300 Chapter 5 Higher Dimensions and Other Coordinates
Because the subregionVwas arbitrary, we conclude that the integrand must
be 0 at every point:
−∇ ·q+g−ρc∂u
∂t=0inR,^0 <t. (4)The argument goes this way. If the integrand were not identically 0, we could
find some subregion ofRthroughout which it is positive (or negative). The
integral over that subregion then would be positive (or negative), contradict-
ing Eq. (3), which holds for any subregion.
The vector form of Fourier’s law of heat conduction says that the heat flow
rate in an isotropic solid (same properties in all directions) is negatively pro-
portional to the temperature gradient,
q=−κ∇u. (5)Again, the minus sign makes the heat flow “downhill” — from hotter to colder
regions. Assuming that the conductivityκis constant, we find, on substituting
Fourier’s law into Eq. (4), the three-dimensional heat equation,
κ∇^2 u+g=ρc∂∂ut inR, 0 <t. (6)Of course, we must add an initial condition of the form
u(P, 0 )=f(P) forPinR. (7)Inaddition,ateverypointofthesurfaceBbounding the regionR,some
boundary condition must be specified. Commonly we have conditions such
as those that follow, any one of which may be given onBor some portion of
it,B′.
(1) Temperature specified,u(P,t)=h 1 (P,t), forPany point inB′,where
h 1 is a given function.
(2) Heat flow rate specified. The outward heat flow rate through a small
portion of surface surrounding pointPonB′isq(P,t)·nˆtimes the area. If this
is controlled, then by Fourier’s law∇u·nˆis controlled. But this dot product
is just the directional derivative ofuin the outward normal direction at the
pointP. Thus, this type of boundary condition takes the form
∂u
∂n(P,t)=h 2 (P,t) forPonB′, (8)whereh 2 is a given function.
(3)Convection.Ifapartofthesurfaceisexposedtoafluidattemperature
T(P,t), then an accounting of energy passing through a small piece of surface
centered atPleads to the equation
q(P,t)·nˆ=h(
u(P,t)−T(P,t))
forPonB′.