On the existence of free-surface azimuthal equatorial flows

This paper is concerned with the governing equations (together with the free boundary and the bottom boundary conditions) for an inviscid, incompressible fluid, written in spherical coordinates which are fixed at a point on the rotating Earth. For a special type of a steady flow moving only in the azimuthal direction, with no variation in this direction, we provide a functional-analytic study of the link between the pressure at the free surface of the ocean and the distortion of that surface.     

两平行球面间扩散层流进口段流动阻力的分析*

本文首先将В.В.Голубев方法推广到两平行球面间的扩散层流流动,由球坐标下边界层运动方程式,导出了平行球面间进口段层流边界层动量与能量积分关系式.再对动量积分关系式采用Picard逐次逼近法,求得进口段通道长随边界层厚度而改变的近似表达式.然后对进口段效应诸系数进行分析与计算.     

Study of storm surge due to Typhoon Linda (1997) in the Gulf of Thailand using a three dimensional ocean model

Numerical integrations using the three dimensional ocean model based on the princeton ocean model (POM) were applied for the study of both sea level elevation and ocean circulation patterns forced by the wind fields during typhoons that moved over the Gulf of Thailand (GoT). The simulation concerned a case of Typhoon Linda which occurred during November 1-4, 1997. Typhoon Linda was one of the worst storms that passed the Gulf of Thailand and hit the southern coastal provinces of Thailand on November 3, 1997. It caused flooding and a strong wind covering large areas of agriculture and fisheries, which destroyed households, utilities and even human lives. The model is the time-dependent, primitive equation, Cartesian coordinates in a horizontal and sigma coordinate in the vertical. The model grid has 37 × 97 orthogonal curvilinear grid points in the horizontal, with variable spacing from 2 km near the head of the GoT to 55 km at the eastern boundary, with 10 sigma levels in the vertical conforming to a realistic bottom topography. Open boundary conditions are determined by using radiation conditions, and the sea surface elevation is prescribed from the archiving, validation and interpretation of satellite oceanographic data (AVISO). The initial condition is determined from the spin up phase of the first model run, which was executed by using wind stress calculated from climatological monthly mean wind, restoring-type surface heat and salt and climatological monthly mean freshwater flux. The model was run in spin up phase until an ocean model reached an equilibrium state under the applied force. A spatially variable wind field taken from the European Centre for Medium-Range Weather Forecasts (ECMWF) is used to compute the wind stress directly from the velocity fluctuations. Comparison of tendency between the sea surface elevations from model and the observed significant wave heights of moored buoys in the Gulf of Thailand under Seawatch project is investigated. The model predicts the sea level elevation up to 68.5 cm at the Cha-Am area located in the north of where the typhoon strands to the shore. Results of sea level elevation show that there is an area of peak set-up in the upper gulf, particularly in the western coast, and the effects of the storm surge are small at the lower gulf. During the entire period of this study, the surge in the gulf was induced by the northeasterly wind blowing over it.     

Solution of the Basic Boundary Value Problems of Stationary Thermoelastic Oscillations for Domains Bounded by Spherical Surfaces

The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.     

Mathematical model of fuel layer degradation when the laser target is heated by thermal radiation in the reactor working chamber

We propose a mathematical model of the changes occurring in the geometrical properties of the deuterium–tritium layer on the laser target in the process of its insertion into the reactor working chamber. The model is a parabolic equation of general form in spherical coordinates with nonlinear boundary conditions on a moving boundary. We show that under physically justified assumptions this problem may be regarded as a Stefan problem for a singularly perturbed parabolic equation. The first terms of the solution series are written out. Numerical calculations of the fuel layer degradation time are presented for a real target.     

Resonant wind-driven oceanic motions

We are interested here in describing the linear response of a rapidly rotating fluid to some surface stress, possibly due to the wind. The distinctive feature of the model considered here lies in the fact that the stress admits fast time oscillations and may be resonant with the Coriolis force. In addition to the usual Ekman layer, the model exhibits another – much larger – boundary layer, and some global vertical profile. We prove, in particular, that for large times, the wind effect is no longer localized in the vicinity of the surface. To cite this article: A.-L. Dalibard, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On interfacial transition conditions in two phase gravity flow

A layer of ice and sediment is modelled as a mixture of two nonlinear, very viscous, constant density fluids interacting mechanically via Darcy- and Pick-type forces. An inclined layer of this mixture overlain by a layer of ice modelled as a viscous fluid is considered with boundary conditions of no-slip or viscous sliding at the base and no stress at the free surface. The interface is treated as a singular surface across which the jump conditions of mass and momentum for the constituents are assumed to hold. Furthermore, because the components are viscous fluids, a kinematic condition for the continuity of the tangential velocity is formulated. The momentum jump conditions involve surface production terms requiring additional surfacial constitutive relations.We show that the posed physical problem admits a mathematical solution only in the case that the interface momentum production is non-zero.Dedicated to Hans Roethlisberger on the occasion of his seventieth birthday.     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

Surface tension effects in the equatorial ocean dynamics

By investigating a viscous boundary value problem, we show that for capillary waves the level of the thermocline marking the interface of the strongly stratified equatorial flows, decreases as the strength of the wind above the ocean surface increases.     

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.     

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

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

Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.     

On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Dynamics of a coupled atmosphere–ocean model

We consider a coupled atmosphere–ocean model, which involves hydrodynamics, thermodynamics and nonautonomous interaction at the air–sea interface. First, we show that the coupled atmosphere–ocean system is stable under the external fluctuation in the atmospheric energy balance relation. Then, we estimate the atmospheric temperature feedback in terms of the freshwater flux, heat flux and the external fluctuation at the air–sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Finally, we prove that the coupled atmosphere–ocean system has time-periodic, quasiperiodic and almost periodic motions, whenever the external fluctuation in the atmospheric energy balance relation is time-periodic, quasiperiodic and almost periodic, respectively.     

Mathematical Model of Ice Melting on Transmission Lines

During ice storms, ice forms on high voltage electrical lines. This ice formation often results in downed lines and has been responsible for considerable damage to life and property as was evidenced in the catastrophic ice storm of Quebec recently. There are two main aspects, viz., the formation of ice and its timely mitigation. In this paper, we mathematically model the melting of ice due to a higher current applied to the transmission wire. The two dimensional cross-section contains four layers consisting of the transmission wire, water due to melting of ice, ice, and the atmosphere. The model includes heat equations for the various regions with suitable boundary conditions. Heat propagation and ice melting are expressed as a Stefan-like problem for the moving boundary between the layers of ice and water. The model takes into account gravity which leads to downward motion of ice and to forced convection of heat in the water layer. In this paper, the results are applied to the case when the cross-sections are concentric circles to yield melting times for ice dependent on the increase in intensity of the electrical flow in the line. This research has been supported in part by Manitoba Hydro and NSERC.     

Classroom Notes

This note gives a simple‐minded approach to the two‐dimensional boundary layer equations. The pressure is eliminated from the equations of motion and the resulting equation is simplified by assuming that certain derivatives in the direction of the boundary are small compared with those at right angles to it. The simplified equation is then integrated to give a single boundary layer equation which, together with the stress rate of strain law and the continuity equation, is sufficient (in theory at least) to predict the flow.

The boundary layer equation as given does not depend on a particular form for the stress rate of strain law and could possibly form the basis for a non‐Newtonian investigation. The viscous boundary layer is given as a special case.     

Three-dimensional turbulent boundary layer on wind turbine blades

The momentum integral technique for predicting the boundary–layer growth in three–dimensional flow has been extended to include the entrainment equation as the closure model. Special attention has been devoted to those terms in the differential equations that change the boundary–layer structure from that of cvasi two–dimensional steady flow, at outboard locations, to a tree–dimensional flow pattern. It is concluded that the stall is delayed due to the boundary–layer reattachment at inboard sections in conjunction with the onset of a spanwise vortex like structure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear inertial effects on the instability of a single long wave of finite amplitude at the interface of two viscous fluids

The evolution of a single long wave of finite amplitude at the interface of two immiscible fluids of different viscosities and densities, between two horizontal plates is solved, using a boundary layer flow approximation for the equation of motion in each fluid layer. It is found that when the nonlinear inertial effects are taken into account in a moderate manner, at least in the frame of the boundary layer approximation, the initial unperturbed flow with smooth interface is stable to a single wave perturbation at the interface, even in the presence of adverse density and viscosity stratifications. However, when the nonlinear effects are increased in a specific way, and the magnitudes of the parameters involved are kept within the order of magnitude established for the present theory, an unstable flow configuration can be obtained.     

Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling of flow in a very small surface separation

When a fluid flows in a very small surface separation, the very thin boundary layer physically adhering to the solid surface will participate in the flow, while between the two boundary layers is a continuum fluid flow. An analysis is here presented for this multiscale flow. The continuum fluid is treated as Newtonian. The physical adsorbed boundary layer is treated as non-continuum across the layer thickness. The interfacial slippage can occur on the adsorbed layer-solid surface interface, while it is absent on the adsorbed layer-fluid interface. Three flow equations are derived respectively for the two adsorbed layers and the intermediate continuum fluid. They together govern the multiscale flow in such a small surface separation.