Analytical solution to the unsteady one-dimensional conduction problem with two time-varying boundary conditions: Duhamel’s theorem and separation of variables

2010 ◽  
Vol 46 (7) ◽  
pp. 707-716 ◽  
Author(s):  
Dominic Groulx
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Separation Of Variables ◽  
Time Varying ◽  
One Dimensional

Discussion of Boundary Conditions of Transpiration Cooling Problems Using Analytical Solution of LTNE Model

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Jianhua Wang ◽  
Junxiang Shi
Keyword(s):  
Temperature Field ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Coolant Temperature ◽  
Transpiration Cooling ◽  
Dimensional Problem ◽  
Gas Temperature ◽  
One Dimensional ◽  
Hot Gas ◽  
Hot Surface

To compare five kinds of different boundary conditions (BCs), an analytical solution of a steady and one-dimensional problem of transpiration cooling described by a local thermal nonequilibrium (LTNE) model is presented in this work. The influence of the five BCs on temperature field and thermal effectiveness is discussed using the analytical solution. Two physical criteria, if the analytical solution of coolant temperature may be higher than hot gas temperature at steady state and if the variation trend of thermal effectiveness with coolant mass flow rate at hot surface is reasonable, are used to estimate the five BCs. Through the discussions, it is confirmed which BCs at all conditions are usable, which BCs under certain conditions are usable, and which BCs are thoroughly unreasonable.

scholarly journals ANALYTICAL SOLUTION OF A 2D TRANSIENT HEAT CONDUCTION PROBLEM USING GREEN´S FUNCTIONS

2020 ◽  
Vol 19 (1) ◽  
pp. 66
Author(s):  
J. R. F. Oliveira ◽  
J. A. dos Santos Jr. ◽  
J. G. do Nascimento ◽  
S. S. Ribeiro ◽  
G. C. Oliveira ◽  
...  
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Heat Conduction Problem ◽  
Transient Heat Conduction ◽  
Heat Fluxes ◽  
Two Dimensional ◽  
One Dimensional ◽  
Transient Heat Conduction Problem

Through the present work the authors determined the analytical solution of a transient two-dimensional heat conduction problem using Green’s Functions (GF). This method is very useful for solving cases where heat conduction is transient and whose boundary conditions vary with time. Boundary conditions of the problem in question, with rectangular geometry, are of the prescribed temperature type - prescribed flow in the direction x and prescribed flow - prescribed flow in the direction y, implying in the corresponding GF given by GX21Y22. The initial temperature of the space domain is assumed to be different from the prescribed temperature occurring at one of the boundaries along x. The temperature field solution of the two-dimensional problem was determined. The intrinsic verification of this solution was made by comparing the solution of a 1D problem. This was to consider the incident heat fluxes at y = 0 and y = 2b tending to zero, thus making the problem one-dimensional, with corresponding GF given by GX21. When comparing the results obtained in both cases, for a time of t = 1 s, it was seen that the temperature field of both was very similar, which validates the solution obtained for the 2D problem.

scholarly journals Analytical Solution for Electrical Problem Forced by a Finite-Length Needle Electrode: Implications in Electrostimulation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Ricardo Romero-Méndez ◽  
Francisco G. Pérez-Gutiérrez ◽  
Francisco Oviedo-Tolentino ◽  
Enrique Berjano
Keyword(s):  
Analytical Solution ◽  
Electric Field ◽  
Boundary Conditions ◽  
Separation Of Variables ◽  
Needle Electrode ◽  
Analytical Models ◽  
Infinite Length ◽  
Principle Of Superposition ◽  
Clinical Procedures ◽  
Sturm Liouville

Needle electrodes, widely used in clinical procedures, are responsible for creating an electric field in the treated biological tissue. This is achieved by setting a constant voltage along the length of their metallic section. In accordance with Laplace’s equation, the electric field is spatially non-uniform around the electrode surface. Mathematical modelling can provide useful information on the spatial distribution of electrical fields. Indeed, exact solutions for the electrical problem are indispensable for validating numerical codes. All the analytical models developed to date to solve the needle electrode electrical problem have been one-dimensional models, which assumed an electrode of infinite length. We here propose the first analytical solution based on a two-dimensional model that considers the real length of the electrode in which the Laplace equation is solved through the method of separation of variables, dealing with the nonhomogeneous source term and boundary conditions by Green’s functions. On assuming a needle electrode of given length, the problem combines boundary conditions on the electrode boundary (of the first and second kind). Since this rules out using the Sturm-Liouville Theorem, the problem is decomposed into two different problems and the principle of superposition is used. The solution obtained can reproduce a reasonable electric field around the electrode, especially the edge effect characterized by an extremely high gradient around the electrode tip.

scholarly journals Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions

2020 ◽  
Vol 1551 ◽  
pp. 012002
Author(s):  
Norazlina Subani ◽  
Faizzuddin Jamaluddin ◽  
Muhammad Arif Hannan Mohamed ◽  
Ahmad Danial Hidayatullah Badrolhisam
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
One Dimensional

scholarly journals Analytical Solution for Free Vibrations of a Moderately Thick Rectangular Plate

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ivo Senjanović ◽  
Marko Tomić ◽  
Nikola Vladimir ◽  
Dae Seung Cho
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Thick Plate ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Mixed Boundary Conditions ◽  
Free Vibrations ◽  
Force Displacement ◽  
Good Agreement

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.

Analytical solutions of some internal boundary value problems of elasticity for domains with hyperbolic boundaries

2018 ◽  
Vol 24 (6) ◽  
pp. 1726-1748 ◽  
Author(s):  
Natela Zirakashvili
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Strain State ◽  
Separation Of Variables ◽  
Internal Boundary ◽  
Test Problems ◽  
Two Dimensional ◽  
Elliptic Coordinates ◽  
Hyperbolic Boundary

An analytical solution of two-dimensional problems of elasticity in the region bounded by a hyperbola in elliptic coordinates is constructed using the method of separation of variables. The stress–strain state of a homogenous isotropic hyperbolic body and that with a hyperbolic cut is studied when there are non-homogenous (non-zero) boundary conditions given on the hyperbolic boundary. The graphs for the numerical results of some test problems are presented.

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