavelengthevolution equation for the depth-averaged temperature fieldthat is spatially isotropic in the absence of an imposed shearflow, but is strongly anisotropic for ‘strong’ shear.We derive in this paper a nonlocal long-wavelength equationthat bridges these two cases, and that contains each case inthe zero-shear and large-shear limits. Using this evolutionequation, we show how the shear flow stabilizes the longitudinalrolls to the zigzag instability, and how a preference for asquare planfonn on a periodic square lattice gives way to apreference for longitudinal rolls near onset. The longitudinalrolls may then become unstable as the Rayleigh number is increased.The analytical work is illustrated by some numerical simulationsof the full three-dimensional Boussinesq Navier-Stokes equations.The problem of pattern selection on a hexagonal lattice is alsodiscussed, and some new results are presented.     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

Unsteady Magnetohydrodynamic Flows

It is easy enough to deduce from the exact solution for a basicunsteady magnetohydrodynamic channel flow that there are twotime scales present. The possible existence of two time scalesenables one to formulate a method of constructing approximatesolutions for several flow problems. The approximation schemeemployed here has the advantage over the more usual methodsof boundary layer analysis in that it yields for each physicalquantity a single representation which is valid for all times.Moreover, it is shown that the error involved in using thisapproximate solution is suitably small.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

On the Non-linear Evolution of Gortler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtlervortices in a non-parallel boundary layer on a curved wall isinvestigated using perturbation methods based on the smallnessof e, the non-dimensional wavelength of the vortices. The crucialstage in the growth or decay of the vortices takes place inan interior viscous layer of thickness O(²) and length O().In this region the downstream velocity component of the perturbationcontains a mean flow correction of the same order of magnitudeas the fundamental which is driving it. Moreover, these functionssatisfy a pair of coupled non-linear partial differential equationswhich must be solved subject to some initial conditions imposedat a given downstream location. It is found that, dependingon whether the boundary layer is more or less unstable downstreamof this location, the initial disturbance either grows intoa finite amplitude Görtler vortex or decays to zero. Forthe Blasius boundary layer on a concave wall it is found thatGörtler vortices can only develop if the rate of increaseof curvature of the wall is sufficiently large. In this casethe finite amplitude solution which develops initially in an-neighbourhood of the position where the disturbance is introducedchanges its structure further downstream. This structure isinvestigated at a distance O() (with 0< <1) downstreamof the above -neighbourhood. In this régime the downstreamfundamental velocity component has an elliptical profile overmost of the flow field. However, in two thin boundary layerslocated symmetrically either side of the centre of the viscouslayer the fundamental velocity component decays exponentiallyto zero. The locations of these layers are determined by aneigenvalue problem associated with the one-dimensional diffusionequation. The mean flow correction persists both sides of theboundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposedinitial conditions and appears to be the ultimate state of anyinitial disturbance. However, in the initial stages of the growthof the vortex, some surprising flows are possible. For example,it is possible to set up a vortex flow similar to that observedby Wortmann (1969) which consists of a sequence of cells inclinedat an angle to the vertical.     

Absolute Instability of Incompressible Boundary Layer over a Compliant Plate

An incompressible boundary layer on a compliant plate is considered. The influence exerted by the tensile stress and bending stiffness of the plate on the stability of the boundary layer is investigated in the limit of high Reynolds numbers on the basis of the triple-deck theory. It is shown that upstream-propagating growing waves can be generated in a certain range of parameters characterizing the plate properties. As a result, the flow becomes absolutely unstable in the conventional sense.     

Ising model in half-space: A series of phase transitions in low magnetic fields

For the Ising model in half-space at low temperatures and for the “unstable boundary condition,” we prove that for each value of the external magnetic field μ, there exists a spin layer of thickness q(μ) adjacent to the substrate such that the mean spin is close to −1 inside this layer and close to +1 outside it. As μ decreases, the thickness of the (−1)-spin layer changes jumpwise by unity at the points μ_q, and q(μ) → ∞ as μ → +0. At the discontinuity points μ_q of q(μ), two surface phases coexist. The surface free energy is piecewise analytic in the domain Re μ > 0 and at low temperatures. We consider the Ising model in half-space with an arbitrary external field in the zeroth layer and investigate the corresponding phase diagram. We prove Antonov’s rule and construct the equation of state in lower orders with the precision of x⁷, x = e^−7ɛ. In particular, with this precision, we find the points of coexistence of the phases 0, 1, 2 and the phases 0, 2, 3, where the phase numbers correspond to the height of the layer of unstable spins over the substrate. Dedicated to Roland L’vovich Dobrushin __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 220–261, November, 2007.     

Stability of uniformly Morse-Smale gradient-like numerical methods for flows

In Garay (1996, Numer. Math., 72, 449–479) and Li (1997b,SIAM J. Math. Anal., 28, 381–388), it was shown that thequalitative properties of a Morse–Smale gradient-likeflow are preserved by its numerical approximations. In thispaper, we show that the qualitative properties of a family ofuniformly Morse–Smale gradient-like numerical methodsare preserved by the approximated flow. The techniques usedin the study of the structural stability theorem for diffeomorphismsare the main tools for this work.     

Multipeak Solutions for the Neumann Problem of an Elliptic System of Fitzhugh-Nagumo Type

For systems of elliptic equations of Fitzhugh–Nagumo typeon bounded domains and with small diffusion in one equation,we construct solutions with multiple sharp peaks close to eachother and close to, but not on, the boundary. This is a strikingcontrast to results for scalar equations. For some symmetric domains, we also construct similar multipeaksolutions except that here the peaks are not close to each other.2000 Mathematics Subject Classification 35J50 (primary), 93C15(secondary).