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.
Permanent Citation