Page 86 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 69

Fig. 2.9

In a Cartesian coordinate system let a particle move from point P(x,y) to point
Q(x + Sx, y + Sy), a distance of 6s in time St (Fig. 2.9). Then the velocity of the
particle is
. 6s ds
]Im- = - = q
6+0 St dt
Going from P to Q the particle moves horizontally through SX giving the horizontal
velocity u = dx/dt positive to the right. Similarly going from P to Q the particle
moves vertically through Sy and the vertical velocity v = dy/dt (upwards positive). By
geometry:

(Ss)2 = (Sx)2 + (Sy)2
Thus
q2=22+v2
and the direction of q relative to the x-axis is a = tan-’ (v/u).
In a polar coordinate system (Fig. 2.10) the particle moves distance 6s from P(r, 0)
to Q(r + Sr, 0 + SO) in time 5t. The component velocities are:
dr
radially (outwards positive) q - -
’ - dt
do
tangentially (anti-clockwise positive) qt = r -
dt
Again
(Ss)2 = (Sr)2 + (rSo)2

Fig. 2.10