Page 99 - Aerodynamics for Engineering Students

82 Aerodynamics for Engineering Students

Similarly, the y component of the net pressure force is given by
aP
--SxSy x I (2.64b)
aY
The evaluation of the x component of Term (v), the net viscous force, is illustrated
in Fig. 2.23. In a similar way as for Eqn (2.64a7b), we obtain the net viscous force in
the x and y directions respectively as

(%+%)6xSy x 1 (2.65a)

(Z+3)SxSy x 1 (2.65b)

We now substitute Eqns (2.61) to (2.65) into Eqn (2.59) and cancel the common
factor SxSy x 1 to obtain
(t : ;;) ax ax ay (2.66a)
p -+u-+v- =pg,--+-+- aP aa,, acxy

(E i: ;;) ay ax ay
aP aa,, ayy
p -+u-+v- =pgy--+-+- (2.66b)
These are the momentum equations in the form of partial differential equations.
For three dimensional flows the momentum equations can be written in the form:

av av av

(2.67~)

where g,, gy, g, are the components of the acceleration g due to gravity, the body
force per unit volume being given by pg.
The only approximation made to derive Eqns (2.66) and (2.67) is the continuum
model, i.e. we ignore the fact that matter consists of myriad molecules and treat it as
continuous. Although we have made use of the incompressible form of the continuity

Fig. 2.23 x-component of forces due to viscous stress acting on infinitesimal control volume