492
Appendix C Useful Theoretical Approaches and Formulas
y
o
'I'=K7
'I'=K6
'I'=K5
'I'=K4
Fig. C.4: Uniform flow in plane Oxy
x
potential function, namely [m 2 s- 1 ]. Equation (C.61) also indicates that along
a streamline 'Ij; is constant, i. e.:
'Ij;(x,y,z) = Const.
(C.62)
A noteworthy and useful property of streamlines is that the discharge between
two streamlines is given by their difference. Consider, for example, a steady
uniform flow in the plane Oxy (Fig. C.4). The velocity of flow, inclined at an
angle 0 with respect to the x-axis, is Vo. Thus, the velocity components should
be:
u(x,y) = Vo cosO, v(x,y) = Vo sinO.
(C.63)
Integration of both sides of the first equation in (C.60), with respect to y,
yields:
'Ij;(x,y) = JUdY = J Vo cos Ody + f(x) = VoycosO + f(x),
(C.64)
and after using the second equation of (C.60) we obtain:
a'lj;
af
- - = - - = VosinO.
ax
ax
(C.65)
Thus:
f ( x) = - Vox sin O.
(C.66)