Solution of the Heat Equation for a Couple in Bed with a Cat

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This Demonstration solves the heat equation for two people and a cat lying in bed. The model assumes that the couple are linear heat sources, that the cat is a point source, and that both are constant and of the same magnitude (even though cats are slightly warmer than humans). The bed has no heat flow at the borders (it is zero at all times) and extends over a square area of 2m by 2m. The people are placed at positions and , with a height of 1.7 m lying between and , and the cat is placed between the feet of the people. The plots show the temperature distribution as a function of time. Only terms in the solution are considered, with a spatial discretization of width .

Contributed by: Enrique Zeleny (July 2014)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The heat equation in this case can be written as

,

with thermal diffusivity . The functions are of the form

and

,

where is the width and is a constant related to body heat, . Then the solution is

,

where and are the dimensions of the bed and the cat term is

.

Reference

[1] "El Problema de Dos Cuerpos… Bajo la Frazada," Saltos Convergentes (blog). (Dec 25, 2013) saltosconvergentes.wordpress.com/2013/12/25/el-problema-de-dos-cuerpos-bajo-la-frazada.



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