IMA Journal of Applied Mathematics - recent issues

Existence, upper and lower solutions and quasilinearization for singular differential equations

O'Regan, D., El-Gebeily, M. — 2008-04-01

In this paper, we discuss existence theorems in the presence of upper and lower solutions as well as the method of quasilinearization (QSL) for general non-linear second-order singular ordinary differential equations. We show the existence of solutions under the assumption of weak continuity of the non-linear part. If the non-linear part is monotone decreasing, a solution may be obtained by the QSL method as the strong limit of a quadratically convergent sequence of approximate solutions. Under stronger assumptions on the linear and the non-linear parts, a solution is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.

Transient diffusion-controlled smoulder propagation: a similarity solution approach

Adler, J., Herbert, D. M. — 2008-04-01

One-, two- and three-dimensional time-dependent smoulder propagation through solid reactants with plane bounding surfaces is considered. Propagation is maintained by the diffusion of oxidizer from the boundaries to the smouldering reaction zone. The resulting burnt solid is assumed to be porous and the unreacted solid is taken to be sufficiently dense for no oxidizer to be present. The activation energy of the reaction is taken to be so large that the exothermic reaction term has a delta-function behaviour. This enables the reaction zone to be approximated by a narrow reaction front and results in the equations of heat and mass transfer being decoupled away from the front. The assumption, based on experimental observations, that the reaction fronts propagate with a speed proportional to t^–1/2, where t is the time, permits the introduction of similarity variables. The resulting intermediate asymptotic equations, lying between those for very small and very large times, are solved and the equation of the smouldering fronts determined for each geometry is considered.

Shock initiation of explosives: the idealized condensed-phase model

Sharpe, G. J., Gorchkov, V., Short, M. — 2008-04-01

Many current models of condensed-phase explosives employ reaction rate law models where the form of the rate has a power-law dependence on pressure (i.e. proportional to pⁿ where n is an adjustable parameter). Here, shock-induced ignition is investigated using a simple model of this form. In particular, the solutions are contrasted with those from Arrhenius rate law models as studied previously. A large n asymptotic analysis is first performed, which shows that in this limit the evolution begins with an induction stage, followed by a sequence of pressure runaways, resulting in a forward propagating, decelerating, shockless supersonic reaction wave (a weak detonation). The theory predicts secondary shock and super-detonation formation once the weak detonation reaches the Chapman–Jouguet speed. However, it is found that secondary shock formation does not occur until the weak detonation has reached a point close to the initiating shock, whereas for Arrhenius rate laws the shock forms closer to the piston. Numerical simulations are then conducted for O(1) values of n, and it is shown that the idealized condensed-phase model can qualitatively describe a wide range of experimentally observed behaviours, from growth mainly at the shock, to smooth growth of a pressure pulse behind the shock, to cases where a secondary shock and possibly a super-detonation form. The numerics are used to reveal the different evolutionary mechanisms for each of these cases. However, the evolution is found to be sensitive to n, with the whole range of behaviours covered by varying n from about 3 to 5. The simulations also confirm the predictions of the theory that pressure-dependent rate laws are unable to describe homogeneous explosive scenarios where a super-detonation forms very close to the point of initial runaway.

Green's function of the Brinkman equation in a 2D anisotropic case

Kohr, M., Sekhar, G. P. R., Blake, J. R. — 2008-04-01

The purpose of this paper is to obtain the Green's function of the Brinkman equation in a 2D case of hydrodynamic anisotropy with respect to the permeability. The anisotropic nature of the permeability is assumed to be not space or time dependent. We use the method of Fourier transform which reduces the computation of the Green's function to the computation of the fundamental solution of a fourth-order partial differential equation. This research work has several applications in engineering and medicine to the motion of bodies in anisotropic porous media.

A Stefan problem modelling crystal dissolution and precipitation

van Noorden, T. L., Pop, I. S. — 2008-04-01

A simple 1D model for crystal dissolution and precipitation is presented. The model equations resemble a one-phase Stefan problem and involve non-linear and multivalued exchange rates at the free boundary. The original equations are formulated on a variable domain. By transforming the model to a fixed domain and applying a regularization, we prove the existence and uniqueness of a solution. The paper is concluded by numerical simulations.

The fundamental matrix of the system of linear elastodynamics in hexagonal media. Solution to the problem of conical refraction

Ortner, N., Wagner, P. — 2008-04-01

An explicit integral representation by single definite integrals of the fundamental matrix (Green's tensor) of the time-dependent system of hexagonal elastic media is derived. Thereby the problem of internal conical refraction in such media is solved.

Foreword: Andy King

Needham, D. J., Billingham, J. — 2008-01-30

Gravity-driven thin-film flow using a new contact line model

Billingham, J. — 2008-01-30

In this paper, we consider how a new model for the motion of a contact line, proposed by Shikhmurzaev (1993, Int. J. Multiphase Flow, 19, 589-–610), affects predictions for the gravity-driven flow of a thin film down an inclined plane. We find that for sufficiently thin films, the model reduces to Navier slip with the contact angle equal to its static value, while for thicker films the model has a character of its own, with a slip region that becomes larger, the thicker the film and a contact angle that increases as the thickness of the film increases.

Surface-tension-driven flow in a slender cone

Decent, S. P., King, A. C. — 2008-01-30

After a droplet has broken away from a slender thread or jet of liquid, the tip of the thread or jet recoils rapidly. At the moment of break-off, the tip of the thread/jet is observed to have the shape of a cone close to the bifurcation point. In this paper, we study the evolution of an ideal fluid which is initially conical, where the only force acting on the fluid is due to surface tension. We find an asymptotic solution to the problem in terms of the aspect ratio of the cone which is assumed to be small. Using a similarity transformation, which is valid for small times after the bifurcation, we identify a rapidly oscillating non-linear wave which propagates away from the tip, as observed in experiments.

Population-scale modelling of cellular chemotaxis and aggregation

Fozard, J. A., King, J. R. — 2008-01-30

Motivated by chemotaxis of, and especially aggregation within, populations of cells, we examine an extension of the Becker–Döring aggregation equations in which monomers undergo diffusion and advection in one spatial dimension, as well as attaching themselves to clusters of all sizes. We restrict our attention to irreversible aggregation, particularly for power-law rate coefficients. We examine the large-time behaviour of the initial-value problem on an infinite domain, both in the purely diffusive case and with advection. We also determine the large-time behaviour on a semi-infinite domain, with a non-zero Dirichlet condition imposed on the monomer concentration at the boundary. The asymptotic results are confirmed by numerical simulations.

Numerical solutions of a model for the propagation of a surface-catalysed flame in a tube

Sharpe, G. J., Falle, S. A. E. G., Billingham, J. — 2008-01-30

Numerical simulations of a surface-catalysed flame in a tube are performed, corresponding to an experiment where a premixed fuel is fed into a tube whose inner surface is coated with a catalyst. In these experiments, subsequent to ignition, a reaction wave can be seen as a red-hot region which propagates back along the tube towards the inlet, and is due to low temperature combustion occurring only on the inner surface of the tube where the catalyst is present. The solutions of a mathematical model for this behaviour show that initial-value problems do indeed result in such steadily propagating waves. The numerically obtained wave speeds and steady solution are compared to a previous large Damköhler number (D_a) asymptotic analysis using a simple reaction rate model, and agreement is very good even for moderately large values of D_a. However, for such Damköhler numbers, the wave speeds are found to be much larger than observed experimentally. Indeed, the simulations show that O(1) values of D_a are required to obtain the lower experimental wave speeds. Nevertheless, the wave speeds as a function of flow rate through the tube do not agree well with the preliminary experimental results for any choice of the parameters. A more realistic, Arrhenius reaction rate model is then considered. The Arrhenius model predicts a rapid change in temperature at the wave front, in much better agreement with the experiments than for the simpler reaction model.

Thermoelasticity and generalized thermoelasticity viewed as wave hierarchies

Scott, N. H. — 2008-01-30

It is seen how to write the standard form of the four partial differential equations in four unknowns of anisotropic thermoelasticity as a single equation in one variable, in terms of isothermal and isentropic wave operators. This equation, of diffusive type, is of the eighth order in the space derivatives and seventh order in the time derivatives and so is parabolic in character. After having seen how to cast the 1D diffusion equation into Whitham's wave hierarchy form, it is seen how to recast the full equation, for unidirectional motion, in wave hierarchy form. The higher order waves are isothermal and the lower order waves are isentropic or purely diffusive. The wave hierarchy form is then used to derive the main features of the solution of the initial-value problem, thereby bypassing the need for an asymptotic analysis of the integral form of the exact solution. The results are specialized to the isotropic case. The theory of generalized thermoelasticity associates a relaxation time with the heat flux vector and the resulting system of equations is hyperbolic in character. It is also seen how to write this system in wave hierarchy form which is again used to derive the main features of the solution of the initial-value problem. Simpler results are obtained in the isotropic case.

Non-classical shallow water flows

Edwards, C. M., Howison, S. D., Ockendon, H., Ockendon, J. R. — 2008-01-30

This paper deals with violent discontinuities in shallow water flows with large Froude number F. On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory, we show that, over a certain time-scale, this discontinuity may be described by a delta shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted. For flows on a sloping base, we show that for flow with an aspect ratio of O(F^–2) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a delta shock. The physical manifestation of this discontinuity is a small ‘tube’ of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base

The evolution of travelling wave-fronts in a hyperbolic Fisher model. I. The travelling wave theory

Needham, D. J., Leach, J. A. — 2008-01-30

In this paper, we consider an initial-value problem for a hyperbolic Fisher equation. In particular, we consider the corresponding travelling wave problem and establish the conditions under which permanent-form travelling wave solutions exist. Numerical simulations of the initial-value problem are also presented.

A combined BIE-FE method for the Stokes equations

Lukyanov, A. V., Shikhmurzaev, Y. D., King, A. C. — 2008-01-30

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’.

The role of thermal instability in star formation

Falle, S. A. E. G. — 2008-01-30

The observations tell us that the density in the giant molecular clouds in which stars are formed is inhomogeneous on a variety of scales, but it seems unlikely that this is due to the action of gravitational instability. This paper describes numerical calculations using an adaptive mesh refinement magnetohydrodynamics code that show that thermal instability may have an important role to play in the formation of this structure

Free convection boundary layers on a vertical surface in a heat-generating porous medium

Mealey, L., Merkin, J. H. — 2008-01-30

The natural convection boundary-layer flow on a solid vertical surface with heat generated within the boundary layer at a rate proportional to (T – T)^p (p ≥ 1) is considered. The surface is held at the ambient temperature T except near the leading edge where it is held at a temperature above ambient. The behaviour of the flow as it develops from the leading edge is examined and is seen to become independent of the initial heat input; however, it does depend strongly on the exponent p. For 1 ≤ p ≤ 2, the local heating eventually dominates at large distances and there is a convective flow driven by this mechanism. For p ≥ 4, the local heating does not have a significant effect, the fluid temperature remains relatively small throughout and the heat transfer dies out through a wall jet flow. For 2 < p < 4, the local heating has a significant effect at relatively small distances, with a thermal runaway developing at a finite distance along the surface.

Influence of rapid changes in a channel bottom on free-surface flows

Binder, B. J., Dias, F., Vanden-Broeck, J.-M. — 2008-01-30

Two-dimensional non-linear free-surface flows in a channel bounded below by an uneven bottom with rapid changes are considered. Numerical solutions are computed by a boundary integral equation method similar to that first introduced by King & Bloor (1987, J. Fluid Mech., 182, 193–208). Free-surface flows past localized disturbances, steps and sluice gates are calculated. In addition, weakly non-linear solutions are discussed.

The development of slugging in two-layer hydraulic flows

Needham, D. J., Billingham, J., Schulkes, R. M. S. M., King, A. C. — 2008-01-30

In this paper, we develop a hydraulic theory to describe the occurrence and structure of slugging in a confined two-layer gas–liquid flow generated by prescribed, constant, upstream volumetric flow rates in each layer. A linearized theory for the uniform flow is established, after which we use bifurcation theory to study fully non-linear periodic travelling wave structures. We find that a two-parameter family of such travelling wave solutions exists. Under given conditions, the volumetric flow rate constraint provides a relation between these two parameters. To select a unique periodic travelling wave solution, we require a further relation. We first investigate the conjecture that the periodic travelling wave solution selected in the initial value problem has the same wavelength as the linearly most temporally unstable mode. To do this, we solve the initial value problem numerically on a periodic domain. We find that the separation of the liquid slugs that form is much longer than the wavelength of the most unstable temporal mode. We then develop a different conjecture based on the convective instability of the long ‘tails’ of the available periodic travelling wave solutions, which leads to a better understanding of the wavelength selection process.

Method of integral equations for systems of difference equations in diffraction theory

Antipov, Y. A., Silvestrov, V. V. — 2007-11-19

A system of difference equations of the first order with meromorphic coefficients (not necessary periodic ones) subject to certain conditions of symmetry is analysed. These conditions arise in model problems of diffraction theory. A method for the constructive solution of the system of difference equations is proposed. It consists of three steps. First, it requires factorization of a certain function. Then, scalar integral equations with the same kernel and different right-hand sides should be solved. The kernel of the equations has a fixed singularity, and the solution belongs to a class of functions with a power singularity. Finally, arbitrary constants are fixed from some conditions which guarantee the equivalence of the original system of difference equations and the integral equations. The method is illustrated by solving a model electromagnetic problem of a plane wave diffracted from an anisotropic impedance half-plane at oblique incidence.

Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation

Bartolucci, D., Pistoia, A. — 2007-11-19

We prove the existence of nodal solutions for – u = sinh u with Dirichlet boundary conditions in bounded, 2D, smooth and non-smooth domains. Indeed, for positive and small enough, we show that there exist at least two pairs of solutions, which change sign exactly once, whose nodal lines intersect the boundary.

Surface waves supported by thin-film/substrate interactions

Steigmann, D. J., Ogden, R. W. — 2007-11-19

A systematic approximation to the linear equations for small-amplitude surface waves in an elastic half space, interacting with a residually stressed thin film of different material bonded to its plane boundary, is developed in powers of the film thickness, assuming the latter to be small compared to the wavelength of the disturbance. The theory is illustrated by calculating asymptotic expansions of the wave speeds for Love and Rayleigh waves valid to second order in the dimensionless film thickness for a transversely isotropic film bonded to an isotropic substrate.

A variational method for identifying a spacewise-dependent heat source

Johansson, B. T., Lesnic, D. — 2007-11-19

The inverse problem of determining a spacewise-dependent heat source for the parabolic heat equation using the usual conditions of the direct problem and information from one supplementary temperature measurement at a given instant of time is studied. This spacewise-dependent temperature measurement ensures that this inverse problem has a unique solution, but the solution is unstable and hence the problem is ill-posed. We propose a variational conjugate gradient-type iterative algorithm for the stable reconstruction of the heat source based on a sequence of well-posed direct problems for the parabolic heat equation which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterative procedure at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented which have the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure yields stable and accurate numerical approximations after only a few iterations.

The stability of the boundary layer on a compliant rotating disc

John, J.-A. L., Stephen, S. O. — 2007-11-19

The boundary layer over a infinite rotating disc is 3D and of finite depth. The breakdown and eventual transition of flow over the surface is preceded by the emergence of crossflow vortices that are stationary with respect to the disc. These result from an inviscid instability mechanism associated with an inflexion point within the boundary layer's velocity profile or a mechanism induced by the balance between viscous and Coriolis forces. It has been seen in past studies that compliance can substantially postpone the onset of transition, therefore the aim of this research is to investigate whether compliance can be used as a useful tool to do so here. We use numerical and asymptotic methods to predict possible behaviour by calculating growth rates and producing neutral solutions for the wave number and orientation of both inviscid and viscous modes. The results obtained suggest that the inviscid mode of instability will be stabilized by compliance but the viscous mode will be greatly destabilized.

Lambert function and a new non-extensive form of entropy

Shafee, F. — 2007-11-19

We propose a new way of defining entropy of a system, which gives a general form that is non-extensive like Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a conceptually novel but simple origin and is mathematically easy to define by a very simple expression involving a derivative. It leads to a probability distribution function involving the Lambert function resulting from optimizing the entropy, which has hitherto never appeared in this context, and is somewhat more complex than the Shannon or Boltzmann form, but is nevertheless mathematically quite tractable. We have compared it numerically with the Tsallis and Shannon entropies. We have also considered constraints on the energy spectra imposed by the properties of the Lambert function, which are absent in the Shannon form. It may turn out to be a more appropriate candidate in a physical situation where the probability distribution does not suit any of the previously defined forms, especially when the probability density function sought is expected to be stiffer than that resulting from maximizing the other entropies. We consider the problem of defining free energy and other thermodynamic functions when the entropy is given as a general function of the probability distribution, including that for non-extensive forms. We then find that the free energy, which is central to the determination of all other quantities of interest in a thermodynamic context, can be obtained uniquely, at least numerically, even when it is the root of a transcendental equation. In particular, we examine the cases of the Tsallis form and the new form proposed by us. We compare the free energy, the internal energy and the specific heat of a simple system of two energy states for each of these forms and find significant departures for some quantities, while some others are less sensitive to the parametrization.

Spreading speed and travelling wave solutions of a partially sedentary population

Volkov, D., Lui, R. — 2007-11-19

In this paper, we extend the population genetics model of Weinberger (1978, Asymptotic behavior of a model in population genetics. Nonlinear Partial Differential Equations and Applications (J. Chadam ed.). Lecture Notes in Mathematics, vol. 648. New York: Springer, pp. 47–98.) to the case where a fraction of the population does not migrate after the selection process. Mathematically, we study the asymptotic behaviour of solutions to the recursion u_n₊₁ = Q_g[u_n], where In the above definition of Q_g, K is a probability density function and f behaves qualitatively like the Beverton–Holt function. Under some appropriate conditions on K and f, we show that for each unit vector R^d, there exists a c^*_g() which has an explicit formula and is the spreading speed of Q_g in the direction . We also show that for each c ≥ c^*_g(), there exists a travelling wave solution in the direction which is continuous if gf '(0) ≤ 1.

Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering

Liu, H., Zou, J. — 2007-11-19

Some novel interlacing properties of the zeros for the Bessel and spherical Bessel functions are first presented and then applied to prove an interesting uniqueness result in inverse acoustic obstacle scattering. It is shown that in the resonance region, the shape of a sound-soft/sound-hard ball in R³ or a sound-soft/sound-hard disc in R² is uniquely determined by a single far-field datum measured at some fixed spot corresponding to a single incident plane wave.

Numerical and analytical studies of non-linear gravity capillary waves in fluid layers under normal electric fields

Papageorgiou, D. T., Vanden-Broeck, J.-M. — 2007-11-19

Non-linear gravity–capillary waves travelling at constant speed are considered in the presence of a normal electric field. The fluid, which is assumed to be inviscid, irrotational and a perfect dielectric, is bounded below by a solid plate electrode held at constant voltage, and the region above the free surface is a hydrodynamically passive perfect dielectric, e.g. air. A second parallel flat plate electrode is placed laterally far away and drives a uniform normal electric field there. Electrohydrodynamic coupling occurs at the free surface through the Maxwell stresses which act to modify the normal stress balance and consequently the Bernoulli equation boundary condition there. Three harmonic problems in deforming domains need to be solved, one for the hydrodynamics and one each for the electrostatics above and below the free surface, respectively. We derive and implement an accurate boundary integral method to compute travelling waves of arbitrary wavelength and amplitude. In addition, we consider a long-wave non-linear model of the full problem and compare solutions with the direct simulations. In both cases, we establish the existence of multiple families of solutions extending the classical theory of gravity–capillary waves. An asymptotic theory is also developed to construct periodic waves with ripples. These are used in comparisons with the numerical calculations and the agreement is very good.

Spatial decay in a cross-diffusion problem

Payne, L. E., Song, J. C. — 2007-11-19

In this paper, the authors investigate the decay of end effects for a cross-diffusion problem defined on a semi-infinite cylindrical region. With homogeneous Dirichlet or Neumann conditions prescribed on the lateral surface of the cylinder, it is shown that for fixed finite time and under certain restrictions on the coefficients, solutions decay point-wise as the distance d from the finite end of the cylinder tends to infinity at least of order e^–k^d². Under less restrictive conditions, it is shown that solutions decay in L₂ at least as fast as e^–kd. In both cases, k is a computable function of time.

Steric hindrance effects in thin reaction zones: applications to BIAcore

Edwards, D. A. — 2007-11-19

Many biological and industrial processes have reactions which occur in thin zones of densely packed receptors. Understanding the rate of such reactions is important, and the BIAcore surface plasmon resonance biosensor for measuring rate constants has such a geometry. However, interpreting biosensor data correctly is difficult since large ligand molecules can block multiple receptor sites, thus skewing the kinetics. General mathematical principles are presented for handling this phenomenon, and a receptor layer model is presented explicitly. An integro-partial differential equation results. Using perturbation techniques, the problem can be simplified somewhat. In the limit of small Damköhler number, the non-local nature of the system becomes evident in the association problem, while other experiments can be modelled using local techniques. Explicit and asymptotic solutions are constructed for large-molecule cases motivated by experimental design. The analysis provides insight into surface–volume reactions occurring in various contexts. In particular, this steric hindrance effect can often be quantified with a single dimensionless parameter.

Micro/nano thermal boundary layer equations with slip creep jump boundary conditions

Matthews, M. T., Hill, J. M. — 2007-11-19

At the micro- and nanoscale, the standard continuity boundary conditions at fluid–solid interfaces of classical transport phenomena do not apply and must be replaced by boundary conditions that allow discontinuities. In this study, the classical thermal laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition for tangential velocity and continuous temperature boundary conditions replaced by non-linear slip–creep–jump boundary conditions. These boundary conditions contain an arbitrary index parameter, denoted by n > 0, which appears in the coefficients of the coupled ordinary differential equations to be solved. As an independent check on the numerical procedure, the case of a boundary layer formed in a convergent channel with a sink, which corresponds to n = 1/2, is solved analytically for various values of the Prandtl number and zero Brinkham number. Other values of n for n > 1/2 which correspond to the thermal boundary layer formed in the flow past a wedge are solved numerically for various values of the Prandtl and Brinkham number and constant coefficients appearing in the non-linear slip–creep–jump boundary conditions. It is found that for 1/2 < n < 2, solutions may be found for all values of the constant coefficients, while for n ≥ 2 the constant coefficient for the creep term must be set to zero.

Pseudoparabolic equations with convection

Ptashnyk, M. — 2007-11-19

The existence of solutions of pseudoparabolic equations with convection by using discretization along characteristics is shown. The uniqueness of the solution of a pseudoparabolic equation is proved for a linear elliptic part and for a space dimension N ≤ 4.

Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

Smith, W. R. — 2007-11-19

A formal perturbation scheme is developed to determine original modulation equations for laminar finite-amplitude non-linear waves in an incompressible fluid. Three idealized problems are analysed. The modulation equations comprise conservation of waves, averaged conditions for conservation of mass, momentum, kinetic energy and angular momentum and the averaged projection of the Navier–Stokes equations onto the vorticity vector. The last of these modulation equations, which is related to vortex stretching, only appears in 3D problems. The technique of Reynolds averaging is also employed to obtain equations for the mean velocities and pressure. The Reynolds-averaged Navier–Stokes equations correspond to the modulation equations for conservation of mass and momentum. However, the Reynolds stress transport equations are shown to be inconsistent with the other necessary modulation equations. In two further idealized problems, exact solutions of the Navier–Stokes equations are obtained by employing the modulation equations.