Environmental Engineering Reference
In-Depth Information
Let
M
and
S
be the point and the boundary of one-, two- or three-dimensional
domain. When the dependent variable
u
is given on the
S
, the boundary condition
reads
u
(
M
,
t
)
|
S
=
ϕ
(
M
,
t
)
or
u
(
M
)
|
S
=
ϕ
(
M
)
,
(1.79)
where
are known functions. Such boundary conditions are called
the
boundary conditions of the first kind
.
ϕ
(
M
,
t
)
and
ϕ
(
M
)
Boundary Conditions of the Second Kind
In many applications, boundary conditions are not given by values of the depen-
dent variable on the boundary, but by its directional derivative along the boundary
normal, the outward-drawn normal in particular. For one-dimensional cases, we can
express a directional derivative by a partial derivative.
Consider the vibration of a string of length
l
. If the string can slide freely along
the
y
-direction at
x
=
0 without any force, a force balance at
x
=
0 yields
x
=
0
=
T
∂
u
0or
u
x
(
0
,
t
)=
0
.
∂
x
This is the boundary condition at
x
=
0. If the string slides along the
y
-direction at
x
=
l
with the action of a force
f
(
t
)
, the boundary condition at
x
=
l
is
x
=
l
=
T
∂
u
f
(
t
)
or
u
x
(
l
,
t
)=
f
(
t
)
/
T
,
∂
x
where
T
is the tension of the vibrating string.
Consider heat conduction in a body. If the heat flux density on the boundary
S
is given, the normal derivative of temperature on
S
is known by the Fourier law of
heat-conduction, say
ϕ
(
M
,
t
)
. The boundary condition is thus
S
=
ϕ
(
∂
u
M
,
t
)
.
∂
n
If the boundary is well insulated such that the heat flux vanishes on the boundary,
the boundary condition reduces to
S
=
∂
u
0
.
∂
n
If the two ends are well insulated for heat conduction in a solid rod of length
l
,the
boundary conditions are
(
,
)=
(
,
)=
.
u
x
0
t
u
x
l
t
0
Search WWH ::
Custom Search