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We present a semi-analytic solution for the non-axisymmetric oscillations of a viscous cylindrical free-standing liquid bridge formed between two coaxial discs of radius R. Even though a streamfunction does not exist, a Helmholtz decomposition is used to obtain an analytic representation of the velocity field. An eigenvalue problem is formulated by projecting the free-surface boundary conditions onto a suitable space of test functions. This is then solved iteratively along with the dispersion relation obtained from the satisfaction of endwall boundary conditions. Extensive comparison with previous theoretical and numerical results, for a range of Reynolds number and bridge slenderness ratio, shows very good agreement in most cases. The present solution generalises that of Tsamopoulos, Chen & Borkar (J. Fluid Mech., vol. 235, 1992, p. 579), which employed a streamfunction formulation and was for the axisymmetric case. [ABSTRACT FROM PUBLISHER]

Our fundamental understanding of steady, stratified flow over two-dimensional (2D) topography rests on the pioneering works of G. Lyra and R. Long. Within linear theory, Lyra established the far-field radiation conditions that determine the downstream pattern of buoyancy waves. Soon after, Long discovered that the steady, nonlinear streamfunction for special cases of stratified, 2D flow could satisfy the same equations as linear theory, subject to an exact topographic boundary condition. Fourier methods are currently used to compute solutions to Long's theory for arbitrary topography in the near-hydrostatic or small-amplitude topographic parameter regimes. It is not generally appreciated however, that these methods encounter difficulties for flows that are both strongly non-hydrostatic and beyond linear amplitudes. By recasting Long's theory into a linear integral equation, this difficulty is shown to be a computational barrier associated with an ill-conditioning of the Fourier method. The problem is overcome through the development of a boundary integral computation which relies on some lesser known solutions from Lyra's original analysis. This method is well-conditioned for strongly non-hydrostatic flows, and is used to extend the exploration of critical overturning flows over Gaussian and bell-shaped ridges. These results indicate that the critical value of the non-dimensional height () asymptotes to a finite value with increasing non-hydrostatic parameter (). [ABSTRACT FROM PUBLISHER]

We compare the results of two-dimensional, biased random walk models of individual swimming micro-organisms with advection–diffusion models for the whole population. In particular, we consider the influence of the local flow environment (gyrotaxis) on the resulting motion. In unidirectional flows, the results of the individual and population models are generally in good agreement, even in flows in which the cells can experience a range of shear environments, and both models successfully predict the phenomena of gravitactic focusing. Numerical results are also compared with asymptotic expressions for weak and strong shear. Discrepancies between the models arise in two cases: (i) when reflective boundary conditions change the orientation distribution in the random walk model from that predicted by the long-term asymptotics used to derive the advection–diffusion model; (ii) when the spatial and temporal scales are not large enough for the advection–diffusion model to apply. We also use a simple two-dimensional flow containing a variety of flow regimes to explore what happens when there are localized regions in which the generalized Taylor dispersion theory used in the derivation of the population model does not apply. For spherical cells, we find good agreement between the models outside the ‘break-down’ regions, but comparison of the results within these regions is complicated by the presence of nearby boundaries and their influence on the random walk model. In contrast, for rod-shaped cells which are reorientated by both vorticity and strain, we see qualitatively different spatial patterns between individual and advection–diffusion models even in the absence of gyrotaxis, because cells are advected between regions of differing rates of strain. [ABSTRACT FROM PUBLISHER]

Continuous dependence upon the elastic moduli of the total energy, and consequently in appropriate measure of the stress, strain and displacement, is established in the initial boundary value problem defined on a semi-infinite cylinder occupied by a linear anisotropic compressible elastic material. Free, constrained and mixed boundary conditions on the lateral surface of the cylinder are discussed. The conclusions are obtained using differential inequalities and partly depend upon previously published results. [ABSTRACT FROM AUTHOR]

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of type Mu = −N∇p + F with suitable matrices M and N with F depending on the right-hand side in the bulk term and in the boundary condition. [ABSTRACT FROM AUTHOR]

We present the results of a combined experimental and theoretical investigation of the surface-tension-driven coalescence of flexible structures. Specifically, we consider the dynamics of the rise of a wetting liquid between flexible sheets that are clamped at their upper ends. As the elasticity of the sheets is progressively increased, we observe a systematic deviation from the classical diffusive-like behaviour: the time to reach equilibrium increases dramatically and the departure from classical rise occurs sooner, trends that we elucidate via scaling analyses. Three distinct temporal regimes are identified and subsequently explored by developing a theoretical model based on lubrication theory and the linear theory of plates. The resulting free-boundary problem is solved numerically and good agreement is obtained with experiments. [ABSTRACT FROM PUBLISHER]

We consider a spectral problem that describes the time-harmonic small-amplitude motion of the mechanical system that consists of a three-dimensional water layer of constant depth and a body (either surface-piercing or totally submerged), freely floating in it. This coupled boundary-value problem contains a spectral parameter – the frequency of oscillations – in the boundary conditions as well as in the equations governing the body motion. It is proved that the total energy of the water motion is finite and the equipartition of energy of the whole system is established. Under certain restrictions on body's geometry the problem is proved to have only a trivial solution for sufficiently large values of the frequency. The uniqueness frequencies are estimated from below. [ABSTRACT FROM PUBLISHER]

The rate of change of the perturbation's kinetic energy E of a perturbed inviscid, incompressible, axisymmetric, columnar and near-critical swirling flow in a finite-length, straight, circular pipe with periodic and non-periodic inlet–outlet conditions is studied using the Reynolds–Orr equation. The perturbation's mode shape and growth rate are computed from the linear-stability eigenvalue problem using a novel asymptotic solution in the case of a flow in a long pipe. This solution technique is general and can be applied to any vortex flow profile, in a range of swirl levels around the critical level, and for various boundary conditions. The solutions are used to analytically estimate the production (or loss) of E at the pipe boundaries and inside the domain and to shed new light on the Wang–Rusak mechanism of exchange of global stability around the critical swirl, that is leading to the vortex breakdown process. It is shown that the production of E inside the domain is modulated by the base flow strain-rate tensor. For the special case of a solid-body rotating flow, this term vanishes and the stability is determined only by the asymmetric transfer of E at the boundaries. For a general base flow, the dominant perturbation's mode shape develops deviations in response to the non-periodic inlet–outlet conditions. These deviations couple with the base flow strain-rate tensor to generate production or loss of E in the bulk. Together with the asymmetric transfer of E at the boundaries, they form a critical balance of production of E and determine the flow stability around the critical state. This behaviour is demonstrated for the Lamb–Oseen and Q vortex models. This analysis reveals a more complicated, as well more realistic, interaction between the perturbed flow in the domain and at the boundaries that dominates vortex flow dynamics. [ABSTRACT FROM PUBLISHER]

We develop an asymptotic theory for marine ice sheets from a first-principles Stokes flow contact problem, in which different boundary conditions apply to areas where ice is in contact with bedrock and inviscid sea water, along with suitable inequalities on normal stress and boundary location constraining contact and non-contact zones. Under suitable assumptions about basal slip in the contact areas, the boundary-layer structure for this problem replicates the boundary layers previously identified for marine ice sheets from depth-integrated models and confirms the results of these previous models: the interior of the grounded ice sheet can be modelled as a standard free-surface lubrication flow, while coupling with the membrane-like floating ice shelf leads to two boundary conditions on this lubrication flow model at the contact line. These boundary conditions determine ice thickness and ice flux at the contact line and allow the lubrication flow model with a contact line to be solved as a moving boundary problem. In addition, we find that the continuous transition of vertical velocity from grounded to floating ice requires the presence of two previously unidentified boundary layers. One of these takes the form of a viscous beam, in which a wave-like surface feature leads to a continuous transition in surface slope from grounded to floating ice, while the other provides boundary conditions on this viscous beam at the contact line. [ABSTRACT FROM PUBLISHER]

A general continuum theory for particle-size segregation and diffusive remixing in polydisperse granular avalanches is formulated using mixture theory. Comparisons are drawn to existing segregation theories for bi-disperse mixtures and the case of a ternary mixture of large, medium and small particles is investigated. In this case, the general theory reduces to a system of two coupled parabolic segregation–remixing equations, which have a single diffusion coefficient and three parameters which control the segregation rates between each pair of constituents. Considerable insight into many problems where the effect of diffusive remixing is small is provided by the non-diffusive case. Here the equations reduce to a system of two first-order conservation laws, whose wave speeds are real for a very wide class of segregation parameters. In this regime, the system is guaranteed to be non-strictly hyperbolic for all admissible concentrations. If the segregation rates do not increase monotonically with the grain-size ratio, it is possible to enter another region of parameter space, where the equations may either be hyperbolic or elliptic, depending on the segregation rates and the local particle concentrations. Even if the solution is initially hyperbolic everywhere, regions of ellipticity may develop during the evolution of the problem. Such regions in a time-dependent problem necessarily lead to short wavelength Hadamard instability and ill-posedness. A linear stability analysis is used to show that the diffusive remixing terms are sufficient to regularize the theory and prevent unbounded growth rates at high wave numbers. Numerical solutions for the time-dependent segregation of an initially almost homogeneously mixed state are performed using a standard Galerkin finite element method. The diffuse solutions may be linearly stable or unstable, depending on the initial concentrations. In the linearly unstable region, ‘sawtooth’ concentration stripes form that trap and focus the medium-sized grains. The large and small particles still percolate through the avalanche and separate out at the surface and base of the flow due to the no-flux boundary conditions. As these regions grow, the unstable striped region is annihilated. The theory is used to investigate inverse distribution grading and reverse coarse-tail grading in multi-component mixtures. These terms are commonly used by geologists to describe particle-size distributions in which either the whole grain-size population coarsens upwards, or just the coarsest clasts are inversely graded and a fine-grained matrix is found everywhere. An exact solution is constructed for the steady segregation of a ternary mixture as it flows down an inclined slope from an initially homogeneously mixed inflow. It shows that for distribution grading, the particles segregate out into three inversely graded sharply segregated layers sufficiently far downstream, with the largest particles on top, the fines at the bottom and the medium-sized grains sandwiched in between. The heights of the layers are strongly influenced by the downstream velocity profile, with layers becoming thinner in the faster moving near-surface regions of the avalanche, and thicker in the slowly moving basal layers, for the same mass flux. Conditions for the existence of the solution are discussed and a simple and useful upper bound is derived for the distance at which all the particles completely segregate. When the effects of diffusive remixing are included, the sharp concentration discontinuities are smoothed out, but the simple shock solutions capture many features of the evolving size distribution for typical diffusive remixing rates. The theory is also used to construct a simple model for reverse coarse-tail grading, in which the fine-grained material does not segregate. The numerical method is used to calculate diffuse solutions for a ternary mixture and a sharply segregated shock solution is derived ... [ABSTRACT FROM PUBLISHER]

A steady sheet flow of an inviscid incompressible fluid along a curvilinear surface ending with a rounded trailing edge is considered in the presence of gravity. The effect of surface tension is ignored. The formulation of the problem is applicable to the study of free-surface flows over obstacles in channels, weirs and spillways, and pouring flows. An advanced hodograph method is employed for solving the problem, which is reduced to a system of two integro-differential equations in the velocity modulus on the free surface and in the slope of the bottom surface. These equations are derived from the dynamic and kinematic boundary conditions. The Brillouin–Villat criterion is applied to determine the location of the point of flow separation from the rounded trailing edge. Results showing the effect of gravity on the flow detachment and the geometry of the free boundaries are presented over a wide range of Froude numbers including both subcritical and supercritical flows. For supercritical flows two families of solutions for an arbitrary bottom shape are reproduced. It is shown that the additional condition requiring the free surface to be flat at a finite distance from the end of the channel selects a unique solution for a given bottom height and geometry for supercritical flows. This solution is continuous in going from the subcritical to the supercritical flow regime. [ABSTRACT FROM PUBLISHER]

We derive boundary conditions at a rigid wall for a granular material comprising rough, inelastic particles. Our analysis is confined to the rapid flow, or granular gas, regime in which grains interact by impulsive collisions. We use the Chapman–Enskog expansion in the kinetic theory of dense gases, extended for inelastic and rough particles, to determine the relevant fluxes to the wall. As in previous studies, we assume that the particles are spheres, and that the wall is corrugated by hemispheres rigidly attached to it. Collisions between the particles and the wall hemispheres are characterized by coefficients of restitution and roughness. We derive boundary conditions for the two limiting cases of nearly smooth and nearly perfectly rough spheres, as a hydrodynamic description of granular gases comprising rough spheres is appropriate only in these limits. The results are illustrated by applying the equations of motion and boundary conditions to the problem of plane Couette flow. [ABSTRACT FROM PUBLISHER]

The liquid lining in small human airways can become unstable and form liquid plugs that close off the airways. Direct numerical simulations are carried out on an airway model to study this airway instability and the flow-induced stresses on the airway walls. The equations governing the fluid motion and the interfacial boundary conditions are solved using the finite-volume method coupled with the sharp interface method for the free surface. The dynamics of the closure process is simulated for a viscous Newtonian film with constant surface tension and a passive core gas phase. In addition, a special case is examined that considers the core dynamics so that comparisons can be made with the experiments of Bian et al. (J. Fluid Mech., vol. 647, 2010, p. 391). The computed flow fields and stress distributions are consistent with the experimental findings. Within the short time span of the closure process, there are large fluctuations in the wall shear stress. Furthermore, dramatic velocity changes in the film during closure indicate a steep normal stress gradient on the airway wall. The computational results show that the wall shear stress, normal stress and their gradients during closure can be high enough to injure airway epithelial cells. [ABSTRACT FROM PUBLISHER]

Using high-resolution 3-D vortex simulations, this study seeks a mechanistic understanding of vorticity dynamics in transverse jets at a finite Reynolds number. A full no-slip boundary condition, rigorously formulated in terms of vorticity generation along the channel wall, captures unsteady interactions between the wall boundary layer and the jet – in particular, the separation of the wall boundary layer and its transport into the interior. For comparison, we also implement a reduced boundary condition that suppresses the separation of the wall boundary layer away from the jet nozzle. By contrasting results obtained with these two boundary conditions, we characterize near-field vortical structures formed as the wall boundary layer separates on the backside of the jet. Using various Eulerian and Lagrangian diagnostics, it is demonstrated that several near-wall vortical structures are formed as the wall boundary layer separates. The counter-rotating vortex pair, manifested by the presence of vortices aligned with the jet trajectory, is initiated closer to the jet exit. Moreover tornado-like wall-normal vortices originate from the separation of spanwise vorticity in the wall boundary layer at the side of the jet and from the entrainment of streamwise wall vortices in the recirculation zone on the lee side. These tornado-like vortices are absent in the case where separation is suppressed. Tornado-like vortices merge with counter-rotating vorticity originating in the jet shear layer, significantly increasing wall-normal circulation and causing deeper jet penetration into the crossflow stream. [ABSTRACT FROM AUTHOR]

We study classes of higher-order singular boundary-value problems on a time scale $\mathbb{T}$ with a positive parameter λ in the differential equations. A homeomorphism and homomorphism ø are involved both in the differential equation and in the boundary conditions. Criteria are obtained for the existence and uniqueness of positive solutions. The dependence of positive solutions on the parameter λ is studied. Applications of our results to special problems are also discussed. Our analysis mainly relies on the mixed monotone operator theory. The results here are new, even in the cases of second-order differential and difference equations. [ABSTRACT FROM PUBLISHER]

Space-charge waves in a relativistic electron beam that completely fills a cylindrical metallic waveguide and is guided by an ion channel are analyzed numerically. Equilibrium consists of a uniform and rigid rotation without betatron oscillations. Using cold fluid equations a differential equation and boundary conditions are derived that constitute an eigenvalue problem. This eigenvalue problem is solved, numerically, with the finite difference scheme using shooting method. Dispersion characteristics and electrostatic potential structures of azimuthally symmetric and nonsymmetric space-charge waves are studied. Perfect agreement with analytical results at asymptotic limit of zero axial velocity is found. It was found that relativistic effects modify the dispersion characteristics of the space-charge waves considerably and can concentrate the electric field energy of the wave into a thin and small shell around the axis. [ABSTRACT FROM PUBLISHER]

We deduce maximum principles for fourth-, sixth- and eighth-order elliptic equations by modifying an auxiliary function introduced by Payne (J. Analyse Math. 30 (1976), 421–433). Integral bounds on various gradients of the solutions of these equations are obtained. [ABSTRACT FROM PUBLISHER]

The stretched-vortex subgrid model is used to run large-eddy simulations of temporal mixing layers at various Reynolds and Schmidt numbers, with different initial and boundary conditions. A self-similar flow is obtained, during which the growth rate, mean velocity and Reynolds stresses are in accord with experimental results. However, predictions of the amount of mixed fluid, and of the variation in its composition across the layer, are excessive, especially at high Schmidt number. More favourable comparisons between experiment and simulation are obtained when the large-scale flow is quasi-two-dimensional; however, such states are not self-similar and not sustainable. Present model assumptions lead to predictions of the continued subgrid spectrum with a viscous cutoff that is dependent on grid resolution. [ABSTRACT FROM AUTHOR]

A dynamic method for prescribing realistic inflow boundary conditions is presented for simulations of spatially developing turbulent boundary layers. The approach is based on the rescaling–recycling method proposed by Lund, Wu & Squires (J. Comput. Phys, vol. 140, 1998, pp. 233–258) and the multi-scale method developed by Araya, Jansen & Castillo (J. Turbul., vol. 10, no. 36, 2009, pp. 1–33). The rescaling process requires prior knowledge about how the velocity and length scales are related between the inlet and recycle stations. Here a dynamic approach is proposed in which such information is deduced dynamically by involving an additional plane, the so-called test plane located between the inlet and recycle stations. The approach distinguishes between the inner and outer regions of the boundary layer and enables the use of multiple velocity scales. This flexibility allows applications to boundary layer flows with pressure gradients and avoids the need to prescribe empirically the friction velocity and other flow parameters at the inlet of the domain. The dynamic method is tested in direct numerical simulations of zero, favourable and adverse pressure gradient flows. The dynamically obtained scaling exponents for the downstream evolution of boundary layer parameters are found to fluctuate in time, but on average they agree with the expected values for zero, favourable and adverse pressure gradient flows. Comparisons of the results with data from experiments, and from other direct numerical simulations that use much longer computational domains to capture laminar-to-turbulence transition, demonstrate the suitability of the proposed dynamic method. [ABSTRACT FROM PUBLISHER]

The creeping flow of an incompressible viscous liquid past a porous approximately spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equations. The flow within the porous annular region of the shell is governed by Brinkman’s model. The boundary conditions used at the interface are continuity of the velocity, continuity of the pressure and Ochoa-Tapia and Whitaker’s stress jump condition. An exact solution for the problem and an expression for the drag on the porous approximately spherical shell are obtained. The drag is evaluated numerically for several values of the parameters governing the flow. [ABSTRACT FROM PUBLISHER]

The problem of stagnation point flow with heat transfer of an electrically conducting fluid impinging normally on a permeable axisymmetric surface in the presence of a uniform transverse magnetic field is analysed. The governing nonlinear differential equations and their associated boundary conditions are reduced to dimensionless form using suitable similarity transformations. Comparison with previously published work shows good agreement. Effects of the injection–suction parameter, magnetic parameter and Prandtl number on the flow and thermal fields are presented. The investigations show that the wall shear stress and heat transfer rate from the surface increase with increased applied magnetic field. An increase in the velocity and thermal boundary layer thicknesses is observed with an increase in the wall injection, while the velocity and thermal boundary layers become thinner when increasing the wall suction and applied magnetic field. [ABSTRACT FROM PUBLISHER]

A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R+1/2 and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R+2/3 in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales. [ABSTRACT FROM AUTHOR]

In a variety of applications, most notably microfluidics design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the ‘effective slip’, which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip. [ABSTRACT FROM AUTHOR]

Direct numerical simulation is performed for the interaction between a deformable free surface and the homogeneous isotropic turbulent flow underneath. The Navier–Stokes equations subject to fully nonlinear free-surface boundary conditions are simulated by using a pseudospectral method in the horizontal directions and a finite-difference method in the vertical direction. Statistically, steady turbulence is generated by using a linear forcing method in the bulk flow below. Through investigation of cases of different Froude and Weber numbers, the present study focuses on the effect of surface deformation of finite amplitude. It is found that the motion of the free surface is characterized by propagating waves and turbulence-generated surface roughness. Statistics of the turbulence field near the free surface are analysed in detail in terms of fluctuations of velocity, fluctuations of velocity gradients and strain rates and the energy budget for horizontal and vertical turbulent motions. Our results illustrate the effects of surface blockage and vanishing shear stress on the anisotropy of the flow field. Using conditional averaging analysis, it is shown that splats and antisplats play an essential role in energy inter-component exchange and vertical transport. [ABSTRACT FROM AUTHOR]

Consider the p(x)-Laplacian-Dirichlet problem with sign-changing non-linearity of the form {-div(|∇u|p(x)-2 ∇u) + m(x)|u|p(x)-2 u = λa(x)f(u) in Ω u = 0 on ∂Ω where Ω ⊂ ℝN is a bounded domain, p ∈ C0(Ω̅) and infx∈Ω̅p(x) > 1, m ∈ L∞(Ω) is non-negative, f : ℝ → ℝ is continuous and f (0) > 0, the coefficient a ∈ L∈(Ω) is sign-changing in Ω. We give some sufficient conditions to assure the existence of a positive solution to the problem for sufficiently small λ > 0. Our results extend the corresponding results established in the p-Laplacian case to the p(x)-Laplacian case. [ABSTRACT FROM AUTHOR]

A procedure for constructing two-dimensional incompressible potential flowfield solutions with separation and a recirculation region is presented. It naturally makes use of complex variable theory and other analysis techniques such as conformal mapping and the generalized Poisson integral formula. Flowfield determination is reduced to solution of a boundary value problem in various simple domains. The entire velocity field is described analytically; stream function and velocity potential contour maps are readily constructed. Example solutions are presented. Solutions for sharp leading edge airfoils at arbitrary angle of attack are completely determined, including the limiting angle of attack for upper-surface flow re-attachment. For other configurations (e.g. circular cylinder, backward-facing step) the analytical solution contains one or more free parameters, whose values may be inferred from boundary layer theory or experiment. [ABSTRACT FROM AUTHOR]

A novel formulation of the wall boundary conditions relying on the asymptotic behaviour of the Taylor microscale λ and its relationship to the homogeneous part of the viscous dissipation rate of the kinetic energy of turbulence ϵh =5?q2/λ2, applicable to near-wall turbulence, is examined. The linear dependence of λ on the wall distance in close proximity to the solid surface enables the wall-closest grid node to be positioned immediately below the edge of the viscous sublayer, leading to a substantial coarsening of the grid resolution. This approach provides bridging of a major portion of the viscous sublayer, higher grid flexibility and weaker sensitivity against the grid non-uniformities in the near-wall region. The performance of the proposed formulation was checked against available direct numerical simulation databases of complex wall-bounded flows featured by swirl and separation. [ABSTRACT FROM AUTHOR]

A cation-selective conducting particle is suspended in an electrolyte solution and is exposed to a uniformly applied electric field. The electrokinetic transport processes are described in a closed mathematical model, consisting of differential equations, representing the physical transport in the electrolyte, and boundary conditions, representing the physicochemical conditions on the particle boundary and at large distances away from it. Solving this mathematical problem would in principle provide the electrokinetic flow about the particle and its concomitant velocity relative to the otherwise quiescent fluid. Using matched asymptotic expansions, this problem is analysed in the thin- Debye-layer limit. A macroscale description is extracted, whereby effective boundary conditions represent appropriate asymptotic matching with the Debye-scale fields. This description significantly differs from that corresponding to a chemically inert particle. Thus, ion selectivity on the particle surface results in a macroscale salt concentration polarization, whereby the electric potential is rendered non-harmonic. Moreover, the uniform Dirichlet condition governing this potential on the particle surface is transformed into a non-uniform Dirichlet condition on the macroscale particle boundary. The Dukhin-Derjaguin slip formula still holds, but with a non-uniform zeta potential that depends, through the cation-exchange kinetics, upon the salt concentration and electric field distributions. For weak fields, an approximate solution is obtained as a perturbation to a reference state. The linearized solution corresponds to a uniform zeta potential; it predicts a particle velocity which is proportional to the applied field. The associated electrokinetic flow is driven by two different agents, electric field and salinity gradients, which are of comparable magnitude. Accordingly, this flow differs significantly from that occurring in electrophoresis of chemically inert particles. [ABSTRACT FROM AUTHOR]

The effect of axial forces on the dynamic properties is formulated and investigated for a 3-PRR planar parallel manipulator with three flexible intermediate links. A dynamic model of the manipulator system is developed based on the assumed mode method with the consideration of the effect of longitudinal forces on lateral stiffness is included. The flexible intermediate links are modeled as Euler-Bernoulli beams with pinned-pinned boundary conditions, which are verified by experimental modal tests. Natural frequencies of bending vibration of the intermediate links are derived as the functions of axial force and rigid-body motion of the manipulator. Dynamic behavior including the effect of axial forces on lateral deformation is investigated, and configuration-dependant frequencies are analyzed. Numerical simulations of configuration-dependent frequency properties and axial forces are performed to illustrate the effect of axial forces on the dynamic behaviors of the flexible parallel manipulator. Simulation results of mode amplitudes, deformations, axial forces, inertial, and coupling forces are presented, and further validate the theoretical derivations. These analyses and results provide a new and valuable insight to the design and control of the parallel manipulators with flexible intermediate links. [ABSTRACT FROM AUTHOR]

We investigate natural convection cooling of the fluid in a drink can placed in a refrigerator by simulating the full combined boundary layer system on the can wall. The cylindrical can is filled with water at initial nondimensional temperature 0, and located within a larger cylindrical container filled with air at initial temperature −1. The outer container walls are maintained at constant temperature −1. Initially both fluids are at rest. Two configurations are examined: the first has the inner can placed vertically in the middle of the outer container with no contact with the outer container walls, and the second has the inner can placed vertically at the bottom of the outer container. The results are compared to those obtained by assuming that the inner can walls are maintained at a constant temperature, showing similar basic flow features and scaling relations, but with very different proportionality constants. [ABSTRACT FROM PUBLISHER]

We consider a non-local filtration equation of the form ut = ΔK(u) + λf(u) (∫Ω f(u) dx)-p and a porous medium equation, in this case K(u) = um, with some boundary and initial data u0, where 0 < p < 1 and f, f′, f″ > 0. We prove blow-up of solutions for sufficiently large values of the parameter λ > 0 and for any u0 > 0, or for sufficiently large values of u0 > 0 and for any λ > 0. [ABSTRACT FROM AUTHOR]

We establish new sampling representations for linear integral transforms associated with arbitrary general Birkhoff regular boundary value problems. The new approach is developed in connection with the analytical properties of Green's function, and does not require the root functions to be a basis or complete. Unlike most of the known sampling expansions associated with eigen-value problems, the results obtained are, generally speaking, of Hermite interpolation type. [ABSTRACT FROM AUTHOR]

The response of the Blasius boundary layer to free-stream vortical disturbances of the convected gust type is studied. The vorticity signature of the boundary layer is computed through the boundary-region equations, which are the rigorous asymptotic limit of the Navier-Stokes equations for low-frequency disturbances. The method of matched asymptotic expansion is employed to obtain the initial and outer boundary conditions. For the case of forcing by a two-dimensional gust, the effect of a wall-normal wavelength comparable with the boundary-layer thickness is taken into account. The gust viscous dissipation and upward displacement due to the mean boundary layer produce significant changes on the fluctuations within the viscous region. The same analysis also proves useful for computing to second-order accuracy the boundary-layer response induced by a three-dimensional gust with spanwise wavelength comparable with the boundary-layer thickness. It also follows that the boundary-layer fluctuations of the streamwise velocity match the corresponding freestream velocity component. The velocity profiles are compared with experimental data, and good agreement is attained. The generation of Tollmien-Schlichting waves by the nonlinear mixing between the two-dimensional unsteady vorticity fluctuations and the mean flow distortion induced by localized wall roughness and suction is also investigated. Gusts with small wall-normal wavelengths generate significantly different amplitudes of the instability waves for a selected range of forcing frequencies. This is primarily due to the disparity between the streamwise velocity fluctuations in the free stream and within the boundary layer. [ABSTRACT FROM AUTHOR]

AbstractLet R(n,?) denote the number of representations of the natural number nas the?sum?of four squares, each composed only with primes not exceeding n?/2. When ?>e?1/3?a?lower bound for R(n,?) of the expected order of magnitude is established, and when ?>365/592, it is shown that R(n,?)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed ?>0. This last result is the key to bound R(n,?)?from?below. [ABSTRACT FROM AUTHOR]

The flow of a viscous incompressible fluid in a circular tube generated by a sudden impulse on the axis is studied on the basis of the linearized Navier-Stokes equations. A no-slip boundary condition is assumed to hold on the wall of the tube. At short time the flow is irrotational and may be described by a potential which varies with the square root of time. At later times there is a sequence of moving and decaying vortex rings. At long times the flow velocity decays with an algebraic long-time tail. The impulse generates a time-dependent pressure difference between the ends of the tube. [ABSTRACT FROM AUTHOR]

We analyse the linear stability and nonlinear evolutions of circular hetons under the quasi-geostrophic approximation. We compare results obtained with a three-layer model and with a model based on a continuous density stratification. Though the models also differ by the vertical boundary conditions, they show a remarkable similarity in the stability properties of the hetons (threshold values of vortex radius for baroclinic instability, dominant modes, growth rates, etc.), and in their nonlinear evolutions (spatial reorganization of potential vorticity by nonlinear processes, endstates of the simulations). The hetons prone to baroclinic instability often break into two hetons drifting in opposite directions, and in more hetons, for wider initial structures. In both models, instability is quite sensitive to the vertical gap between the opposite-signed vortices: as it increases, the instability decreases and shifts to lower azimuthal modes. Finally, though modes l ⩾ 2 (i.e. elliptical and shorter wave deformations) prevail in most of the parameter space, the mode l =1 perturbation (a vertical tilt of the vortex column) exists for hetons with small vertical gaps. Such perturbations are concentrated vertically near the gap, and can only be evidenced in the continuously stratified model. [ABSTRACT FROM AUTHOR]

Based on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725-732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, thenMhas finite topology.We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below. [ABSTRACT FROM AUTHOR]

ApolynomialP ϵ P(kE, F) is left ℓ1-factorable if there are a polynomial Q ϵ P(kE, ℓ1) and an operator L ϵ L(ℓ1, F) such that P = L o Q. We characterise the Radon-Nikodým property by the left ℓ1-factorisation of polynomials on L1(μ). We study the left ℓ1-factorisation of nuclear, compact and Pietsch integral polynomials. For Pietsch integral polynomials, we introduce the left integral ℓ1-factorisation property, obtaining a second polynomial characterisation of the Radon-Nikodým property and showing that it plays a role somehow comparable, in this setting, to nuclearity of operators. A characterisation of ℒ1-spaces is also given in terms of the left compact ℓ1-factorisation of polynomials. [ABSTRACT FROM AUTHOR]

We consider the solution in the time domain of the two-dimensional water-wave scattering by fixed bodies, which may or may not intersect with the free surface. We show how the problem with arbitrary initial conditions can be found from the single-frequency solutions using a generalized eigenfunction expansion, required because the operator has a continuous spectrum. From this expansion we derive simple formulas for the evolution in time of the initial surface conditions, and we present some examples of numerical calculations. [ABSTRACT FROM AUTHOR]

We consider the solution in the time domain of water-wave scattering by arrays of bottom-mounted cylinders. It has already been shown that near trapping occurs for certain arrangements of cylinders and we are especially focused on this phenomenon. We begin with the well-known single-frequency solution to the problem of a group of cylinders, and the extension of this solution to complex frequencies. It has been shown that singularities (scattering frequencies or resonances) occur for certain values of the complex frequency and these singularities are associated with the near-trapped mode. We show that it is possible to approximate the solution near these singularities, and produce a modal shape which is associated with the near-trapped mode. We then consider the time-dependent problem, beginning with the well-known incident plane wave packet solution. We also show how the problem of an arbitrary initial displacement can be found using the single-frequency solutions. This latter result relies on a special inner product which gives a generalized eigenfunction expansion (because the operator has a continuous spectrum). We then consider the approximation of the time-dependent motion using special mode shapes associated with the scattering frequencies. This approximation relies on the scattering frequencies lying close to the real axis. We present numerical results which show that this approximation is accurate for sufficiently large time. [ABSTRACT FROM AUTHOR]

We study numerically the viscous phase of horizontal gravity currents of immiscible fluids in the lock-exchange configuration. A numerical technique capable of dealing with stiff density gradients is used, allowing us to mimic high-Schmidt-number situations similar to those encountered in most laboratory experiments. Plane two-dimensional computations with no-slip boundary conditions are run so as to compare numerical predictions with the 'short reservoir' solution of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43-58), which predicts the front position lf to evolve as t1/5, and the 'long reservoir' solution of Gratton & Minotti (J. Fluid Mech., vol. 210, 1990, pp. 155-182) which predicts a diffusive evolution of the distance travelled by the front xf ~ t½2. In line with dimensional arguments, computations indicate that the self-similar power law governing the front position is selected by the flow Reynolds number and the initial volume of the released heavy fluid. We derive and validate a criterion predicting which type of viscous regime immediately succeeds the slumping phase. The computations also reveal that, under certain conditions, two different viscous regimes may appear successively during the life of a given current. Effects of sidewalls are examined through three-dimensional computations and are found to affect the transition time between the slumping phase and the viscous regime. In the various situations we consider, we make use of a force balance to estimate the transition time at which the viscous regime sets in and show that the corresponding prediction compares well with the computational results. [ABSTRACT FROM AUTHOR]

Makienko's conjecture, a proposed addition to Sullivan's dictionary, can be stated as follows: the Julia set of a rational function R : ℂ∞ → ℂ∞has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko's conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum. [ABSTRACT FROM AUTHOR]

We consider the effect of a wetting condition at the moving contact line on the frequency and damping of surface waves on an inviscid liquid in a circular cylinder. The velocity potential ϕ and the free surface elevation η are sought as complex eigenfunction expansions. The ϕ eigenvalues are the classical ones whereas the η eigenvalues are unknown and have to be computed so as to satisfy the wetting condition on the contact line and the other free surface conditions - these turn out to be complex in general. A projection of the latter conditions on to an appropriate basis leads to an eigenvalue problem, for the complex frequency Ω, which has to be solved iteratively with the wetting condition. The variation of Ω with liquid depth h, Bond number Bo, capillary coefficient γ and static contact angle θc0 is explored for the (1, 0), (2, 0), (0, 1), (3, 0) and (4, 0) modes. The damping vanishes for γ = 0 (pinned-end edge condition) and γ = ∞ (free-end edge condition) with a maximum in the interior while the frequency decreases with increasing γ, approaching limiting values at the endpoints. A comparison with the analytic results of Miles (J. Fluid Mech., vol. 222, 1991, p. 197) for the no-meniscus case and the experimental results of Cocciaro, Faetti, & Festa (J. Fluid Mech., vol. 246, 1993, p. 43), where a meniscus is present, is good. The study provides a simple procedure for calculating the inviscid capillary damping associated with the moving contact line in a circular cylinder of finite depth with meniscus effects also being considered. [ABSTRACT FROM AUTHOR]

Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying ∣H∣ ≤ ε(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S  S0 and M is isometric to a Clifford hypersurface, where e(n) is a sufficiently small constant depending on n and S0 = (Multiple line equation(s) cannot be represented in ASCII text). [ABSTRACT FROM AUTHOR]