142 Chapter 2 The Heat Equation
EXERCISES
1.Give a physical interpretation for the problem in Eqs. (13)–(16).
2.Verify that the following functions are solutions of the heat equation (5):
u(x,t)=exp(
−λ^2 kt)
cos(λx),
u(x,t)=exp(
−λ^2 kt)
sin(λx).3.Suppose that the rod exchanges heat through the cylindrical surface by con-
vection with a surrounding fluid at temperatureU(constant). Newton’s law
of cooling says that the rate of heat transfer is proportional to exposed area
and temperature difference. What isgin Eq. (1)? What form does Eq. (4)
take?
4.Suppose that the end of the rod atx=0 is immersed in an insulated con-
tainer of water or other fluid; that the temperature of the fluid is the same
as the temperature of the end of the rod; that the heat capacity of the fluid
isCunits of heat per degree. Show that this situation is represented math-
ematically by the equationC∂u
∂t(^0 ,t)=κA∂u
∂x(^0 ,t),
whereAis the cross-sectional area of the rod.
5.Put Eq. (10) into Eq. (8) form. Notice that the signs still indicate that heat
flows in the direction of lower temperature. That is, ifu(a,t)>T(t),then
q(a,t)is positive and the gradient ofuis negative. Show that, if the surface
atx=0 (left end) is exposed to convection, the boundary condition would
read
κ∂∂ux( 0 ,t)=hu( 0 ,t)−hT(t).
Explain the signs.
6.Suppose the surface atx=ais exposed to radiation. The Stefan–Boltzmann
law of radiation says that the rate of radiation heat transfer is proportional
to the difference of the fourth powers of theabsolutetemperatures of the
bodies:
q(a,t)=σ(
u^4 (a,t)−T^4)
.
Use this equation and Fourier’s law to obtain a boundary condition for
radiation atx=ato a body at temperatureT.
7.The difference cited in Exercise 6 may be written
u^4 −T^4 =(u−T)(
u^3 +u^2 T+uT^2 +T^3