On the use of equations of the Fuchs class to calculate seepage from channels and irrigation ditches

Several schemes for seepage flows from the channels and ditches of irrigation systems through a layer of soil underlaid by a highly permeable artesian water-bearing table or an impermeable foundation are considered within the framework of the theory of the plane steady seepage of an incompressible liquid oblying to Darcy's law. Mixed multiparameter boundary value problems of the theory of analytic functions are formulated for their investigation, which are solved using Polubarinova–Kochina's method and integration of differential equations of the Fuchs class that are characteristic in problems of subterranean hydromechanics. On the basis of these models, algorithms are developed for calculating the dimensions of the saturation zone in cases when, in the seepage of water from channels and irrigation ditches, there is a need to estimate the combined effect on the pattern of motion of such important factors as the backwater from the underlying artesian water table or confining bed, the soil capillarity and the evaporation of ground waters from the free surface. The results of the calculations for all the flow schemes are compared for identical seepage characteristics.     

Groundwater regime in a soil layer with seepage from a channel allowing for capillarity

We obtain an exact hydrodynamic solution of the problem of steady plane seepage from a channel through a soil layer af finite thickness into an underlying drainage stratum. The solution allows far evaporation from the free surface and for capillarity of the soil. Unique solvability of the system of equations for the unknown mapping parameters is established. The seepage discharge from the channel and the capillary spread of wetness are determined as functions of the depth of the soil layer, the evaporation rate, and the capillarity of the soil. The numerical results are demonstrated by tables and graphs.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 43–47, 1987.     

Investigation of the effect of seepage or evaporation from a free surface by the method of circular polygons

It is proposed to use a technique developed for polygons in polar nets to integrate equations of the Fuchs class that arise when solving a wide range of problems of plane steady seepage flow using the Polubarinova-Kochina method, based on the use of the analytical theory of linear differential equations. It is shown that, for a large class of pentagons in domains where the flows,which are very characteristic of seepage problems when there is infiltration or evaporation from the free surface, have a complex velocity, the solution of the problem of determining the unknown parameters which appear in the conformal mapping can be completed. In this case, the mapping is carried out in closed form in terms of elementary functions and it is simple and convenient for subsequent application. The results obtained are used to solve the problem of seepage from a channel, taking account of the capillarity of the ground when there is evaporation from the free surface. The results of numerical calculations are presented and a hydrodynamic analysis of the effect of the basic physical parameters of the model on the dimensions of the saturation zone is given.     

Some mathematical models related to the Zhukovskii problem of the flow around a sheet pile

The seepage under a Zhukovskii sheet pile through a layer of soil underlain by a highly permeable pressurized horizon is considered. The left semi-infinite part of the roof of this horizon is simulated by an impermeable foundation. The flow when the velocity on the edges of the sheet pile is equal to infinity and, on the two water permeable parts of the boundary of the domain of motion, the flow rate takes extremal values, is investigated. The limiting cases, associated with the absence of both a backwater and an impermeable inclusion, are mentioned. The problem of seepage from a foundation pit formed by two Zhukovskii sheet piles is solved within the limits of a flow with a highly permeable pressurized stratum lying below. In the case when there is no infiltration onto the free surface, a solution of the well-known Vedernikov problem is obtained. A contact scheme, arising when there are no such indicated critical points, is considered; it is described outside the scope of the constraints imposed on the unknown conforming mapping parameters ensuring the realization of the basic mathematical model. Solutions are given for two schemes of motion in a semi-inverse formulation. The classical Zhukovskii problem is the limiting case of one of them. The special features of such models are mentioned. The Polubarinova-Kochina method is used to study all the above-mentioned flows. This method enables exact analytical representations of the elements of the motion to be obtained. The results of numerical calculations and an analysis of the effect of all the physical factors on the seepage characteristics are presented.     

Investigation of the squeezing out of the current in some flows in coastal water-bearing layers

Mathematical models of certain flows of fresh ground waters, in a semi-infinite pressurized water-bearing layer, to a salt water sea (basin, reservoir, pot hole, etc.), above the surface of which there is a layer of fresh water, are considered within the framework of the two-dimensional theory of steady seepage. To investigate them, mixed boundary-value problems in the theory of analytic functions are formulated and solved using Polubarinova-Kochina's method. On the basis of these models, algorithms are developed for calculating the squeezing out (that is, the process of the forcing out of the seeping fresh waters by the heavier salt waters, leading to deformation of the interface of the liquids) in cases when the ground water flows enter the sea from the side and from below. A detailed analysis of the structure and characteristic features of the processes, as well as of the effect of all the physical characteristics of the models on the nature and degree of the squeezing out of the fresh water, is carried out using the exact analytical relations obtained as well as numerical calculations. In the special case when there is no layer of fresh water above the surface of the sea, a comparison of the results of the calculation is given for both inflow schemes, and the nature of the dependences of the degree of squeezing out of the water from the initial position of contact of the liquids is discussed.     

The Zhukovskii problem of the flow around a sheet pile

The solution of the Zhukovskii problem of the flow around a sheet pile is given using the principles of two-dimensional steady-state seepage in the case when, accompanying the motion of the seeping water, there is a layer of saline ground waters at a certain depth under the sheet pile and this layer is located above an impermeable thickness of rock salt. The mixed boundary-value problem of the theory of analytic functions which arises is solved using Polubarinova-Kochina's method, which is based on the application of the analytical theory of linear differential equations and, also, the method, developed by us, of the conformal mappings of circular polygons in polar meshes, which are extremely typical for the velocity hodograph domains of such flows. While reflecting the specific details and individual properties of such flows, the solution constructed below turns out to be expressed in closed form in terms of elementary functions and, consequently, it is the simplest and most convenient solution. In addition, it is the most general solution for the class of problems being considered. The well known results Zhukovskii, Vedernikov and others are obtained from it as special and limiting çases A detailed hydrodynamic analysis and the specific features of the seepage process being considered, as well as the effects of all the physical parameters of the model on the pattern of the phenomenon, are presented using this solution and by numerical calculations.     

Seepage from a channel in an irrigated soil layer in the presence of an inclusion in the underlying highly permeable zero-head layer

We investigate the boundary-value problem describing plane steady seepage through a soil layer into an underlying zero-head layer capped by an impervious section. Uniform infiltration is assumed on the free surface. The solution of the problem is used to construct a direct computer algorithm; the computation results are illustrated with tables and diagrams.Translated from Vychislitei'naya i Prikladnaya Matematika, No. 62, pp. 52–56, 1987.     

Capillarity problems with nonlocal surface tension energies

We explore the possibility of modifying the classical Gauss free energy functional used in capillarity theory by considering surface tension energies of nonlocal type. The corresponding variational principles lead to new equilibrium conditions which are compared to the mean curvature equation and Young’s law found in classical capillarity theory. As a special case of this family of problems we recover a nonlocal relative isoperimetric problem of geometric interest.     

The design of the iso-velocity contour for the flow past the base of a dam with a confining bed

The underground contour of a submerged rectangular dam, the angles of which are rounded off along curves of constant magnitude of the seepage flow velocity is constructed in the case when the water-permeable base is underlain by a confining bed, consisting of one horizontal and two curved sections, which are the iso velocity lines of the seepage flow. The corresponding multiparameter mixed problem in the theory of analytical functions is solved using the Riemann-Schwartz symmetry principle and a semi-inverse version of the velocity hodograph method, first proposed by Polybarinova-Kochina and Kochina. The results of numerical calculations are presented and a hydrodynamic analysis of the effect of the basic physical parameters of the model on the shape and dimensions of the underground contour of the dam and of the horizontal and curved sections of the confining bed is given.     

Finite element modelling of surface waves in a channel

A variational formulation of the vertically-integrated differential equations for free surface wave motion is presented. A finite element model is derived for solving this nonlinear system of hydrodynamic equations. The time integration scheme employed is discussed and the results obtained demonstrate its good stability and accuracy.Several applications of the model are considered: the first problem is an open channel of uniform depth and the second an open channel of linearly varying depth. The ‘inflow’ boundary condition is prescribed in terms of the velocity which represents a wavemaker and/or a flow source, while the ‘outflow’ boundary condition is specified in terms of the water elevation. The outflow condition is adjusted for two cases, a reflecting boundary (finite channel) and a non-reflecting boundary (open-ended channel). The latter boundary condition is examined in some detail and the results obtained show that the numerical model can produce the non-reflecting boundary that is similar to the analytical radiation condition for waves. Computational results for a third problem, involving wave reflection from a submerged cylinder, are also presented and compared with both experimental data and analytical predictions.The simplicity and the performance of the computational model suggest that free surface waves can be simulated without excessively complicated numerical schemes. The ability of the model to simulate outflow boundary conditions properly is of special importance since these conditions present serious problems for many numerical algorithms.     

Analytical modelling for a three-dimensional hydrofoil with winglets operating beneath a free surface

The current study focuses on establishing a theoretical lifting surface model for predicting the hydrodynamic loads acting on the three-dimensional hydrofoil with winglets, which is considerably influenced by the proximity to the free surface through finding the three-dimensional Green’s function for the planar and vertical horseshoe vortices operating below a free surface. The hydrofoil surface is decomposed into a finite number of elements along the span direction and the chord directions, each of which can then be represented by a horseshoe vortex. The linearized free surface boundary condition is applied to analyze the influence of the free surface on the hydrofoil as well as the winglets. The thickness problem is considered using the source distribution among the hydrofoil and winglets surfaces and the analytical Green’s function that satisfies the linearized free surface boundary condition is used. As a sample application, numerical examples were conducted to show the performance of the hydrodynamic characteristics for the hydrofoil with winglets as a function of the Froude number. It was concluded that there are significant efficiency benefits from using winglets inside the free surface proximity effect. These results are substantiated by the comparison with the available published data.     

Mathematical modeling of systematic drainage of irrigated soil over an impermeable layer

We solve the problem of plane steady-state seepage of groundwater in a homogeneous isotropic soil layer from a periodic system of irrigation canals under conditions of both infiltration and horizontal drainage. A detailed hydrodynamic analysis is performed of the structure and properties of seepage flow and the effect of physical parameters of the flow scheme.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 72–75, 1990.     

具有蒸发液滴两相介质近壁流动的数学分析

在双连续介质理论框架下，采用匹配渐进展开方法导出并求解了具有蒸发液滴的汽雾流中层流边界层方程，给出了控制汽雾流的相似判据。对于沿曲面的流动，边界层方程的形式取决于是否存在液滴的惯性沉积。给出了热钝体驻点附近蒸汽－液滴边界层的数值计算结果。它们表明：由于蒸发，在边界层内近壁处形成了一个无液滴区域；在该区上边界处，液滴半径趋于零而液滴数密度急剧增高。液滴蒸发及聚集的联合效应造成了表面热流的显著增加，甚至在自由来流中液滴质量浓度很低时此效应依然存在。     

Thermal radiation effects on magnetohydrodynamic free convection heat and mass transfer from a sphere in a variable porosity regime

A mathematical model is presented for multiphysical transport of an optically-dense, electrically-conducting fluid along a permeable isothermal sphere embedded in a variable-porosity medium. A constant, static, magnetic field is applied transverse to the cylinder surface. The non-Darcy effects are simulated via second order Forchheimer drag force term in the momentum boundary layer equation. The surface of the sphere is maintained at a constant temperature and concentration and is permeable, i.e. transpiration into and from the boundary layer regime is possible. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite difference scheme. Increasing porosity (ε) is found to elevate velocities, i.e. accelerate the flow but decrease temperatures, i.e. cool the boundary layer regime. Increasing Forchheimer inertial drag parameter (Λ) retards the flow considerably but enhances temperatures. Increasing Darcy number accelerates the flow due to a corresponding rise in permeability of the regime and concomitant decrease in Darcian impedance. Thermal radiation is seen to reduce both velocity and temperature in the boundary layer. Local Nusselt number is also found to be enhanced with increasing both porosity and radiation parameters.     

Flexural‐Gravity Wave Scattering by a Circular‐Arc‐Shaped Porous Plate

The interaction of flexural‐gravity waves with a thin circular‐arc‐shaped permeable plate submerged beneath the ice‐covered surface of water with uniform finite depth is considered under the assumption of linear theory. The problem is reduced to a second kind hypersingular integral equation for the potential difference across the plate which is solved approximately by an expansion–collocation method. Utilizing the solution, the reflection and the transmission coefficients and the hydrodynamic forces are evaluated numerically. The focus of the paper is to illustrate the effect of a porous curved plate submerged in finite depth water with an ice‐cover on the normally incident waves. Numerical results for a circular‐arc‐shaped plate for different configurations are derived and represented graphically. Also, by choosing an appropriate set of parameters, the known results for a circular‐arc‐shaped rigid plate submerged in deep water and a semicircular porous plate submerged in finite depth water with a free surface are recovered as special cases.     

Higher-order asymptotic solutions in fluids of various configurations

Applying perturbation methods, symbolic computation, and generalizing the solution method, higher-order asymptotic solutions are constructed in Lagrangian variables for several models describing 2D standing wave motions in fluids of various configurations. Three main parameters of the fluid configuration, depth, capillarity, and stratification layer, are considered. The frequency-amplitude dependences are obtained and compared with those known in the literature in Eulerian and Lagrangian variables. The comparison shows that the analytical frequency-amplitude dependences are in complete agreement with previous results known in the literature and with the results obtained for other models. A generalization allows us to investigate critical phenomena for standing waves in fluids of various configurations. Namely, special attention is focused on critical values of one parameter, the fluid depth. The frequency-amplitude dependences are analyzed from the point of view of critical values: critical points and critical curves are determined for several models describing standing waves in fluids of various configurations.     

Analytical approach to heat and mass transfer in MHD free convection from amoving permeable vertical surface

An analytical study is performed on heat and mass transfer in MHD‐free convection from a moving permeable vertical surface and the results are compared with previous works on this phenomenon to test the validity. The coupled equations of boundary layer are transformed from their non‐linear form to ordinary form using similarity transformation and then are solved by a newly developed method, homotopy analysis method. Having different base functions, homotopy analysis method provides us with great freedom in choosing the solution of a nonlinear problem. Solving the boundry layer equations, the effects of different parameters such as magnetic field strength parameter (M), Prandtl number (Pr), Schmidt number (Sc), buoyancy ratio and suction/blowing parameter (f_w) on velocity, temperature, and concentration profiles are taken into consideration. Obtained results show that increment of magnetic field strength parameter (M) leads to decrease in velocity profile. Copyright © 2011 John Wiley & Sons, Ltd.     

Reflection of acoustic waves in a cylindrical channel from the perforated part

The reflection and transmission of harmonic waves and waves of finite duration through the boundary of the perforated part of a cylindrical channel (a lined borehole), filled with a fluid and surrounded by a permeable porous medium, is investigated. A model of the plane time-varying fluid flow in the cylindrical channel in a quasi-one-dimensional approximation and of the seepage absorption of the fluid in the porous medium surrounding the channel is presented. The effect of the collector characteristics of the porous medium surrounding the channel and the quality of the perforation (the length of the perforation channels) on the evolution of the waves when they are reflected from the boundary of the perforated part of the wall are investigated.     

Stability of a thin liquid layer,spreading on a quasi–rotating disk

In many industrial processes solids are coated to obtain specific surface properties, as e.g. corrosion resistance, mechanical (wear) resistance, optical, or electrical properties. Even today many coating processes are not fully understood and the choice of parameters is largely based on experience. Hence, a prediction of the complete hydrodynamic process and the appearance of instabilities in its dependency on the parameters appears highly desirable. This would serve to optimize the quality of the coating. A common coating technique is the so-called spin coating. The coating agent is dissolved or suspended in a liquid, brought onto the solid, spread by rotation, and the carrier liquid is finally removed by evaporation or by chemical reactions. In this article an evolution equation is derived from lubrication theory, valid for thin liquid layers. The model involves a dynamic contact angle, centrifugal, capillary, and gravitational forces. The evolution equation can be solved analytically, provided the capillary number is small. Then a coupled linear stability analysis of the contact line and the free interface is performed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the sequential gradient-restoration algorithm to thermal convective instability problems

The problem of the thermal stability of a horizontal incompressible fluid layer with linear and nonlinear temperature distributions is solved by using the sequential gradient-restoration algorithm developed for optimal control problems. The hydrodynamic boundary conditions for the layer include a rigid or free upper surface and a rigid lower surface. The resulting disturbing equations are solved as a Bolza problem in the calculus of variations. The results of the study are compared with the existing works in the literature.The authors acknowledge valuable discussions with Dr. A. Miele.