Third-type Boundary Conditions Research Articles

Modeling of Chemical Vapor Infiltration for Fiber-Reinforced Silicon Carbide Composites Using Meshless Method of Fundamental Solutions

In this study, the Method of Fundamental Solutions (MFSs) is adopted to model Chemical Vapor Infiltration (CVI) in a fibrous preform. The preparation of dense fiber-reinforced silicon carbide composites is considered. The reaction flux at the solid surface is equal to the diffusion flux towards the surface. The Robin or third-type boundary condition is implemented into the MFS. From the fibers’ surface concentrations obtained by MFS, deposition rates are calculated, and the geometry is updated at each time step, modeling the pore filling over time. The MFS solution is verified by comparing the results to a known analytical solution for a simplified geometry of concentric cylinders with a concentration set at the outer cylinder and a reaction at the inner cylinder. MFS solutions are compared to published experimental data. Porosity transients are obtained by a combination of MFSs with surface deposition to show the relation between the initial and final porosities.

Parasitic Eigenvalues of Spectral Problems for the Laplacian with Third-Type Boundary Conditions

Parasitic Eigenvalues of Spectral Problems for the Laplacian with Third-Type Boundary Conditions

Parasitic Eigenvalues of Spectral Problems for the Laplacian with Third-Type Boundary Conditions

Spectral problems for the Laplacian with Robin and Steklov (third-type) boundary conditions on a smooth boundary of a plane domain are considered. These conditions involve a small parameter and a coefficient of “wrong” sign, giving rise to negative eigenvalues, which are called parasitic. Such problems and eigenvalues arise in numerical schemes when regular variations in boundaries (small nonuniform shifts along the normal) are modeled by perturbations of differential operators in boundary conditions. Asymptotic expansions of some parasitic eigenvalues are constructed and justified, and a priori estimates are obtained, which help to determine their locations on the real axis and the effect exerted on the simulation errors.

A Study of a Thermally-Vigorous Rayleigh-Bénard Convection in a Hybrid Nanofluid Using Non-Classical Boundary Condition

A linear stability analysis of a thermally-vigorous Rayleigh-Bénard convection in a mono and a hybrid nanofluid is carried out using the single-phase model. We make use of non-classical boundary condition on velocity (rough boundary condition) and temperature (third-type boundary condition) instead of a specific boundary condition (free-free isothermal, rigid-rigid isothermal, etc.) which is traditionally considered. The thermophysical properties are calculated using phenomenological laws and mixture theory. A unique and novel combination of a single-term Galerkin technique and the Maclaurin series expansion is used to solve the boundary-eigen-value problem obtained in the problem. The critical value of the wave number and the Rayleigh number that are calculated using the procedure are quite accurate up to 5 and 3 decimal places respectively. A comparative study on the instability in two types of nanofluids is carried out so as to infer which type of nanofluid suits best for thermally vigorous systems. This paper aims to provide a theoretical basis to ease the selection process pertaining to the making of a choice from different types of nanofluids. To validate the study sixteen limiting cases have been obtained. To visualize the flow behavior physically, streamlines have been plotted.

Impact of climate change on groundwater recharge in shallow young glacial aquifers in northern Poland

We investigated the influence of climate change in the period 1951–2020 on shallow aquifers in the Brda and Wda outwash plains (Pomeranian Region, Northern Poland). There was a significant temperature rise (0.3 °C/10 years), which accelerated after 1980 (0.66 °C/10 years). Precipitation became increasingly irregular – extremely rainy years occurred right after or before extremely dry years, and intensive rainfall events became more frequent after 2000. The groundwater level decreased over the last 20 years, even though the average annual precipitation was higher than in the previous 50 years. We carried out numerical simulations of water flow in representative soil profiles for the years 1970–2020 using the HYDRUS-1D model, developed and calibrated during our earlier work at an experimental site in the Brda outwash plain (Gumuła-Kawęcka et al., 2022). We used a relationship between the water head and flux at the bottom of the soil profiles (the third-type boundary condition) to reproduce groundwater table fluctuations caused by recharge variability in time. The calculated daily recharge showed a decreasing linear trend for the last 20 years (0.05–0.06 mm d−1/10 years), and dropping trends in water table level and soil water content in the entire profile of vadose zone. Field tracer experiments were performed to estimate impact of extremely rain events on water flux in vadose zone. The results suggest that tracer travel times are strongly determined by water content in the unsaturated zone which is determined by precipitation amount in span of weeks, rather than extremely high precipitation events.

Mathematical Modelling of Transient Processes in a Three Phase Electric Power System for a Single Phase Short-Circuit

Field approaches are employed to develop a mathematical model of a power network section. The facility consists of two electric power subsystems described with ordinary differential equations and presented as concentrated parameter systems connected with a three-phase power supply line, presented as a distributed parameter system. The model of the electric power line is described with partial differential equations. Mathematically, the supply line model is described utilizing a mixed problem with explicitly indefinite boundary conditions. All electromagnetic state equations of the integrated system are introduced in their matrix-vector forms. The equation of the three-phase long supply line is expressed untraditionally as a system of two first-order differential equations as a function of long line voltage. Since the power supply line is part of the integrated system that includes two subsystems, the boundary conditions at the line’s start and end are implicitly defined, avoiding the traditional application of the Dirichlet first-type boundary condition. An expanded system of ordinary differential equations that describe physical processes in both the supply and loading subsystems is used to calculate the boundary conditions. To this end, third-type boundary conditions, or Poincaré’s conditions, serve to describe the wave equation of the electric power line. Such an integrated model of an electric power system helps analyse transient processes across the supply line when the electric power system is switched on and is single-phase short-circuited at the final point of the electric power line. A comparison of computer simulation results with well-known software packages shows a convergence of approx. 96%.

Solution of an inverse heat conduction problem with third-type boundary conditions

An algorithm is presented for the solution of an inverse heat conduction problem: reconstruct the steady-state distribution of the convection heat transfer coefficient on one wall of a slab from the temperature distribution on a plane embedded in the slab, assuming that the thermal boundary conditions on the other walls are known. The algorithm is based on the finite volume discretization of the slab and on the construction and subsequent inversion of square matrices linking the distributions of wall temperature and wall heat flux to that of temperature at the internal plane. The algorithm is first applied to a simulated temperature distribution, for which it is assessed for accuracy and compared with a simpler method based on the assumption of one-dimensional heat conduction. It is then applied to real experimental temperature distributions obtained by digitally processing thermochromic liquid crystals images in spacer-filled channels simulating membrane distillation modules. Tolerance to noise is demonstrated in both cases provided an adequate amount of filtering is applied to the primitive, high-resolution temperature distribution.

Mathematical Modelling of Transient Processes in an Asynchronous Drive with a Long Shaft Including Cardan Joints

Beginning with the classic methods, a mathematical model of an electromechanical system is developed that consists of a deep bar cage induction motor that, via a complex motion transmission with distributed mechanical parameters, drives a working machine, loading the drive system with a constant torque. The electromagnetic field theory serves to create the motor model, which allows addressing the displacement of current in the rotor cage bars. Ordinary and partial differential equations are used to describe the electromechanical processes of energy conversion in the motor. The complex transmission of the drive motion consists of a long shaft with variable geometry cardan joints mounted on its ends. Non-linear electromechanical differential equations are presented as a system of ordinary differential equations combined with a mixed problem of Dirichlet first-type and Poincaré third-type boundary conditions. This system of equations is integrated by discretising partial derivatives by means of the straight-line methods and successive integration as a function of time using the Runge–Kutta fourth-order method. Starting from there, complicated transient processes in the drive system are analysed. Results of computer simulations are presented in the graphic form, which is analysed.

Determining the critical insulation thickness of breathing wall: Analytical model, key parameters, and design

Breathing wall (BW) based on air-permeable porous medium offers an alternative solution for utilizing the conductive heat loss of building envelope to preheat the infiltration ventilation airflow within porous medium. However, current studies neglect the influence of pressure drop within porous medium on the energy performance of BW, which may lead to an overestimation or non-optimal design. In this study, we proposed a framework to determine the critical insulation thickness of BW for minimizing its convective heat loss and pressure drop related energy loss. An analytical model was developed and validated to calculate the heat loss of BW under third-type boundary condition. Darcy's law was applied to estimate the pressure drop of infiltration airflow and its associated energy loss. Case studies were conducted to identify the critical insulation thickness of BW. The critical thickness of BW under different scenarios was investigated. The results demonstrate the existence of critical thickness, which yields the lowest overall heat loss of BW. A larger infiltration airflow rate or lower air permeability of porous medium will result in a downward trend of this critical thickness. The outcomes of this study can provide a design guideline of BW for maximizing its energy saving potential.

Heat Transfer in Composite Spheroids

AbstractHeat transfer from a composite prolate spheroid under the third-type boundary condition is investigated using a Legendre series expansion. The model is verified against published data on cooling boiled eggs and also against the asymptotic solution of a composite sphere. The impact of Biot number on the heat transfer in spheroids with realistic dimensions and properties, such as eggs and olives, is investigated. The results are also presented for varying conductivity ratios and fractional volume of the inner part of the spheroid.

2-D Darcian flow in vicinity of permeable fracture perturbing unidirectional flow in homogeneous formation

A thin planar fracture with a highly permeable porous filling disturbs an incident unidirectional one-phase flow in a formation/aquifer and makes it essentially 2-D. The main objective of this paper is to get an analytical evaluation of the quantity of fluid intercepted, conveyed, and released back by the fracture and assessment of the near-fracture zone where 2-D effects are essential. The impact of the fracture is modelled by the Robin (third-type) boundary condition imposed along a mathematical cut. Flow in the fracture is 1-D and along the fracture axis. For a fracture of a rectangular shape, we use a conformal mapping of a physical plane with a cut onto the exterior of a unit disc in a reference plane and transform the Robin boundary condition into an infinite system of linear equations. Using the Kantorovich–Krylov theory, we rigorously prove that the system is entirely regular, i.e. it can be solved by the method of reduction (truncation to a finite system). A method of iterations is used to solve a reduced system. Convergence with respect to the truncation order is investigated. For non-rectangular (curvilinear) planar fractures with a straight axis, we use an inverse approach of Pilatovskii, viz., specifying the flow rate function along the fracture axis, i.e. the quantity of fluid which is intercepted by the fracture from the ambient rock. The fracture geometry is obtained as a part of solution. Pilatovskii’s function is involved as a weight of a singular integral with the Cauchy kernel. The corresponding integro-differential equation is a special case of the Prandtl equation for a thin airfoil in an ideal fluid flow with the circulation along the airfoil cord as a weight. The derivative of the Pilatovskii function is expanded into a series of Chebyshev’s orthonormal functions. This series is truncated, giving, in particular, the Pilatovskii fracture as a limit. Flow nets for both rectangular and curvilinear fractures are plotted. The dependences of the capture zone (quantity of intercepted–released fluid) on the ratios of permeabilities of the bulk rock-soil/fracture’s filling and fracture’s length/thickness, as well as the flow nets, are of interest to reservoir engineers and groundwater hydrologists working with Darcian flows in heterogeneous porous media.

Janus surface concept for three-dimensional turbulent flows

Three-dimensional numerical simulations of turbulent flow over a partial super-hydrophobic cylinder, hereinafter referred to as the “Janus cylinder”, are performed using large-eddy simulation (LES) with OpenFOAM solver. The slip-wall velocity at the super-hydrophobic surface is evaluated from a third-type (Robin) boundary condition whose coefficient is adjusted based on the available experimental data. Compared to no-slip smooth cylinder, the Janus cylinder has a lower drag coefficient and a lower root-mean-square (rms) lift coefficient, whereas its dominant vortex shedding frequency is higher. The power density distribution of the oscillations, obtained from the Fourier transform of the time-history of the lift coefficient, of the Janus cylinder has a less-sharp and broader distribution compared to the one of the smooth cylinder. No significant difference is observed between the wall shear stress distributions of a Janus cylinder, where super-hydrophobic surface covers only the frontal half of the cylinder, and a totally super-hydrophobic cylinder. This helps the potential manufacturers to reduce the manufacturing costs by partially processing the surface while achieving the highest possible drag reduction.

Reactive Solute Transport In An Asymmetrical Fracture-Rock Matrix System

Abstract The understanding of reactive solute transport in a single fracture–rock matrix system is the foundation of studying transport behavior in the complex fractured porous media. When transport properties are asymmetrically distributed in the adjacent rock matrixes, reactive solute transport has to be considered as a coupled three-domain problem, which is more complex than the symmetric case with identical transport properties in the adjacent rock matrixes. This study deals with the transport problem in a single fracture–rock matrix system with asymmetrical distribution of transport properties in the rock matrixes. Mathematical models are developed for such a problem under the first-type and the third-type boundary conditions to analyze the spatio–temporal concentration and mass distribution in the fracture and rock matrix with the help of Laplace transform technique and de Hoog numerical inverse Laplace algorithm. The newly acquired solutions are then tested extensively against previous analytical and numerical solutions and are proven to be robust and accurate. Furthermore, a water flushing phase is imposed on the left boundary of system after a certain time. The diffusive mass exchange along the fracture/rock matrixes interfaces and the relative masses stored in each of three domains (fracture, upper rock matrix, and lower rock matrix) after the water flushing provide great insights of transport with asymmetric distribution of transport properties. This study has the following findings: 1) Asymmetric distribution of transport properties imposes greater controls on solute transport in the rock matrixes. However, transport in the fracture is mildly influenced. 2) The mass stored in the fracture responses quickly to water flushing, while the mass stored in the rock matrix is much less sensitive to the water flushing. 3) The diffusive mass exchange during the water flushing phase has similar patterns under symmetric and asymmetric cases. 4) The characteristic distance which refers to the zero diffusion between the fracture and the rock matrix during the water flushing phase is closely associated with dispersive process in the fracture.

Reactive solute transport in an asymmetrical fracture–rock matrix system

Abstract The understanding of reactive solute transport in a single fracture–rock matrix system is the foundation of studying transport behavior in the complex fractured porous media. When transport properties are asymmetrically distributed in the adjacent rock matrixes, reactive solute transport has to be considered as a coupled three-domain problem, which is more complex than the symmetric case with identical transport properties in the adjacent rock matrixes. This study deals with the transport problem in a single fracture–rock matrix system with asymmetrical distribution of transport properties in the rock matrixes. Mathematical models are developed for such a problem under the first-type and the third-type boundary conditions to analyze the spatio–temporal concentration and mass distribution in the fracture and rock matrix with the help of Laplace transform technique and de Hoog numerical inverse Laplace algorithm. The newly acquired solutions are then tested extensively against previous analytical and numerical solutions and are proven to be robust and accurate. Furthermore, a water flushing phase is imposed on the left boundary of system after a certain time. The diffusive mass exchange along the fracture/rock matrixes interfaces and the relative masses stored in each of three domains (fracture, upper rock matrix, and lower rock matrix) after the water flushing provide great insights of transport with asymmetric distribution of transport properties. This study has the following findings: 1) Asymmetric distribution of transport properties imposes greater controls on solute transport in the rock matrixes. However, transport in the fracture is mildly influenced. 2) The mass stored in the fracture responses quickly to water flushing, while the mass stored in the rock matrix is much less sensitive to the water flushing. 3) The diffusive mass exchange during the water flushing phase has similar patterns under symmetric and asymmetric cases. 4) The characteristic distance which refers to the zero diffusion between the fracture and the rock matrix during the water flushing phase is closely associated with dispersive process in the fracture.

SPH-FDM boundary for the analysis of thermal process in homogeneous media with a discontinuous interface

A SPH-FDM boundary method is proposed for the analysis of thermal process in homogeneous media with a discontinuous interface in this study, in which the smoothed particle hydrodynamics (SPH) method is used in the inner computational domain; and the finite difference method (FDM) is used as the function approximation near the boundary. This mixed method not only can improve the calculation accuracy under the first-type boundary conditions (i.e., Dirichlet), but also can convert the second- and third-type boundary conditions (i.e., Neumann and Robin) into the first-type boundary conditions in solving heat conduction problems of homogeneous media. As a result, a second-order accuracy can be achieved in the entire solution domain. The proposed SPH-FDM boundary method is applicable to the analysis of heat conduction in various media, including the problems with discontinuous interface in the computational domain and the solidification of materials with a moving phase transition boundary. Numerical results show that the proposed SPH-FDM boundary method overcomes the difficulties of the conventional SPH method in dealing with the second- and third-type boundary conditions and has a very high calculation accuracy.

Modeling the Formation of Grain Boundaries as a Result of Two-Sided Crystallization Using Molecular Dynamics

This article considers the issues related to the correct description of the internal structure of grain boundaries in metals. We offered a mathematical model describing grain boundaries formation as a result of two-sided crystallization by applying molecular dynamics method with third-type boundary conditions for heat extraction (Newton-Richman law). In the construction of the interatomic potential, the embedded atom method (EAM) is used. The work offers an algorithm for generation of initial conditions for two adjacent grains with different crystal lattices orientation and melt between them. To detect defects and defective areas we use a central symmetry parameter. The system energy before and after the crystallization process is estimated.

Heat conduction in convectively cooled eccentric spherical annuli: A boundary integral moment method

In this paper heat conduction equation for an eccentric spherical annulus with the inner surface kept at a constant temperature and the outer surface subjected to convection is solved analytically. Eccentric problem domain is first transformed into a concentric domain via formulating the problem in bispherical co-ordinate system. Since an analytical Green?s function for the heat conduction equation in bispherical co-ordinate for an eccentric sphere subject to boundary condition of third type can not be found, an analytical Green's function obtained for Dirichlet boundary condition is employed in the solution. Utilizing this Green's function yields a boundary integral equation for the unknown normal derivative of the surface temperature distribution. The resulting boundary integral equation is solved analytically using method of moments. The method has been applied to heat generating eccentric spherical annuli and results are compared to the simulation results of FLUENT CFD code. A very good agreement was observed in temperature distribution computations for various geometrical configurations and a wide range of Biot number. Variation of heat dissipation with radii and eccentricity ratios are studied and a very good agreement with FLUENT has been observed

An exponential boundary-layer solution to the convection-dispersion equation of solute transport in soils

The dispersion coefficient (D) and retardation factor (R) are key parameters in convection–dispersion equation (CDE). The boundary-layer theory provides a simple and convenient method to solve the CDE for a third-type boundary condition under steady water flow. However, the present boundary-layer solutions cannot accurately describe the solute concentration at higher average pore-water velocities for a long-period solute transport process. In this study, an improved exponential solution to the CDE was developed using boundary-layer theory with the assumption that the resident solute concentration in soil was an exponential function related to the position under steady water flow. The accuracies of three boundary-layer solutions (parabolic polynomial, cubic polynomial, and exponential) were evaluated by comparing with the exact solution in concentration predictions for a third-type boundary condition under steady water flow. The solute concentration distributions calculated from the three boundary-layer solutions were close to those from the exact solution. At higher average pore-water velocities, the exponential solution was better than the polynomial solutions in D and R estimations. Moreover, the feasibility of the three boundary-layer solutions was verified using a column experiment. This study provides an effective way for estimating D and R in laboratory or field studies.

Critical conditions for thermal explosion in a plate with a nonlinear heat source

The exact analytical solution of the nonstationary heat conduction problem with a nonlinear internal heat source at the nonsymmetric third-type boundary conditions was obtained by the method of variable separation. The relations between the boundary condition parameters and a heat source separating the stationary processes from processes of an unlimited temperature increase (thermal explosion) were found. For known physical properties of medium, the maximum allowable value of the specific power of a heat source has been found. If nonsymmetric boundary conditions are exceeded on wall surfaces, it can lead to thermal destruction.

Existence, uniqueness and behaviour of solutions for a nonlinear diffusion equation with third type boundary value condition

In this work, we investigate a mixed problem with boundary condition of third type for a nonlinear diffusion equation having nonlocal term. Existence and uniqueness of a solution of the posed problem are proved under fairly general conditions. Moreover, we obtain some results on the behaviour of the solution and the existence of an absorbing set for the problem under consideration.