Engineering Optimization: Theory and Practice, Fourth Edition

12.2 Calculus of Variations 673

depend on the shape of the body and the relative velocity in a very complex manner.
However, if the density of the fluid is sufficiently small, the normal pressure (p)acting
on the solid body can be approximately taken as [12.3]

p= 2 ρν^2 sin^2 θ (E 1 )

whereρis the density of the fluid,vthe velocity of the fluid relative to the solid body,
andθthe angle between the direction of the velocity of the fluid and the tangent to the
surface as shown in Fig. 12.3.
Since the pressure (p) acts normal to the surface, thex-component of the force
acting on the surface of a slice of lengthdxand radiusy(x)shown in Fig. 12.4 can
be written as

dP=(normal pressure) (surface area) sinθ

=( 2 ρv^2 sin^2 θ )( 2 πy

√

1 +(y′)^2 dx) sinθ (E 2 )

wherey′= dy/dx. The total drag force,P, is given by the integral of Eq. (E 2 ) as

P=

∫L

0

4 ρνπ^2 y ins^3 θ

√

1 +(y′)^2 dx (E 3 )

whereLis the length of the body. To simplify the calculations, we assume thaty′≪ 1
so that

sinθ=

y′

√

1 +(y′)^2

≃y′ (E 4 )

Thus Eq. (E 3 ) can be approximated as

P= 4 πρν^2

∫ L

0

(y′)^3 y dx (E 5 )

Nowthe minimum drag problem can be stated as follows.

Figure 12.3 Solid body of revolution translating in a fluid medium.