The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximat...The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.展开更多
It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. T...It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.展开更多
文摘The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.
文摘It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.
