2.5 Several Special Cases of the Euler Equation and Their Integrals
139
O
x
y
L
P
R
θ
Fig. 2.9 The airflow resistance diagram of a body of rotation
caused by air velocity in the normal direction of the surface of the body of rotation
is
p = 2ρu
2 sin
2
θ
(1)
where, ρ is the density of the air; θ is the included angle between the air velocity and
the tangential direction of the surface of the body of rotation. On the torus surface
that the arc length is ds =
1 + y dx and the radius is y, the component of the
pressure along the x axial direction namely the resistance is
dF = 2ρu
2 sin
2
θ × 2πy
1 + y sin θ dx
(2)
Assume that y
is lesser, and
sin θ =
y
1 + y
≈ y
(3)
Thus all resistance acting on the x direction of the surface of the body of rotation
is the functional
J [y] =
F
dF = 4πρu
2
L
0
yy
dx
(4)
The boundary conditions are y(0) = 0, y(L) = R. Because the integrand of the
functional does not contain x, so the first integral of the Euler equation is
yy
− y
(3yy
) = −2c
3
(5)
139
O
x
y
L
P
R
θ
Fig. 2.9 The airflow resistance diagram of a body of rotation
caused by air velocity in the normal direction of the surface of the body of rotation
is
p = 2ρu
2 sin
2
θ
(1)
where, ρ is the density of the air; θ is the included angle between the air velocity and
the tangential direction of the surface of the body of rotation. On the torus surface
that the arc length is ds =
1 + y dx and the radius is y, the component of the
pressure along the x axial direction namely the resistance is
dF = 2ρu
2 sin
2
θ × 2πy
1 + y sin θ dx
(2)
Assume that y
is lesser, and
sin θ =
y
1 + y
≈ y
(3)
Thus all resistance acting on the x direction of the surface of the body of rotation
is the functional
J [y] =
F
dF = 4πρu
2
L
0
yy
dx
(4)
The boundary conditions are y(0) = 0, y(L) = R. Because the integrand of the
functional does not contain x, so the first integral of the Euler equation is
yy
− y
(3yy
) = −2c
3
(5)