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A.7 ORTHOGONAL CURVILINEAR COORDINATES - UCSB Engineering Flipbook PDF
Orthogonal Curvilinear Coordinates. 571 . S AND . TENSORS (A.7-3) m s;, so that . 3; 1 . and the corres nds on position
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569
~NSORS
Orthogonal Curvilinear Coordinates
)osition
ated by converting its components (but not the unit dyads) to spherical coordinates, and integrating each over the two spherical angles (see Section A.7). The off-diagonal terms in Eq. (A.6-13) vanish, again due to the symmetry.
1 r+r',
(A.6-6)
A.7 ORTHOGONAL CURVILINEAR COORDINATES (A.6-7) (A.6-8)
to func he usual st calcu :lrSe and
ial prop ve obtain (A.6-9)
(A.6-1O)
(A.6-11)
Enormous simplificatons are achieved in solving a partial differential equation if all boundaries in the problem correspond to coordinate surfaces, which are surfaces gener ated by holding one coordinate constant and varying the other two. Accordingly, many special coordinate systems have been devised to solve problems in particular geometries. The most useful of these systems are orthogonal; that is, at any point in space the vectors aligned with the three coordinate directions are mutually perpendicular. In gen eral, the variation of a single coordinate will generate a curve in space, rather than a straight line; hence the term curvilinear. In this section a general discussion of orthogo nal curvilinear systems is given first, and then the relationships for cylindrical and spher ical coordinates are derived as special cases. The presentation here closely follows that in Hildebrand (1976).
Base Vectors Let (Ul, U2' U3) represent the three coordinates in a general, curvilinear system, and let e i be the unit vector that points in the direction of increasing u i• A curve produced by varying U;, with uj (j =1= i) held constant, will be referred to as a "u; curve." Although the base vectors are each of constant (unit) magnitude, the fact that a U; curve is not gener ally a straight line means that their direction is variable. In other words, ei must be regarded as a function of position, in general. This discussion is restricted to coordinate systems in which (e l , e2 , e3 ) is an orthonormal and right-handed set. At any point in space, such a set has the properties of the base vectors used in Section A.3, namely,
e;·ej
= Su'
e; X ej = 2:s;jkek'
tion dyad lting each , the con ie integra Itegrals of
(A.6-12)
(A.6-13) e negative r is evalu-
(A.7-1) (A.7-2)
k
Recalling that the multiplication properties of vectors and tensors are derived from these relationships (and their extensions to unit dyads), we see that all of the relations in Section A.3 apply to orthogonal curvilinear systems in general, and not just to rectangu lar coordinates. It is with spatial derivatives that the variations in ei come into play, and the main task in this section is to show how the various differential operators differ from those given in Section AA for rectangular coordinates. In the process, we will obtain general expressions for differential elements of arc length, volume, and surface area.
Arc Length The key to deriving expressions for curvilinear coordinates is to consider the arc length along a curve. In particular, let Si represent arc length along a u; curve. From Eq. (A.6 2), a vector that is tangent to a U i curve and directed toward increasing U i is given by
570
VECTORS AND TENSORS
Orthogonal Curvilinear C
=S27T. Using
x = r sin e cos cf>,
y = r sin e sin cf>,
z = r cos
e
(A.7-34)
the position vector is expressed as r = r sin
e cos cf>ex + r sin e sin cf>ey + r cos eez .
(A.7-35)
The position vector is also given by
r= re r .
(A.7-36)
Either expression indicates that Jrl = 1; consistent with our usual notation. Employing Eq. (A.7-35) in the general relationships for curvilinear coordinates, the scale factors and base vectors for spherical coordinates are evaluated as
576
VECTORS AND TENSORS
hr=l,
he=r,
Surface Geometry
(A.7-37)
he/>=rsin e,
TABLE A.4 Differential Operations if
er = sin e cos ¢ex+ sin e sin ¢ey+ cos ee z '
(A.7-38a)
ee = cos e cos ¢ex+ cos e sin ¢ey- sin eez ,
(A.7-38b)
(I)
ee/> = - sin ¢ex+ cos ¢eY'
(A.7-38c)
(2)
The dependence of the three base vectors on e and ¢ is shown in Eq. (A.7-38); the expression for er confirms the equivalence of Eqs. (A.7-35) and (A.7-36). The comple mentary expressions for the rectangular base vectors are
(3)
Vxv=_I_ r sin 6
(4)
e cos ¢er + cos e cos ¢ee - sin ¢ee/>' ey= sin e sin ¢e + cos e sin ¢ee + cos ¢ee/>'
(A.7-39b)
(5)
ez = cos ee r - sin eee'
(A.7-39c)
(6)
ex = sin
r
(A.7-39a)
Finally, the differential volume and surface elements are evaluated as
dV=r 2 sin e dr de d¢ dS r = r 2 sin e de d¢,
(7)
(A.7-40)
dS e = r sin e dr d¢,
dSe/>=r dr de.
(A.7-4l)
A summary of differential operations in spherical coordinates is presented in Table A-4. As already mentioned, several other quantities, including "ij2 v, V· T, and v· Vv, may be obtained from the tables in Chapter 5. Many other orthogonal coordinate systems have been developed. A compilation of scale factors, differential operators, and solutions Of Laplace's equation in 40 such sys tems is provided in Moon and Spencer (1961).
(8) (9) (10)
(II) (12)
A.8 SURFACE GEOMETRY
(13)
In Section A.5 a number of integral transformations were presented involving vectors which are normal or tangent to a surface. The objective of this section is to show how those vectors are computed and how they are used to define such quantities as surface gradients. This completes the information needed to understand the integral forms of the various conservation equations. Much of the material in this section is adapted from Brand (1947).
a In
these relationships f is any differ
=~ al3
(11)
(Vv) 1>r
(12)
(Vv) 1>0
(13)
=_I_av,_~ r sin 13 a
involving vectors is to show how mtities as surface >gral forms of the is adapted from
r al3
(9)
(10)
A compilation of 1 in 40 such sys
r sin 13 a