Page 95 - Aerodynamics for Engineering Students

78 Aerodynamics for Engineering Students

Fig. 2.19

But here $ is a function of (r, Q) and again
a$
6$ = -6r -t -60 (2.58)
dr dB
and equating terms in Eqns (2.57) and (2.58)
a$
qt 1 -- (2.58a)
dr

(2.58b)

these being velocity components at a point r, Q in a flow given by stream function $.
In general terms the velocity q in any direction s is found by differentiating the
stream function $ partially with respect to the direction II normal to q where n is
taken in the anti-clockwise sense looking along q (Fig. 2.19):
q=- a$
dn

2.6 The momentum equation

The momentum equation for two- or three-dimensional flow embodies the applica-
tion of Newton’s second law of motion (mass times acceleration = force, or rate of
change of momentum = force) to an infinitesimal control volume in a fluid flow (see
Fig. 2.8). It takes the form of a set of partial differential equations. Physically it states
that the rate of increase in momentum within the control volume plus the net rate at
which momentum flows out of the control volume equals the force acting on the fluid
within the control volume.
There are two distinct classes of force that act on the fluid particles within the
control volume.
(i) Body forces. Act on all the fluid within the control volume. Here the only body
force of interest is the force of gravity or weight of the fluid.
(ii) Surface forces. These only act on the control surface; their effect on the fluid
inside the control volume cancels out. They are always expressed in terms of
stress (force per unit area). Two main types of surface force are involved namely:
(a) Pressure force. Pressure, p, is a stress that always acts perpendicular to the control
surface and in the opposite direction to the unit normal (see Fig. 1.3). In other words
it always tends to compress the fluid in the control volume. Although p can vary
from point to point in the flow field it is invariant with direction at a particular point
(in other words irrespective of the orientation of the infinitesimal control volume the