Differentiation and Integration of Fourier Series
4.7 METHOD FOR PROBLEMS IN
CYLINDRICAL COORDINATES
For the most general type of problem in cylindrical coordinates. = When there is no volumetric source term, the dimensionless energy or species
equation is
We will be concerned only with steady two-dimensional and unsteady one-dimensional problems involving z), or t). For the eigenvalue problems in
have the same differential operator as in rectangular coordinates, and they will be identical to those considered in Section 4.3. For the eigenvalue problems in
again have the same differential operator; the only distinctive feature of these
problems has to do with periodic boundary conditions, as discussed at the end of this section. However, with or t), the eigenvalue problems in involve a differential equation not yet considered, Bessel's equation. Because they are
the most unique aspect of cylindrical problems, the solutions of this equation are dis- cussed first.
,
Bessel Functions
Eigenvalue problems in result in
is a equation with = and = [compare with
For problems involving annular regions, such that a b with a a eigenvalue problem results when any of the three types of boundary conditions from (4.3-5) are applied at a and For domains containing the origin, all that is required at r=O is that and be finite, as discussed in Section 4.6.
Equation (4.7-2) is a particular form of Bessel's equation, which is written more generally as
with boundary conditions as in are self-adjoint with respect to the weighting
function problems of this form are called problems
[see, for example, (1963) and Greenberg
The conditions given in (4.3-5) are not the only ones which yield self-adjoint eigenvalue problems. Also leading to such problems is the periodicity re- quirement,
Inspection of (4.6-7) indicates that if u and v satisfy Eq. (4.6-lo), then the boundary terms will vanish in the integration by parts, as required for a self-adjoint problem. Equation (4.6- 10) arises, for example, with problems in cylindrical coordinates involving the angle in which the domain is a complete circle. One other way a self-adjoint eigenvalue problem is obtained is if vanishes at one or both ends of the interval. If, for example, and and are both finite, then the corresponding bound-
ary term in (4.6-7) will again vanish. This type of condition arises in certain prob- lems in cylindrical or spherical coordinates (see Sections 4.7 and 4.8).
All Sturm-Liouville problems yield eigenvalues which are real and eigenfunctions which are orthogonal. A proof that A is real is given, for example, in Churchill (1963). The orthogonality of the eigenfunctions is a direct consequence of the fact that the Sturm-Liouville operator is as is shown now. Considering two different eigenfunctions and self-adjointness implies that
From the definition of the eigenvalue problem and the properties of the inner product (Section 4.3) we obtain
Subtracting (4.6-13) from leads to
Because A, this implies that
In other words, the solutions to problems are
In Sturm-Liouville problems with boundary conditions from or with at one of the endpoints, the eigenfunctions are unique. That is, there is only one linearly independent corresponding to a given eigenvalue. A proof of this, with a discussion of the exception which occurs with periodic boundary condi- tions, given by Churchill (1963).
applying the method, the most awkward term to in the
equation is that involving the second-order operator in the basis function ,
general (1803-1855) and J. Liouville ( 1 French
folowing Fourier.
coordinate. Taking advantage of the integration by parts performed for the
operator in (4.6-7), the of that is written generally as
The eigenvalue problems introduced in Section 4.3 and applied in Section 4.5 correspond to w = p = and q = The operators and eigenfunctions in some of the more problems in cylindrical and spherical coordinates are discussed in Sec- tions 4.7 and 4.8. Eigenvalue problems not encompassed by classical
theory, such as those where w and p are discontinuous, are discussed in and
Amundson 1985) and et al. (1979). A few such situations are examined in the problems at the end of this chapter.
4.7
METHOD FOR PROBLEMS IN
CYLINDRICAL COORDINATES
For the most general type of problem in cylindrical coordinates. = When there is no volumetric source term, the dimensionless energy or species
equation is
We will be concerned only with steady two-dimensional and unsteady one-dimensional problems involving z), or t). For the eigenvalue problems in
have the same differential operator as in rectangular coordinates, and they will be identical to those considered in Section 4.3. For the eigenvalue problems in
again have the same differential operator; the only distinctive feature of these problems has to do with periodic boundary conditions, as discussed at the end of this section. However, with or t), the eigenvalue problems in involve a differential equation not yet considered, Bessel's equation. Because they are
the most unique aspect of cylindrical problems, the solutions of this equation are dis- cussed first.
,
Bessel Functions
Eigenvalue problems in result in
is a equation with = and = [compare with
For problems involving annular regions, such that a b with a a eigenvalue problem results when any of the three types of boundary conditions from (4.3-5) are applied at a and For domains containing the origin, all that is required at r=O is that and be finite, as discussed in Section 4.6.
Equation (4.7-2) is a particular form of Bessel's equation, which is written more generally as
where is a and any real constant. In (4.7-2), m = A and The properties of the solutions to Bessel's equation have been studied extensively (Watson, 1944). The two linearly independent solutions are usually written as and
and are known as of order of the first and second kind,
Values of Bessel functions of integer order are available from numerous sources, includ- ing software written for personal computers, making calculations involving these func- tions quite routine. The solutions to are the Bessel functions of order zero, and The first derivatives and the integrals of and can both be expressed in terms of the corresponding Bessel functions of order one, and
so that we need be concerned here only the properties of J,, and Graphs of these functions are shown in Fig. 4-12. As the plots suggest, of the func- tions have an infinite number of roots. Two values worth noting are 1 and
An important distinction between Bessel functions of the first and second kinds is that, whereas and are finite, and are not.
Of the many identities involving Bessel functions, ones which are particularly helpful to us in evaluating derivatives and integrals are
Consider an eigenvalue problem involving (4.7-2) on the interval r 1, in which one boundary condition is The fact that is unbounded requires that the
be excluded, so that
X
4-12. functions of orders zero and one.
where a is a constant. The boundary condition at = yields the characteristic equation,
(4.7-7)
That is, the eigenvalues are the roots of The normalization condition for the functions is
Notice the inclusion of the weighting function = in the inner product.
To determine a,, we need to evaluate the integral in Eq. (4.7-8). Integrals like this occur in any eigenvalue problem involving so that it is worthwhile to derive a gen- eral result for the interval valid for any of the three types of boundary conditions at = This is done by first recalling that satisfies Bessel's equation with m = A, and SO that
The next step is to multiply both sides of Eq. (4.7-9) by The left-hand side gives
Integrating (4.7-10) from = to and using (4.7-4a) to evaluate the tive of we obtain
The modified right-hand side of Eq. (4.7-9) is rearranged as
Integrating by parts gives
.
Equating the results from (4.7-1 1) and (4.7-13) leads to the desired identity, which is
Either form of Eq. (4.7-14) may be more convenient, depending on the boundary condi- tion at =
A second integral which may as well be evaluated now is one which arises when representing any constant as a Fourier-Bessel series involving The integral is
where is a and any real constant. In (4.7-2), m = A and The properties of the solutions to Bessel's equation have been studied extensively (Watson, 1944). The two linearly independent solutions are usually written as and
and are known as of order of the first and second kind,
Values of Bessel functions of integer order are available from numerous sources, includ- ing software written for personal computers, making calculations involving these func- tions quite routine. The solutions to are the Bessel functions of order zero, and The first derivatives and the integrals of and can both be expressed in terms of the corresponding Bessel functions of order one, and
so that we need be concerned here only the properties of J,, and Graphs of these functions are shown in Fig. 4-12. As the plots suggest, of the func- tions have an infinite number of roots. Two values worth noting are 1 and
An important distinction between Bessel functions of the first and second kinds is that, whereas and are finite, and are not.
Of the many identities involving Bessel functions, ones which are particularly helpful to us in evaluating derivatives and integrals are
Consider an eigenvalue problem involving (4.7-2) on the interval r 1, in which one boundary condition is The fact that is unbounded requires that the
be excluded, so that
X
4-12. functions of orders zero and one.
where a is a constant. The boundary condition at = yields the characteristic equation,
(4.7-7)
That is, the eigenvalues are the roots of The normalization condition for the functions is
Notice the inclusion of the weighting function = in the inner product.
To determine a,, we need to evaluate the integral in Eq. (4.7-8). Integrals like this occur in any eigenvalue problem involving so that it is worthwhile to derive a gen- eral result for the interval valid for any of the three types of boundary conditions at = This is done by first recalling that satisfies Bessel's equation with m = A, and SO that
The next step is to multiply both sides of Eq. (4.7-9) by The left-hand side gives
Integrating (4.7-10) from = to and using (4.7-4a) to evaluate the tive of we obtain
The modified right-hand side of Eq. (4.7-9) is rearranged as
Integrating by parts gives
.
Equating the results from (4.7-1 1) and (4.7-13) leads to the desired identity, which is
Either form of Eq. (4.7-14) may be more convenient, depending on the boundary condi- tion at =
A second integral which may as well be evaluated now is one which arises when representing any constant as a Fourier-Bessel series involving The integral is
4-2
Sequences of Functions from Certain Eigenvalue Problems in Cylindrical Coordinatesa
Case Boundary condition Basis functions
of the functions shown satisfy (4.7-2) on the [0, few for cases and as follows:
Equation makes use of the identity given as (4.7-4b).
Returning now to the eigenvalue problem for a full cylinder with a Dirichlet boundary condition at = we find from (4.7-6), and (4.7- 14) that
A very similar procedure is used to derive the orthonormal basis functions corresponding to Neumann or Robin conditions. Table 4-2 summarizes the three types of basis func-
tions arising in the analysis of conduction or diffusion inside a complete cylinder of radius For problems involving annular regions, where is not in the domain and the solution cannot be excluded, the basis functions will be linear combinations
and
4.7-1 Conduction in an Electrically Heated Wire Consider a long, cylin- drical wire, initially at ambient temperature, which is subjected to a constant rate of heating for due to passage of an electric current. Heat from the to the surroundings is described using a convection boundary condition, and the Biot number is not necessarily large small. This is a transient problem leading to the steady state analyzed in Example 2.8-1.
the dimensional quantities used in that example, we choose and R as the temperature
and length scales, respectively. Then, the dimensionless problem for is
The only eigenvalue problem derivable from (4.7-17) is one involving r. The domain includes
r = so that the basis functions contain only From case of Table 4-2, the basis functions suitable for the Robin boundary condition in (4.7-19) are
with eigenvalues given by the positive roots of
Inspection and the graphs of the Bessel functions in Fig. 4-12 indicates that zero is not an eigenvalue for any
The transformed temperature is defined as
Notice again the inclusion of the weighting factor Using (4.6-16) to transform the term in (4.7-17) containing the derivatives in r gives
The boundary vanish because of the homogeneous boundary conditions in (4.7-19).
Transforming the term in (4.7-17) and use of we
obtain
The transformation of the time derivative and the initial condition is straightforward. It is found
that the transformed temperature must satisfy
The solution to (4.7-25) is
and the solution is
The form of the overall solution is clarified by rewriting the steady-state part. From Exam-
? ple 2.8-1, the steady-state solution is
4-2
Sequences of Functions from Certain Eigenvalue Problems in Cylindrical Coordinatesa
Case Boundary condition Basis functions
of the functions shown satisfy (4.7-2) on the [0, few for cases and as follows:
Equation makes use of the identity given as (4.7-4b).
Returning now to the eigenvalue problem for a full cylinder with a Dirichlet boundary condition at = we find from (4.7-6), and (4.7- 14) that
A very similar procedure is used to derive the orthonormal basis functions corresponding to Neumann or Robin conditions. Table 4-2 summarizes the three types of basis func-
tions arising in the analysis of conduction or diffusion inside a complete cylinder of radius For problems involving annular regions, where is not in the domain and the solution cannot be excluded, the basis functions will be linear combinations
and
4.7-1 Conduction in an Electrically Heated Wire Consider a long, cylin- drical wire, initially at ambient temperature, which is subjected to a constant rate of heating for due to passage of an electric current. Heat from the to the surroundings is described using a convection boundary condition, and the Biot number is not necessarily large small. This is a transient problem leading to the steady state analyzed in Example 2.8-1.
the dimensional quantities used in that example, we choose and R as the temperature
and length scales, respectively. Then, the dimensionless problem for is
The only eigenvalue problem derivable from (4.7-17) is one involving r. The domain includes
r = so that the basis functions contain only From case of Table 4-2, the basis functions suitable for the Robin boundary condition in (4.7-19) are
with eigenvalues given by the positive roots of
Inspection and the graphs of the Bessel functions in Fig. 4-12 indicates that zero is not an eigenvalue for any
The transformed temperature is defined as
Notice again the inclusion of the weighting factor Using (4.6-16) to transform the term in (4.7-17) containing the derivatives in r gives
The boundary vanish because of the homogeneous boundary conditions in (4.7-19).
Transforming the term in (4.7-17) and use of we
obtain
The transformation of the time derivative and the initial condition is straightforward. It is found
that the transformed temperature must satisfy
The solution to (4.7-25) is
and the solution is
The form of the overall solution is clarified by rewriting the steady-state part. From Exam-
? ple 2.8-1, the steady-state solution is
Modified
Another differential equation which commonly arises in problems involving cylindrical coordinates is the equation, written generally as
where is a parameter and is any real constant. Equations (4.7-3) and (4.7-30) differ only in the sign of the term. The solutions to (4.7-30) are written as
and and are called Bessel functions of order of the first and second
kind, respectively. As with the "regular" Bessel functions, our concern is with the func- tions corresponding to and 1. Graphs of these modified Bessel functions are shown in Fig. 4-13. The most obvious difference between Bessel functions and modified Bessel functions is that the latter do not display oscillatory behavior or possess multiple roots. The limiting values of the modified Bessel functions are
Identities which are particularly helpful in evaluating derivatives and integrals are
X
4-13. Modified Bessel functions of orders zero one.
Example 4.7-2 Steady Diffusion with a First-Order Reaction in a Cylinder Consider a one- dimensional, steady-state problem involving diffusion and a homogeneous reaction in a long cylin- der. The reaction is first order and irreversible, and the reactant is maintained at a constant concen- tration at the surface. The number (Da) is not necessarily large or small. The conserva- tion equation and boundary for the reactant are written as
dr
A comparison of (4.7-30) and (4.7-35) shows the latter differential equation to be a modified Bessel equation of order zero, with m Accordingly, the general solution to Eq. (4.7-35) is
= r)
+
r), (4.7-36)where a and b are constants. Given that is not finite at that solution is excluded. Using the condition at to evaluate a, the final solution is found to be
This solution was stated without proof in Example 3.7-2, in discussing the results of the singular perturbation analysis of this problem for
Example 4.7-3 Steady Conduction in a Hollow Cylinder Modified Bessel functions arise also
in solutions, as this example will show. Assume that a hollow cylinder has a constant temper- ature at its base, and a different constant temperature at its top and its inner and outer curved surfaces (see Fig. 4-14). The steady, two-dimensional conduction problem involving
z) is
4-14. Steady, two-dimensional
heat conduction in a hollow cylinder. The base is maintained at a different
, temperature than that of the top or the I
curved surfaces. Only half of the hollow
t
cylinder is shown. I
1
Modified
Another differential equation which commonly arises in problems involving cylindrical coordinates is the equation, written generally as
where is a parameter and is any real constant. Equations (4.7-3) and (4.7-30) differ only in the sign of the term. The solutions to (4.7-30) are written as
and and are called Bessel functions of order of the first and second
kind, respectively. As with the "regular" Bessel functions, our concern is with the func- tions corresponding to and 1. Graphs of these modified Bessel functions are shown in Fig. 4-13. The most obvious difference between Bessel functions and modified Bessel functions is that the latter do not display oscillatory behavior or possess multiple roots. The limiting values of the modified Bessel functions are
Identities which are particularly helpful in evaluating derivatives and integrals are
X
4-13. Modified Bessel functions of orders zero one.
Example 4.7-2 Steady Diffusion with a First-Order Reaction in a Cylinder Consider a one- dimensional, steady-state problem involving diffusion and a homogeneous reaction in a long cylin- der. The reaction is first order and irreversible, and the reactant is maintained at a constant concen- tration at the surface. The number (Da) is not necessarily large or small. The conserva- tion equation and boundary for the reactant are written as
dr
A comparison of (4.7-30) and (4.7-35) shows the latter differential equation to be a modified Bessel equation of order zero, with m Accordingly, the general solution to Eq. (4.7-35) is
= r)
+
r), (4.7-36)where a and b are constants. Given that is not finite at that solution is excluded. Using the condition at to evaluate a, the final solution is found to be
This solution was stated without proof in Example 3.7-2, in discussing the results of the singular perturbation analysis of this problem for
Example 4.7-3 Steady Conduction in a Hollow Cylinder Modified Bessel functions arise also
in solutions, as this example will show. Assume that a hollow cylinder has a constant temper- ature at its base, and a different constant temperature at its top and its inner and outer curved