Boundary Conditions Of First Kind Research Articles

Approximate Solution of Mixed Problem for Telegrapher Equation with Homogeneous Boundary Conditions of First Kind Using Special Functions

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

A class of numerical methods for the solution of fourth-order nonlinear ordinary differential equations on a graded mesh with boundary conditions of first kind

We propose difference schemes of order 2 and 3 on a graded mesh for fourth-order nonlinear ordinary differential equations (ODEs). The values of u and (du/dx) are prescribed at two boundary points. Our methods involve only 3-grid points. A graded mesh is generated by introducing a mesh ratio parameter σ. Numerical values of (du/dx) are obtained as a by-product of the proposed methods. The stability and convergence analysis are discussed in brief. The modified version of the method works well for singular problems. We have tested our methods on seven benchmark problems to demonstrate the usefulness of the proposed methods.

On the Applicability of the Ostwald–De Waele Model in Solving Applied Problems

By comparing the results of calculation of power-law fluid flow in a slit with the Ellis fluid flow, the inaccuracy of the power-law model is shown as much in describing the velocity profile as in determining the non-Newtonian fluid flow rate. Moreover, the error of the power-law model in the description of temperature distribution in the case of dissipative heating of non-Newtonian fluid over a stabilized segment is shown. The boundary condition of first kind was used.

Heat Exchange in a Liquid with Energy Dissipation

A method of analytical solution of the problem on the heat exchange in a liquid moving in a tube with dissipation of its energy at the first-kind boundary conditions, i.e., under the conditions of heat shock at the walls of the tube, was developed on the basis of the integral method of heat balance. The approach proposed involves the introduction of an additional desired function and additional boundary conditions, which allows one to reduce the solution of the partial differential equation to the integral ordinary differential equation for the additional desired function determined by only the longitudinal space variable. In this method there is no need to perform integration of the differential equation with respect to the transverse coordinate, involving the use of the heat-balance integral, which enables it to be used for solving the problem on the heat exchange in a medium with variable physical properties as well as nonlinear boundary-value problems.

Boundary Value Problem for the Time-Fractional Telegraph Equation with Caputo Derivatives

In this paper the Green formula for the operator of fractional differentiation in Caputo sense is proved. By using this formula the integral representation of all regular in a rectangular domains solutions is obtained in the form of the Green formula for operator generating the time-fractional telegraph equation. The unique solutions of the initial-boundary value problem with boundary conditions of first kind is constructed. The proposed approach can be used to study the more general evolution FPDE as well as ODE with Caputo derivatives.

Nonstationary thermal field in the parallelepiped in the mode of heat conduction under boundary conditions of first kind

Analytical study of the processes of heat conduction is one of the main topics of modern engineering research in engineering, energy, nuclear industry, process chemical, construction, textile, food, geological and other industries. Suffice to say that almost all processes in one degree or another are related to change in the temperature condition and the transfer of warmth. It should also be noted that engineering studies of the kinetics of a range of physical and chemical processes are similar to the problems of stationary and nonstationary heat transfer. These include the processes of diffusions, sedimentation, viscous flow, slowing down the neutrons, flow of fluids through a porous medium, electric fluctuations, adsorption, drying, burning, etc. There are various methods for solving the classical boundary value problems of nonstationary heat conduction and problems of the generalized type: the method of separation of variables (Fourier method) method; the continuation method; the works solutions; the Duhamel's method; the integral transformations method; the operating method; the method of green's functions (stationary and non-stationary thermal conductivity); the reflection method (method source). In this paper, based on the consistent application of the Laplace transform on the dimensionless time θ and finite sine integral transformation in the spatial coordinates X and Y solves the problem of unsteady temperature distribution on the mechanism of heat conduction in a parallelepiped with boundary conditions of first kind. As a result we have the analytical solution of the temperature distribution in the parallelepiped to a conductive mode free convection, when one of the side faces of the parallelepiped is maintained at a constant temperature, and the others with the another same constant temperature.

Тепловой начальный участок в плоском канале при течении среды в режиме идеального вытеснения с граничными условиями первого рода

Тепловой начальный участок в плоском канале при течении среды в режиме идеального вытеснения с граничными условиями первого рода

Method of Boundary Characteristics

A new integral method based on the system of identical equalities of 2n-multiple integrals of the desired temperature function and the corresponding boundary characteristics is presented. This method was used for the approximate solution of problems of a semi-bounded space and a lengthy plate with first-kind boundary conditions. The results obtained show that the integral method of boundary characteristics surpasses the analogous integral methods by several orders of magnitude in accuracy and convergence of approximate solutions. The solutions constructed with this method are exact in essence because the errors in determining parameters in a wide range of their values comprise hundredths and ten-thousands of a percent.

Exact Solution of Heat Transfer in a Slab with Time-Dependent Boundary Conditions of First Kind

The shifting function method is utilized to find the exact solution for the one dimensional heat conduction of a slab at two surfaces with time-dependent boundary conditions of the first kind. The proposed method is shown to be simple and accurate. Two examples chosen from the literature are given to reveal the methodology. The convergence rate of the present analysis is very fast and we find that when the dimensionless Fourier number is greater than 1.0, the error for the three-term approximation solution of the infinite series solution can be less than 2%. Moreover, we use alternative shifting functions to solve the problem, but we find that more terms of the infinite series solution are required for numerical calculation.

Mathematical Simulation of the Locally Nonequilibrium Heat Transfer in a Body with Account for its Nonlocality in Space and Time

Differential equations for the locally nonequilibrium heat transfer in an infinite plate, in which the nonlocality of this plate in space and time is taken into account, have been derived. Exact analytical solutions of these equations were obtained and analyzed in detail. This analysis has shown that, in the case where the nonlocality of the plate in space and time is taken into account in the indicated equations, they do not give abrupt changes in the temperature of the plate and negative temperatures for it. It was established that, in locally nonequilibrium processes, the first-kind boundary conditions (a heat shock) cannot be realized instantaneously at any conditions of heat exchange between a body and the environment and that, consequently, in the process of this exchange, the coefficient of heat transfer cannot exceed any limiting values determined by the physical properties (including the relaxation ones) of the body.

Hybrid Analytical-Numerical Analysis of SAE 4150 Alloy Steel Rods Cooling

In this work, a hybrid analytical-numerical study was performed in cooling of rectangular rods made from SAE 4150 alloy steel (0.50% carbon, 0.85% chrome, 0.23% molybdenum, and 0.30% silicon). The analysis can be represented by the solution of transient diffusive problems in rectangular cylinders with variable thermo-physical properties in its domain under the boundary conditions of first kind (Dirichlet condition) and uniform initial condition. The diffusion equation was linearized through the Kirchhoff Transformation on the temperature potential to make the analytical treatment easier. The Generalized Integral Transform Technique (GITT) was applied on the diffusion equation in the domain in order to determine the temperature distribution. The physical parameters of interest were determined for several aspect ratios and compared with the results obtained through numerical simulations using the commercial software ANSYS/FluentTM15.

\Transient Heat Diffusion Problems with Variable Thermal Properties Solved by Generalized Integral Transform Technique

In this work, studies of two-dimensional transient heat diffusion problems in domains of rectangular and elliptical geometries, submitted to boundary conditions of first kind, were carried out. For the problem formulation, diffusive means with variable thermophysical properties were considered. The differential equation that governs the energy conservation is non-linear. In this context, the diffusion equation was linearized by the Kirchhoff Transformation. Transformations of the coordinate systems were performed in order to facilitate the boundary conditions application. The resulting differential equation after these transformations does not allow the application of the variable separation techniques. Thus, the Generalized Integral Transform Technique (GITT) was used to solve the energy equation. As result of this transformation a coupled ordinary differential equation system that can be solved through classic numerical methods was obtained. Therefore, for the determination of the temperature field evolution, the inversion formulas of all the performed transformations were used. Physical parameters of interest were, then, calculated and compared for several cylindrical cross section geometries.

Plasmastatic models of galathea traps with magnetically transparent boundaries

Mathematical models and results of calculation of plasma equilibrium in a circular cylinder with three helical or straight imbedded current-carrying conductors (i.e., in a straightened analog of a toroidal Galathea trap) are presented. The equilibrium is described in the framework of two-dimensional boundary value problems with plane and helical analogs of the Grad-Shafranov equation for the scalar magnetic flux function. Problems with first-kind boundary conditions corresponding to a magnetically transparent boundary of the cylinder and problems with second-kind boundary conditions and a given value of the electric current flowing in plasma (in addition to those flowing in the conductors) are considered. Deformations of magnetoplasma configurations in the cylinder for different formulations of the above-specified problems are investigated numerically.

Calculation of the efficiency of circular porous channels by the modified Guhman method

Results of calculation of the efficiency of circular porous channels in the transient region 2300≤ Resm ≤ 10,000 of movement of an incompressible coolant (water) in smooth-wall channels compared with the porouschannels at the first-kind boundary conditions are presented. The efficiencies of circular porous channels and smooth-wall ones were compared using the modified Guhman method, in accordance with which the efficiency of a short porous channel, in which a coolant is rapidly heated, is determined by comparison of this channel with a long smooth-wall channel. As a criterion for selection of the length of a circular porous channel, the attainment of the value 0.8 by the average dimensionless temperature of the liquid at the output of this channel was used. The influence of different design and regime parameters of the channels being compared on their efficiencies was investigated and the parameters, at which the gain in heat transfer superpasses the loss in hydraulics, were determined.

The local discontinuous Galerkin finite element method for a class of convection–diffusion equations

In this paper, we study the local discontinuous Galerkin (LDG) finite element method for solving a class of convection–diffusion equations with the first-kind boundary conditions. Based on the Hopf–Cole transformation, we transform the original equation into a linear heat equation with the same kind boundary conditions. Then the heat equation is solved by the LDG finite element method with a suitably chosen numerical flux. Theoretical analysis shows that this method is stable and has a (k+1)-th order of convergence rate when the polynomials Pk are used. Finally, numerical experiments for one-dimensional and two-dimensional convection–diffusion equations are given to confirm the theoretical results.

Entropy generation due to mixed convection between vertical parallel plates under isothermal boundary conditions

This paper presents the results of an investigation of the buoyancy effects on the entropy generation due to laminar mixed convection in vertical channels between parallel plates. The buoyancy effects, in terms of the buoyancy parameter (Gr/Re) as well as the effects of other operating parameters on the entropy generation are obtained and analysed. The analysis revealed the existence of optimum values of the buoyancy parameter Gr/Re at which the total entropy generation assumes its minimum values. These values are obtained analytically for isothermal boundary conditions of first kind at different values of the two plate's temperature ratio, θT.

Two-dimensional non-Fourier heat conduction with arbitrary initial and periodic boundary conditions

In this paper, the (2+1)-dimensional hyperbolic heat conduction equation is analytically solved under the influence of arbitrary initial conditions for a rectangular plate with homogeneous boundary conditions of first-kind. The temperature field is obtained as a double Fourier series. The presented solution is valid even for discontinuous but integrable initial conditions. Afterwards, the solution is generalized by means of a transformation to cover problems with inhomogeneous first-kind boundary conditions. Another interesting issue is that the obtained solution can be considered as a solution to the Klein–Gordon equation under the influence of arbitrary initial conditions by means of a simple transformation.

Heat transfer in laminar flow of non-Newtonian fluids in ducts of elliptical section

Laminar forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumption used in this work is a laminar flow of a power flow inside elliptical tube, under a boundary condition of first kind with constant physical properties and negligible axial heat diffusion (high Peclet number). To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm–Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number and the average Nusselt number for various cross-section aspect ratios.

Conditions for pressure build-up due to buoyancy effects on forced convection in vertical eccentric annuli under thermal boundary condition of first kind

This paper presents a numerical investigation for the conditions at which the buoyancy effects (represented by the buoyancy parameter (Gr/Re)) result in pressure build-up due to mixed convection in vertical eccentric annuli under thermal boundary conditions of first kind. In this regard, the critical values of buoyancy parameter (Gr/Re)crt at which the pressure gradient vanishes and starts to become positive leading to the pressure build-up are obtained numerically for radius ratio N=0.5 and eccentricity E=0.1–0.7. Results of practical applications such as the locations at which the negative pressure gradient becomes zero changing its sign to be positive, the locations of zero pressure defect and the fully developed length under different operating conditions are drawn and presented. For sufficiently large values of Gr/Re≫(Gr/Re)crt, possibilities and locations of flow reversal incipient are determined. Information of technical relevance is presented.

INTEGRAL TRANSFORM METHOD FOR LAMINAR HEAT TRANSFER CONVECTION OF HERSCHEL-BULKLEY FLUIDS WITHIN CONCENTRIC ANNULAR DUCTS

Related momentum and energy equations describing the heat and fluid flow of Herschel-Bulkley fluids within concentric annular ducts are analytically solved using the classical integral transform technique, which permits accurate determination of parameters of practical interest in engineering such as friction factors and Nusselt numbers for the duct length. In analyzing the problem, thermally developing flow is assumed and the duct walls are subjected to boundary conditions of first kind. Results are computed for the velocity and temperature fields as well as for the parameters cited above with different power-law indices, yield numbers and aspect ratios. Comparisons are also made with previous work available in the literature, providing direct validation of the results and showing that they are consistent.