scholarly journals Integral formulation of solutions of boundary-value problems of heat and mass transfer in domains with moving boundaries

2021 ◽  
Vol 29 (1) ◽  
pp. 73-91
Author(s):  
Valentin V. Shevelev
Keyword(s):  
Heat Conduction ◽  
Boundary Value Problems ◽  
Moving Boundary ◽  
Boundary Value ◽  
Phase Region ◽  
Integral Representations ◽  
Two Phase ◽  
Self Similar ◽  
The Fourier Transform ◽  
And Diffusion

Using the Fourier transform, integral representations of solutions to boundary value problems of heat conduction and diffusion in a two-phase region with a moving interface are obtained. The proposed approach makes it possible to obtain the equation of motion of the interface without the need to first find the temperature and (or) concentration fields. This makes it possible to study the stability of the interface with respect to disturbances in its shape. The validity of the proposed approach is demonstrated by the example of self-similar growth of a spherical crystal in a supercooled melt and crystallization of the melt on a substrate of the same substance. On the basis of the obtained equation, which determines the rate of self-similar motion of the interface, the features of the kinetics of crystallization of the melt on the substrate are analyzed. The conditions of applicability of the developed approach to the solution of boundary value problems of heat conduction and diffusion in regions separated by a moving boundary are briefly discussed.

scholarly journals A new transform method II: the global relation and boundary-value problems in polar coordinates

2010 ◽  
Vol 466 (2120) ◽  
pp. 2283-2307 ◽  
Author(s):  
E. A. Spence ◽  
A. S. Fokas
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Representations ◽  
New Method ◽  
Polar Coordinates ◽  
Nonlinear Pdes ◽  
Boundary Values ◽  
Transform Method ◽  
Global Relation ◽  
The Fourier Transform

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In a previous paper, a simplified way of obtaining these representations was presented. In the current paper, first, the second ingredient of the new method, namely the derivation of the so-called ‘global relation’ (GR)—an equation involving transforms of the boundary values—is presented. Then, using the GR as well as the IR derived in the previous paper, certain BVPs in polar coordinates are solved. These BVPs elucidate the fact that this method has substantial advantages over the classical transform method.

scholarly journals An axially symmetric forced convection problem

1975 ◽  
Vol 20 (1) ◽  
pp. 1-17
Author(s):  
J. A. Belward
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Convection Heat Transfer ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Original Equation ◽  
Transfer Model ◽  
Axially Symmetric ◽  
Simple Diffusion ◽  
Diffusion Convection

AbstractA simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

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