Page 441 - Advanced engineering mathematics

12.10 Curvilinear Coordinates 421

u 3

c
ds = h dq
k g 3 3 3
d b
a
c u 2
d e ds = h dq 2
2
2
f
e = (q , q , q )
1 2 3
a b a = (q + ds , q , q )
1 1 2 3
b = (q + ds , q + ds , q ) u 1
1 1 2 2 3
FIGURE 12.32 Calculating the curl in
FIGURE 12.31 Calculating the divergence
curvilinear coordinates.
in curvilinear coordinates.
Similarly, the fluxes across the other two pairs of opposite faces are
∂ ∂
(F 2 h 1 h 3 )dq 1 dq 2 dq 3 and (F 3 h 1 h 2 )dq 1 dq 2 dq 3 .
∂q 2 ∂q 3
We obtain the divergence, or flux per unit volume, at a point by adding these three expressions for
the flux across pairs of opposite sides, and dividing by the volume h 1 h 2 h 3 dq 1 dq 2 dq 3 to obtain
∇· F(q 1 ,q 2 ,q 3 )

1 ∂ ∂ ∂
= (F 1 h 2 h 3 ) + (F 2 h 1 h 3 ) + (F 3 h 1 h 2 ) .
h 1 h 2 h 3 ∂q 1 ∂q 2 ∂q 3
Laplacian
Knowing the divergence, we immediately have the Laplacian, since
2
∇ f =∇ ·∇ f
for a scalar field f . Then
2
∇ f (q 1 ,q 2 ,q 3 ) =∇ ·∇ f (q 1 ,q 2 ,q 3 )
1 ∂ h 2 h 3 ∂ f ∂ h 1 h 3 ∂ f ∂ h 1 h 2 ∂ f
= + + .
h 1 h 2 h 3 ∂q 1 h 1 ∂q 1 ∂q 2 h 2 ∂q 2 ∂q 3 h 3 ∂q 3
Curl

For the curl in curvilinear coordinates, we will use the interpretation of (∇×F)·n as the rotation
or swirl of a fluid with velocity field F about a point in a plane having unit normal n.
At P, the component of ∇× F in the direction u 1 is
1
lim F,
A→0 A
C
where C may be taken as a rectangle about P in the u 2 − u 3 plane at P (Figure 12.32). Compute
the integral over each side of this rectangle. On side a,

F ≈ F 2 (q 1 ,q 2 ,q 3 )h 2 (q 1 ,q 2 ,q 3 )dq 2 ,
a

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October 14, 2010 14:53 THM/NEIL Page-421 27410_12_ch12_p367-424