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quite sophisticated modeling to determine its value. When applying (7.28) in one-
dimensional problems, one must remember what @T =@n is, i.e., @T =@x or
@T =@x,
as explained above.
Some names are commonly associated with the three boundary conditions men-
tioned here:
Prescribed T is called a
Dirichlet
8
-
condition
Prescribed heat flow,
k
@
@n
, is called a
Neumann
9
condition
-
The cooling law (7.28) is called a
Robin
10
condition
-
Energy transfer by electromagnetic radiation can also heat bodies. Mathematically,
this is often modeled by a nonlinear boundary condition containing the temperature
to the fourth or fifth power. This subject is not discussed further here.
At the initial time, t
D
0, the complete temperature field must be prescribed:
T.x;0/
D
I.x/,whereI.x/ is a specified function.
7.3.3
Viscous Fluid Flow
Fluid flows are frequently dominated by convection, but one can occasionally
encounter flows displaying solely diffusive effects. This happens when the geometry
of the fluid has certain symmetry properties. The following example or derivation
will give an outline of what we mean.
We assume that the fluid is incompressible and homogeneous, which are relevant
assumptions for a range of fluids, including water, air (at flow speeds less than about
350 km/h), and oil. The flow of a homogeneous, incompressible fluid is governed by
the Navier
11
-Stokes
12
equations. These can be written in the form
r
v
D
0;
(7.29)
%
@
v
@t
C
.
v
r
/
v
Dr
p
C
r
2
v
C
%
g
:
(7.30)
8
Johann Peter Gustav Lejeune Dirichlet, 1805-1859, was a German mathematician. His contri-
butions to mathematics are remarkable, including work on Fourier series and the mathematical
theory of heat. On Gauss's death in 1855, Dirichlet was appointed to fill Gauss's vacant chair at
the University of Gottingen.
9
Carl Gottfried Neumann, 1832-1925, was a German mathematician known to be one of the ini-
tiators of the theory of integral equations, and is famous for the very important boundary condition
that bears his name.
10
Victor Gustave Robin, 1855-1897, was a French mathematician who worked in particular in the
field of thermodynamics. Today he is mostly known for a widely used boundary condition in heat
transfer problems.
11
Claude-Louis Navier, 1785-1836, was a French engineer and physicist who made important
contributions to mechanics, in particular by formulating the PDE¡s that govern fluids and solids.
12
Sir George Gabriel Stokes, 1819-1903, was a British mathematician and physicist who
performed important research in the fields of fluid dynamics, optics, and mathematical physics.
