Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

Summary. We derive a set of asymptotically exact coupled amplitude-streaming flow ({CASF}) equations governing the evolution of weakly nonlinear nearly inviscid multimode Faraday waves and the associated streaming flow in finite geometries. The streaming flow is found to play a particularly important role near mode interactions. Such interactions come about either through a suitable choice of parameters or through breaking of degeneracy among modes related by symmetry. An example of the first case is provided by the interaction of two nonaxisymmetric modes in a circular container with different azimuthal wavenumbers. The second case arises when the shape of the container is changed from square to slightly rectangular, or from circular to slightly noncircular but with a plane of symmetry. The generation of streaming flow in each of these cases is discussed in detail and the properties of the resulting CASF equations are described. A preliminary analysis suggests that these equations can resolve discrepancies between existing theory and experimental results in the first two of the above cases.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Instability of Gortler Vortices to Nonlinear Travelling Waves

Author to whom correspondence should be addressed Recent theoretical work by Hall & Seddougui (1989) has shownthat strongly nonlinear high-wavenumber Görtler vorticesdeveloping within a boundary layer flow are susceptible to asecondary instability which takes the form of travelling wavesconfined to a thin region centred at the outer edge of the vortex.This work considered the case in which the secondary mode couldbe satisfactorily described by a linear stability theory, andin the current paper our objective is to extend this investigationof Hall & Seddougui (1989) into the nonlinear regime. Wefind that, at this stage, not only does the secondary mode becomenonlinear, but it also interacts with itself so as to modifythe governing equations for the primary Görtler vortex.In this case, then, the vortex and the travelling wave driveeach other, and indeed the whole flow structure is describedby an infinite set of coupled nonlinear differential equations.We undertake a Stuart-Watson type of weakly nonlinear analysisof these equations and conclude, in particular, that on thisbasis there exist stable flow configurations in which the travellingmode is of finite amplitude. Implications of our findings forpractical situations are discussed, and it is shown that thetheoretical conclusions drawn here are in good qualitative agreementwith available experimental observations.     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

Structure of Equations Solvable by the Inverse Scattering Transform for the Schrödinger Operator

We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as x±. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form.     

Vector Eigenfunction Expansions for Plane Channel Flows

The full nonlinear initial-boundary value problem for the evolution of disturbances in plane Poiseuille flow is considered. The problem is formulated in vector form using the normal velocity and normal vorticity as components. The solution is presented as an expansion in linear eigenmodes. These modes consist of both Orr-Sommerfeld modes and modes of the normal vorticity (Squire) equation. The case of degenerating eigenmodes is also considered and it is shown that the Benney-Gustavsson normal velocity-normal vorticity resonance is a special case of a degeneracy between the vector eigenmodes. The solution to the nonlinear problem is presented as an expansion in the linear eigenmodes as well as in modes of the self-adjoint part of the linear equation. The full nonlinear solution is further reduced to small systems of coupled amplitude equations using the center manifold theorem.     

Existence of a weak solution for the fully coupled Navier–Stokes/Darcy-transport problem

This paper analyzes the surface/subsurface flow coupled with transport. The flow is modeled by the coupling of Navier–Stokes and Darcy equations. The transport of a species is modeled by a convection-dominated parabolic equation. The two-way coupling between flow and transport is nonlinear and it is done via the velocity field and the viscosity. This problem arises from a variety of natural phenomena such as the contamination of the groundwater through rivers. The main result is existence and stability bounds of a weak solution.     

Splitting methods with complex times for parabolic equations

Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real time are known to involve large and/or negative time steps, which destabilizes the overall procedure, the key point of our analysis is, we develop splitting methods that use complex time steps having positive real part: going to the complex plane allows to considerably increase the accuracy, while keeping small time steps; on the other hand, restricting our attention to time steps with positive real part makes our methods more stable, and in particular well adapted in the case when the considered evolution equation involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations. We provide a thorough analysis in the case of linear equations posed in general Banach spaces. We also numerically investigate the nonlinear situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.     

Stable self-similar blowup in the supercritical heat flow of harmonic maps

We consider the heat flow of corotational harmonic maps from \(\mathbb {R}^3\) to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

Contraction of convex hypersurfaces in Euclidean space

We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.This work was carried out while the author was supported by an Australian Postgraduate Research Award and an ANUTECH scholarship.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

Perturbations of Soliton Solutions to the Unstable Nonlinear Schrödinger and Sine-Gordon Equations

The adiabatic evolution of soliton solutions to the unstable nonlinear Schrödinger (UNS) and sine-Gordon (SG) equations in the presence of small perturbations is reconsidered. The transport equations describing the evolution of the solitary wave parameters are determined by a direct multiple-scale asymptotic expansion and by phase-averaged conservation relations for an arbitrary perturbation. The evolution associated with a dissipative perturbation is explicitly determined and the first-order perturbation fields are also obtained.     

On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations

We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension.     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

Initial value problems for creeping flow of Maxwell fluids

We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem locally in time. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data; after that we extend our results to some 3D flow problems with large initial data.We solve our problem using an iteration between linear subproblems. The limit of the iteration provides the solution of our original problem.     

Limit behaviour of focusing solutions to nonlinear diffusions

The paper is concerned with the behaviour of focusing solutions to nonlinear diffusion problems. These solutions describe the movement of a flow filling a hole and have consequences for the qualitative theory of degenerate nonlinear parabolic equations. The general equation under study is theso-called doubly nonlinear diffusion equation a²with parameters m > 0 and p > 1 such that m(p - 1) > 1 so that the finite propagation property holds and free boundaries occur. Well-known particular cases are the Porous Medium Equation and the evolutionary p-

Laplacian Equation. We study the behaviour of the families of selfsimilar focusing solutions as the parameters m and p tend to their limiting values and identify the limit problems these limits solve. In the case m(p - 1) -+ 1 we find as appropriate asymptotic problems a family of Hamilton-Jacobi equations. When we let m + ffi we obtain in the limit the Hele-Shaw problem. When p + cc we

obtain linear travelling waves with arbitrary speed, solutions of a certain ∞-Laplacian evolution problem.     

Transition layer solutions of a higher order equation in an infinite tube

A general stability and convergence theorem is established for generalized solutions of a family of nonlinear evolution equations with non-local diffusion in one space dimension. As the first application we justify the motion by crystalline energy as a limit of regularized problems. As the sec-ond application we show the convergence of crystalline algorithm for general curvature flow equations. Our general results are also important to explain that geometric evolution of crystals depends continuously on temperature even if facets appear.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

Sediment Transport via Dam-Break Flows over Sloping Erodible Beds

When a semi-infinite body of homogeneous fluid initially at rest behind a vertical retaining wall is suddenly released by the removal of the barrier the resulting flow over either a horizontal or a sloping bed is referred to as a dam-break flow. When resistance to the flow is neglected the exact solution, in the case of a stable horizontal bed with or without "tail water," may be obtained on the basis of shallow-water theory via the method of characteristics and the results are well known. Discrepancies between these shallow-water based solutions and experiments have been partially accounted for by the introduction of flow resistance in the form of basal friction. This added friction significantly modifies the wave speed and flow profile near the head of the wave so that the simple exact solutions no longer apply and various asymptotic or numerical approaches must be implemented to solve these frictionally modified depth-averaged shallow-water equations. When the bed is no longer stable so that solid particles may be exchanged between the bed and the water column the dynamics of the flow becomes highly complex as the buoyancy forces vary in space and time according to the competing rates of erosion and deposition. It is our intention here to study dam-break flows over erodible sloping beds as agents of sediment transport taking into account basal friction as well as the effects of particle concentrations on flow dynamics including both erosion and deposition. We consider shallow flows over initially dry beds and investigate the effects of changes in the depositional and erosional models employed as well as in the nature of the drag acting on the flow. These models include effects hitherto neglected in such studies and offer insights into the transport of sediment in the worst case scenario of the complete and instantaneous collapse of a dam.