Mixed Type Boundary Conditions Research Articles

Substantiation of the Darcy law for a porous medium with condition of partial adhesion

A study is made of a stationary Stokes's system in a periodically perforated domain with boundary conditions of mixed type, which describes the motion of a viscous incompressible fluid in a porous medium in the presence of friction between the fluid and the walls of the pores. The relation between the leading terms of the asymptotic expansions with respect to for the fluid velocity and the pressure is obtained, where is the parameter characterizing the fineness of the porous structure.

Hydraulics of a partially penetrating well: solution to a mixed-type boundary value problem via dual integral equations

New semi-analytic solutions are obtained for well response to the pumping test and slug test performed on a partially penetrating well. The solutions account not only for the wellbore storage, infinitesimal skin, and aquifer anisotropy, but also for the mixed-type boundary condition at the well face, which is novel. The solutions are obtained via the method of dual integral equations (DE). The new solutions are computationally robust and efficient, about one to two orders of magnitude faster than the corresponding finite difference solutions. Existing approximate solutions obtained with flux–flux discontinuous boundary conditions are compared to our DE solutions. The accuracy of the approximate solutions appears to be adequate for slender well screens. Our DE solution is computationally more efficient than the approximate solutions. In the range where the approximate solutions are less accurate the DE solution is about an order of magnitude faster. More important, the new solutions provide the correct distribution of the point flux (local velocity) along the well screen, unlike all existing solutions. This feature is essential in cases where vertical variations of hydraulic conductivity are sought (e.g. in flowmeter tests), and for tracer tests.

A new method for the interpretation of the constant-head well permeameter

A novel semi-analytical solution for the interpretation of the constant-head permeameter test is introduced, which accounts for the correct mixed-type boundary condition at the wellbore, unlike all published analytical solutions. Capillarity can also be accounted for. The simplifications are that flow from the bottom of the borehole is neglected (therefore the solution is applicable to slender boreholes, where the ponding depth is at least 10 times the radius) and capillarity can be modeled with a quasi-linear approach. The Green's function approach leads to an integral equation, the solution of which does not show significant ill-posedness. Two sub-cases are presented: the first neglects capillary effects (the all-saturated approximation) and the second (general solution) takes them into account. The all-saturated solution is successfully tested against finite element simulations. The corresponding values of the borehole shape factor C are slightly larger than the ones obtained with approximate analytical solutions from the literature. When capillarity is accounted for, C changes of a factor of 10 when the dimensionless sorptive number A goes from typical values for fine soils to typical values for coarse soils (about two orders of magnitude of variation for A). This range shifts to lower values of A as the dimensionless borehole depth increases. Consequently, the all-saturated solution is a good approximation of the soil behavior for boreholes with large ponding depth, and coarse soils. The proposed semi-analytical solution is fast to compute and thus it is possible to use it in an automated optimization technique to fit field data and estimate the field-saturated hydraulic conductivity and the sorptive number; this would not be feasible using a numerical solution.

Heat content asymptotics for higher order Laplacians

Let M be a compact Riemannian manifold with smooth boundary ϖM. We study the heat content asymptotics for higher order Laplacians with boundary conditions of mixed type.

Boundary element analysis of gear teeth fracture

The steady state thermoelastic boundary element approach is extended to accommodate mixed-type boundary conditions in two-dimensional domains which may be definite functions of some global variables. This technique is applied to fractured gear teeth in order to evaluate numerically the fracture characterizing parameters. Singular crack elements in conjunction with multidomain formulation are utilized for the computation of mode I and II stress intensity factors. Results are given for some severe crack orientations with respect to crack depth and gear geometry. The present results are very close to those obtained by the finite element method. The influence of heat convection characteristics and mechanical loading conditions are also examined. Stress intensity factors are evaluated for a wide range of operating parameters useful for design purposes.

Development of a modular nodal method for solving three dimensional spherical harmonic equations

A nodal method based upon the least squares minimization technique has been developed for solving the neutron transport equation using spherical harmonics approximation in 3-dimensional Cartesian coordinates. The second order form of the spherical harmonic differential equations based upon the even partly form of the transport equation are utilized. The governing equations are casted into a set of elliptic type differential equations for which the coefficients can be generated for any spherical harmonics approximation order automatically. The coefficients of the mixed type boundary conditions are obtained through recurrence relationships for any order as well. The approximation order to be used in the nodes can vary according to the physics of the problem and determined by the computer code.

Elastic equilibrium of a layer with a cavity for boundary conditions of mixed type on the faces

On the basis of the method of homogeneous solutions we study the problem of the stress state of an isotropic layer with a cavity in the case of boundary conditions of mixed type on the faces. By numerical studies we establish the laws of variation of the stresses as functions of the variation of the relative thickness, type of load, and boundary conditions on the faces.

The stress state of thick plates under homogeneous boundary conditions of mixed type on the faces

We propose a method of using homogeneous solutions in the case of mixed boundary conditions on the faces of thick plates. We obtain the exact solution of the problem of the stress state of a lamina weakened by a cylindrical cavity.

Symmetry-Breakings for Semilinear Elliptic Equations on Finite Cylindrical Domains

We study the existence and multiplicity of asymmetric positive solutions of a semilinear elliptic equation on finite cylinders with mixed type boundary conditions. By using a Nehari-type variational method, we prove that the numbers of asymmetric positive solutions are increasing without bound when the lengths of cylinders are increasing. On the contrary, by using the blow up technique, we obtain an a priori bound for positive solutions and then prove that all positive solutions must be symmetric when the cylinders are short enough

Symmetry-breakings for semilinear elliptic equations on finite cylindrical domains

We study the existence and multiplicity of asymmetric positive solutions of a semilinear elliptic equation on finite cylinders with mixed type boundary conditions. By using a Nehari-type variational method, we prove that the numbers of asymmetric positive solutions are increasing without bound when the lengths of cylinders are increasing. On the contrary, by using the blow up technique, we obtain an a priori bound for positive solutions and then prove that all positive solutions must be symmetric when the cylinders are short enough.

Some variational inequalities for age-dependent population dynamics

A variational inequality for an equation of age-dependent population diffusion is studied. A mixed-type boundary condition is prescribed on time-and age-dependent parts of the boundary. The rate of mortality is allowed to have a divergency property at the maximal age.

A modular spherical harmonics approach to the neutron transport equation

A modular nodal method was developed for solving the neutron transport equation in 2-D xy coordinates. The spherical harmonic expansion was used for approximating the second-order even-parity form of the neutron transport equation. The boundary conditions of the spherical harmonics approximation were derived in a form to have forms analogous to the partial currents in the neutron diffusion equation. Relations were developed for generating both the second-order spherical harmonic equations and the boundary conditions in an automated computational algorithm. Nodes using different orders of the spherical harmonics approximation to the transport equation were interfaced through mixed-type boundary conditions. The determination of spherical harmonic orders implemented in the nodes were determined by the scheme in an automated manner. Results of the method compared favorably to benchmark problems.

Navier-Stokes modelling of coupled interfacial and convective transport during the Ge epitaxial growth experiments of D1 mission

The 2-D Navier-Stokes species and energy conservation equations are solved to model the transport process during Ge crystal growth in GeI 2/Gel 4 system in closed cell. The coupling between convection and interface transport is taken into account through mixed type boundary conditions in which the net production term is deduced from the surface kinetics involving a limiting step of GeI 2 adsorption. It is found that: (a) for the real flight deposit temperature, the substrate evaporates as observed in the D1 experiment (15); (b) for the planned deposit temperature, the model shows a very light evaporation of the substrate; (c) for 3 K beneath the planned temperature, a deposit occurs on the substrate; and (d) the velocity on the substrate (deposition or evaporation) for similar thermal conditions depends on the gravity level. This numerical model, in which hydrodynamics and surface physics are really coupled, seems to be a good investigation tool.

Numerical solutions for some coupled systems of nonlinear boundary value problems

In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.

Pressure of a stamp of almost annular planform on an elastic half-space

The generalization of the problem of the impression of an annular stamp without friction into an elastic half-space /1, 2/ is considered. The contact domain has an axis of symmetry and is a ring bounded by curves of almost circular shape. The half-space material is isotropic and homogeneous. Determination of the pressure under the stamp reduces to finding two functions of a complex variable, analytic in a circle, by means of boundary conditions of mixed type. The unknown constants on the right-hand sides of the boundary conditions are determined under the assumption that the dimensions of the holes in the stamp are small. The results from /3, 4/, referring to the case of annular or almost circular stamps, are essentially used here.

Second Order Nonlinear Singular Perturbation Problems with Boundary Conditions of Mixed Type

We study nonlinear turning point problems that admit boundary and/or interior layers at positions that are not determined a priori. Our study differs from previous investigations in that for positive “viscosity” first order derivative terms are allowed in the boundary operator. Under certain conditions, shown in a sense to be sharp, we characterize the viscous limit of such problems and prove that they are identical to those limit solutions obtained from the pure Dirichlet problem.

On a unified computer program for influence function methods in elasticity

By a simple compilation of the influence functions of plane elasticity, a compact unified computer program is developed which allows for an arbitrary choice of formulation for influence function methods. The idea is based on the analysis of the behavior of influence functions and certain relationships which exist among them. It is shown that these influences can be classified into three groups and that their application in elasticity problems is limited to only two options. The program can be used as a solution technique to compute stresses and displacements in a region subjected to mixed type boundary conditions and as comparison tool to compare the merits and drawbacks of the various singularities used in the method. Sample problems and a list of the computer program (in FORTRAN) are included.

Unsteady-State Heat Conduction in Semi-Infinite Regions With Mixed-Type Boundary Conditions

Presentation d'une solution de forme close pour la conduction thermique non stationnaire dans un domaine semi-infini avec des conditions aux limites mixtes isotherme-isoflux

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.

Boundary element method in nonlinear transient heat transfer of reactor solids with convection and radiation on surfaces

This paper deals with the boundary (integral) element method for non-steady conduction problems of solids, subject to non-linear convective and radiation conditions on surfaces. Boundary integral equations for the mixed-type and non-linear boundary conditions, both for the case with constant and variable heat conductivity are derived, modelled by non-conforming boundary elements, while domain integrals are evaluated within triangular cells. A test case is included to illustrate the described procedure.