This preview shows pages 1–3. Sign up to view the full content.
Solution Procedure for the SOV Method
1.
Assume
∏
=
i
i
x
f
F
)
(
)
(
x
Where:
F is the dependent variable
x
= [x
1
x
2
x
3
…. x
n
] is a vector of independent variables
For example:
)
(
)
(
)
,
(
Fo
h
x
g
Fo
x
=
θ
2.
Substitute product into differential equation
For example:
Fo
x
∂
∂
=
∂
∂
2
2
h
g
hg
′
=
'
'
where g(x) only and h(Fo) only
3.
Group terms containing the same independent variables by dividing by
gh
For example:
h
h
g
g
gh
h
g
gh
hg
′
=
′
=
'
'
'
'
4.
For the above to be true, each part is equal to a constant; therefore, introduce
, which will become a set of eigenvalues, to satisfy the differential equation.
2
ζ
For example:
)
(
2
'
'
±
=
g
g
)
(
2
±
=
′
h
h
5.
Set up the corresponding ordinary differential equation (ODE).
For example:
0
)
(
2
'
'
=
±
−
g
g
0
)
(
2
=
±
−
′
h
h
6.
Choose the sign on the
term such that a set of periodic orthogonal functions
exists for the variable that has both homogeneous B.C.’s.
2
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document For example:
1
)
0
,
(
)
,
1
(
0
,
1
,
0
=
−
=
∂
∂
=
∂
∂
=
=
x
Fo
Bi
x
x
Fo
x
Fo
x
θ
where x must be periodic
This transforms to
:
1
)
0
(
)
1
(
)
1
(
0
)
0
(
=
−
=
′
=
′
h
g
Bi
g
g
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 07/25/2008 for the course ME 410 taught by Professor Benard during the Spring '08 term at Michigan State University.
 Spring '08
 BENARD
 Heat Transfer

Click to edit the document details