Component velocities

In general the local velocity in a flow is inclined to the reference axes O. v, Oy and it is usual to resolve the velocity vector v (magnitude q) into two components mutually at right-angles.

Fig. 2.8 An infinitesimal control volume in a typical two-dimensional flow field

Fig. 2.9

In a Cartesian coordinate system let a particle move from point P(x, y) to point Q(jc + 8x, у + 6y), a distance of 6s in time St (Fig. 2.9). Then the velocity of the particle is

.. 6s ds lim— = ~r = q s-> о 6t d /

Going from P to Q the particle moves horizontally through 8x giving the horizontal velocity и = dx/dt positive to the right. Similarly going from P to Q the particle moves vertically through 6y and the vertical velocity v = dy/dt (upwards positive). By geometry:

(&)2 = (&c)2 + («y)2

Thus

q2 = i? + v2

and the direction of q relative to the jc-axis is a = tan-1 (v/k).

In a polar coordinate system (Fig. 2.10) the particle moves distance 6s from P(r, в) to Q(r + 8r,6 + 69) in time St. The component velocities are:

radially (outwards positive) qn = ^

d t

tangentially (anti-clockwise positive) qt = r^-

dt

Again

(8s)2 – (8rf + (r69)2

0 (r+8л,9-h 8 в)

Fig. 2.11

Thus

# = <& + £

and the direction of q relative to the radius vector is given by

/3 = tan-1 ^