On similarity solution of an unsteady laminar boundary layer along a flat plate

Summary A unified analysis has been made to obtain all possible similarity solutions of the steady and unsteady, forced flow, inside a boundary layer along a flat plate. Though previously, attempts were made to obtain similarity solutions of a steady boundary layer flow neglecting viscous dissipation term in the energy conservation equation but the treatments were not complete. Here we have taken account of the viscous dissipation term. In the steady case it has been shown that for a similarity solution of both velocity and temperature, there should be a relation between the undisturbed flow outside the boundary layer and the temperature of the plate. It has been shown that the similarity solution exists in the unsteady case if we neglet the viscous dissipation term in the energy equation.     

Some exact solutions of unsteady boundary layer equations-I

Summary Unsteady boundary layer flows generated in a homogeneous, non-rotating viscous fluid are considered. The method of Laplace transform is used to obtain exact solutions of the unsteady boundary layer equations in a more general situation. The structures of the unsteady velocity field and the associated boundary layers are determined. Several particular solutions are recovered as special cases of the present general theory. The physical implications of the mathematical results are investigated.     

Some exact solutions of unsteady boundary layer equations-II

Summary Some exact solutions are presented for the unsteady boundary layer flows of a homogeneous, viscous, incompressible fluid bounded by (i) an infinite rigid oscillating flat plate or (ii) two parallel rigid oscillating flat plates. An explicit representation of the velocity fields for both the configurations has been given. The structures of the associated periodic boundary layers are determined with physical interpretations. Several results of interest have been recovered as special cases of this general theory. The Heaviside operational calculus along with the theory of residues of analytic functions is adopted in finding the solutions.     

On the growth of unsteady boundary layers on porous flat plates

Summary An investigation is carried out on the unsteady boundary layer induced in an incompressible, homogeneous, viscous fluid bounded by (i) an infinite horizontal porous plate aty=0, or (ii) two parallel horizontal rigid porous plates aty=0 andy=d. The unsteady motion is generated in the above fluid configurations by moving the plate(s) impulsively in its (their) own plane with a prescribed time-dependent velocity. Solutions for the unsteady velocity field are exactly solved by the Laplace transform treatment combined with the theory of residues. The structures of the associated boundary layers are determined. The effects of suction on the solutions and the boundary layers are investigated in detail, the limiting behaviour of the unsteady solution as timet is examined and Physical interpretations of the mathematical results are given. Finally, the frictional stresses on the plates are stated without detailed calculation.     

On unsteady motion of a rotating fluid bounded by flat porous plates

Summary In this paper, an investigation is made of the unsteady flow generated in a viscous, incompressible and homogeneous fluid bounded by (i) an infinite horizontal porous plate atz=0, or (ii) two infinite horizontal porous plates atz=0 andz=D. The fluid together with the plate(s) is in a state of solid body rotation with a constant angular velocity about the z-axis normal to the plate(s), and additionally, the plate(s) performs non-torsional elliptic harmonic oscillations in its (their) own plane(s). A uniform suction or injection is introduced in the configurations through the porous plate(s) and its influence on the unsteady flow and the associated boundary layers is examined. The unsteady flow field as well as the associated boundary layers is obtained explicitly. In contrast to the unsteady rotating flow without suction, solutions of the present problem with suction exhibit no resonant phenomena. It is shown that the suction is responsible for making the boundary layers thinner and for the elimination of the resonant phenomena. It is confirmed that the velocity field and the associated multiple boundary layers are significantly modified by suction. Physical significances of the mathematical results are discussed. Several limiting cases of interest are recovered from this analysis. The initial value problem for both the configurations is exactly solved by the Heaviside operational calculus combined with the theory of residues.     

Dynamic response of a flexible plate on saturated soil layer

An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot's two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances.     

P波入射时含圆柱形空腔板弹性波散射与动应力集中

采用波函数展开法给出了板的横截面处孔洞在平面P波入射下动应力集中问题的解。本文使用大圆弧假定法来满足板中的自由表面的边界条件。数值结果表明,上表面和孔洞间的距离与下表面和孔洞间的距离都对结果有影响。如果孔洞半径相对板的厚度很小,解近似等于孔洞全空间解。所给出的解是半解析解。因此能被用来验证诸如BEM,FEM,FDM等数值法。     

Horizontal impedance of square foundation in layered soil

This paper presents an extensive investigation into the influence of key mechanical and geometrical parameters on horizontal impedance of square foundations resting on or embedded in a two-layer soil deposit. The parameters investigated are the ratio of shear-wave velocities, the thickness of the top layer, the depth of embedment and the degree of contact between the footing-sidewall with backfill-soil. The results are presented in the form of simple and versatile dimensionless graphs, which should prove to be useful in understanding the steady-state harmonic response of square foundations in layered soils due to horizontal excitation. The investigation was conducted using a rigorous boundary element algorithm incorporating isoparametric boundary elements. Higher order quadratic elements were used since they can model the wavy nature of the dynamic problem investigated more accurately.     

One-dimensional dispersion in unsteady flow in an adsorbing porous medium: An analytical solution

Analytical solutions describing the concentration distribution along one-dimensional unsteady seepage flow through adsorbing saturated finite porous medium have been obtained. An exponential function concentration is enforced at the source of the dispersion, while the change in the concentration is zero at the other boundary. A new time variable has been introduced to solve the unsteady flow problem and the solution is illustrated.     

NUMERICAL SIMULATION OF SEDIMENT TRANSPORT IN DYNAMIC FLOW CONDITIONS

1 INTRODUCTION Rivers as a source of life can at the same time impose devastating conditions on the environment. It is , therefore, imperative to analyse and predict river behaviour for different given conditions and engineering activities. Therefore, the use of simulation tools in this field has become a necessity. Many computational tools for simulation of sediment transport in rivers are now available that can be used for prediction and design under different flow conditions. However, …     

求解薄板频率特性的边界元法解析研究

根据弹性薄板理论和振动理论,运用边界元法(BEM)分析了薄板弯曲自由振动的频率特性,从薄板自由振动的微分方程式出发,采用动态基本解推导出其相应的边界积分方程式,计算中对边界积分方程进行缩减,应用频率扫描的方法研究薄板的动态特性,准确、有效地得到了其相应的固有频率、给出的数值算例证明了本文的方法简单、可行,且具有良好的解析计算精度。     

Unsteady hydromagnetic flow in an annulus with porous Walls when the inner boundary starts moving from rest

Summary The problem of unsteady hydromagnetic flow in a porous annulus when the inner boundary starts moving from rest with a constant longitudinal velocity is considered. For the case of equal Reynolds number and magnetic Reynolds number Laplace transforms of the solutions for the axial components of velocity and magnetic field are obtained in terms of modified Bessel functions. The inversions are effected for the cases of very large and small Reynolds number when the pressure gradient vanishes.     

A nonlinear,resistive boundary layer in rotating hydromagnetic flow

A new nonlinear boundary layer in rotating hydromagnetic flows is presented. The purpose of this layer is to provide a smooth transition between the magnetic field at a rigid, electrically insulating boundary and that far from the boundary. The boundary layer problem is solved in a cylindrical geometry, assuming that the variables obey the hydromagnetic analog of the von Karman similarity. The primary momentum balance within this boundary layer is between pressure, Coriolis and hydromagnetic forces; the viscous and nonlinear inertia forces are unimportant. Diffusion of the magnetic field is balanced by advection where the advecting fluid flow is induced by the action of the hydromagnetic body forces within the boundary layer. The structure of the layer depends upon a single parameter which measures the strength of the normal magnetic field. Steady solutions are presented and their uniqueness and temporal stability are analyzed. The relevance of this layer to the core of the Earth is discussed. It is estimated that this boundary layer would be at least as thick as one tenth of the core radius if it exists within the core.     

Modeling nitrogen–carbon cycling and oxygen consumption in bottom sediments

A model framework is presented for simulating nitrogen and carbon cycling at the sediment–water interface, and predicting oxygen consumption by oxidation reactions inside the sediments. Based on conservation of mass and invoking simplifying assumptions, a coupled system of diffusive–reactive partial differential equations is formulated for two-layer conceptual model of aerobic–anaerobic sediments. Oxidation reactions are modeled as first-order rate processes and nitrate is assumed to be consumed entirely in the anoxic portion of the sediments. The sediments are delineated into a thin oxygenated surface layer whose thickness is equal to the oxygen penetration depth, and a lower, but much thicker anoxic layer. The sediments are separated from the overlying water column by a relatively thin boundary layer through which mass transfer is diffusion controlled. Transient solutions are derived using the method of Laplace transform and Green’s function, which relate pore-water concentrations of the constituents to their concentrations in the bulk water and to the flux of decomposable settling organic matter. Steady-state pore-water concentrations are also obtained including expressions for the extent of methane saturation zone and methane gas flux. A relationship relating the sediment oxygen demand (SOD) to bulk water oxygen is derived using the two-film concept, which in combination with the depth-integrated solutions forms the basis for predicting the extent of oxygen penetration in the sediment. Iterative procedure and simplification thereof are proposed to estimate the extent of methane saturation zone and thickness of the aerobic layer as functions of time. Sensitivity of steady-state solutions to key parameters illustrates sediment processes interactions and synergistic effects. Simulations indicate that for a relatively thin diffusive boundary layer, d, oxygen uptake is limited by biochemical processes inside the sediments, whereas for a thick boundary layer oxygen transfer through the diffusive boundary layer is limiting. The results show an almost linear relationship between steady-state sediment oxygen demand and bulk water oxygen. For small d methane and nitrogen fluxes are sediment controlled, whereas for large d they are controlled by diffusional transfer through the boundary layer. It is shown that the two-layer model solution converges to the one-layer model (anaerobic layer) solution as the thickness of the oxygenated layer approaches zero, and that the transient solutions approach asymptotically their corresponding steady-state solutions.     

Numerical simulations of mantle flow around slab edges

A simple kinematic-dynamic model of mantle flow around the slab-edge is constructed in order to understand the flow complexity there. The flow velocity on the top and the small boundary region around the shallow plate boundary is kinematically imposed in order to achieve a subduction-like feature and the flow in other part is dynamically calculated. The geometry of the plate mimics the region around the junction of Aleutian Islands and Kamchatka, that are examples of the convergent-transform fault plate boundaries. In a simple model in which the overlying plate is almost stationary, the lateral flow from the mantle under the subducting slab to the mantle under the neighboring plate is of minor importance, once the slab penetrates into the high viscosity layer where the downward flow encounters the resistance. Similar situation was found when the trench is advancing, that is, the trench moves toward the overlying plate. For the case with retreating trench, that is, the trench moves toward the subducting plate, a lateral flow exists even after the slab penetrates into the high viscosity layer, although its magnitude is significantly smaller than that of the plate velocity. The presence of a low viscosity layer just beneath the subducting plate may promote the emergence of lateral flow. A significant lateral flow is observed when the high temperature anomaly, that is, buoyant and low viscosity block carried by the movement of subducting plate, approaches the slab. These results may have important implications for the possible existence of trench parallel flow in the sub-slab mantle.     

中国及其邻区周围板块作用力的研究

给出了如何根据观测的应力方向，利用平滑方法处理为在规则格点上均匀分布的数据的过程，并利用新编的中国及其邻区地震震源机制解推断的观测应力方向，反演了周围板块作用力的相对大小．结果表明，在周围三大板块中，以印度板块的作用力最大；其次是太平洋板块；而在菲律宾海板块边界上，作用力大小差异较大，其中又以与台湾相邻的边界段受力最大，琉球岛弧受力最小．     

On the geodynamic setting of kimberlite genesis

The emplacement of kimberlites in the North American and African continents since the early Palaeozoic appears to have occurred during periods of relatively slow motion of these continents. The distribution of kimberlites in time may reflect the global pattern of convection, which forces individual plates to move faster or slower at different times. Two-dimensional numerical experiments on a convecting layer with a moving upper boundary show two different regimes: in the first, when the upper boundary velocity is high, heat is transferred by the large-scale circulation and in the second, when the upper boundary velocity is lower, heat is predominantly transferred by thermal plumes rising from the lower boundary layer. For a reasonable mantle solidus, this second regime can give rise to partial melting beneath the moving plate, far from the plate boundaries. The transition between these modes takes place over a small range of plate velocities; for a Rayleigh number of 10⁶ it occurs around 20 mm yr^?1. We suggest that the generation of kimberlite magmas may result from thermal plumes incident on the base of a slowly moving plate.     

中国及其邻区周围板块作用力的研究

给出了如何根据观测的应力方向,利用平滑方法处理为在规则格点上均匀分布的数据的过程,并利用新编的中国及其邻区地震震源机制解推断的观测应力方向,反演了周围板块作用力的相对大小．结果表明,在周围三大板块中,以印度板块的作用力最大;其次是太平洋板块;而在菲律宾海板块边界上,作用力大小差异较大,其中又以与台湾相邻的边界段受力最大,琉球岛弧受力最小．     

The Role of History-Dependent Rheology in Plate Boundary Lubrication for Generating One-Sided Subduction

We have developed a two-dimensional dynamical model of asymmetric subduction integrated into the mantle convection without imposed plate velocities. In this model we consider that weak oceanic crust behaves as a lubricator on the thrust fault at the plate boundary. We introduce a rheological layer that depends on the history of the past fracture to simulate the effect of the oceanic crust. The thickness of this layer is set to be as thin as the Earth's oceanic crust. To treat 1-kilometer scale structure at the plate boundary in the 1000-kilometer scale mantle convection calculation, we introduce a new numerical method to solve the hydrodynamic equations using a couple of uniform and nonuniform grids of control volumes. Using our developed models, we have systematically investigated effects of basic rheological parameters that determine the deformation strength of the lithosphere and the oceanic crust on the development of the subducted slab, with a focus on the plate motion controlling mechanism. In our model the plate subduction is produced when the friction coefficient (0.004–0.008) of the modeled oceanic crust and the maximum strength (400 MPa) of the lithosphere are in plausible range inferred from the observations on the plate driving forces and the plate deformation, and the rheology experiments. In this range of the plate strength, yielding induces the plate bending. In this case the speed of plate motion is controlled more by viscosity layering of the underlying mantle than by the plate strength. To examine the setting of the overriding plate, we also consider the two end-member cases in which the overriding plate is fixed or freely-movable. In the case of the freely-movable overriding plate, the trench motion considerably changes the dip angle of the deep slab. Especially in the case with a shallow-angle plate boundary, retrograde slab motion occurs to generate a shallow-angle deep slab.     

On the application of the boundary layer approximation for the simulation of density stratified flows in aquifers

Stratification of the density in groundwater flow stems from the contact between water which contains minerals in low concentration with water containing a high concentration of minerals. The flow in such a flow field should be simulated by solving simultaneously the equations of continuity, motion and solute transport, because solute concentration affects the dynamics of the flow. Such an approach is generally associated with complicated calculations and numerical schemes subject to problems of convergence and stability, as the basic equations are highly nonlinear.This study applies the phenomenological boundary layer approximation, and suggests a reference to three different zones in the flow field: (a) fresh water zone, (b) transition zone, and (c) mineralized water zone. In zones (a) and (c) it is assumed that the potential flow theory can be applied. In zone (b) the flow is nonpotential but the basic similarity conditions typical to boundary layers exist.The approach suggested in this study simplifies the mathematical models that should be used for the flow field simulation. This approach is especially attractive in cases where the Dupuit approximation is applicable. In such cases very often analytical solutions can be obtained for unidirectional flows. In cases that are too complicated for representation by analytical solutions, the method can be used for the creation of simplified numerical schemes.Various examples in this study demonstrate the application of the method for various field problems associated with steady state as well as unsteady state conditions.The simplicity of the method makes it useful for variety of problems. It can be used even by small institutions and small consulting firms, who have usually access to minicomputers and microprocessors.