This project focuses on the numerical simulation of the Maxwell-Cattaneo’s heat transfer equation with Matlab.
The numerical resolutions are made thanks to the Matlab software. We first start by numerical experiments on the classical Fourier’s then we consider the Maxwell-Cattaneo’s. Several examples and numerical results are given; the last numerical simulations include the velocity of the fluid in the model equations.
The aim of this paper is to present some numerical results for the Maxwell-Cattaneo system.
We first consider the classical heat equation for a linear Fourier law which proposes an interpretation of the heat conduction with infinite propagation. This paradox of instantaneous propagation can be eliminated by changing its parabolic nature into hyperbolic one. Below the Fourier’s law takes the form: q = -grad(T)
where K>0 is the conduction parameter. This paradox has been analyzed by several researchers and lead to the Maxwell-Cattaneo’s law which we will focus on. This equation changes the nature of the heat equation.
Fourier’s law involves a parabolic equation for the temperature field. In Fourier’s law, any initial disturbance in a material body is propagates instantly. To correct this unrealistic feature, the Maxwell-Cattaneo’s law is one of the various modifications of Fourier’s law. Below the Maxwell-Cattaneo’s law writes as:
With denotes the thermal relaxation which must be very small (the order of picoseconds for most metals). In this equation, is called the thermal inertia and avoids the phenomenon of infinite propagation. For our numerical resolution, we will use a liquid material as for gaseous material and for most solids the solutions seems to be unrealistic.