Page 96 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 79

pressure force on any face will be -pSA where 6A is the area of the face) - see
Fig. 1.3. As is evident from Bernoulli's Eqn (2.16), the pressure depends on the flow
speed.
(b) Viscousforces. In general the viscous force acts at an angle to any particular face
of the infinitesimal control volume, so in general it will have two components
in two-dimensional flow (three for threedimensional flow) acting on each face
(one due to a direct stress acting perpendicularly to the face and one shear stress
(two for three-dimensional flow) acting tangentially to the face. As an example let
us consider the stresses acting on two faces of a square infinitesimal control volume
(Fig. 2.20). For the top face the unit normal would be j (unit vector in the
y direction) and the unit tangential vector would be i (the unit vector in the
x direction). In this case, then, the viscous force acting on this face and the side
face would be given by

(ayxi + ayyj)6x x 1, (axxi + axyj)Sy x 1
respectively. Note that, as in Section 2.4, we are assuming unit length in the
z direction. The viscous shear stress is what is termed a second-order tensor -
i.e. it is a quantity that is characterized by a magnitude and two directions
(c.f. a vector or first-order tensor that is characterized by a magnitude and one
direction). The stress tensor can be expressed in terms of four components
(9 for three-dimensional flow) in matrix form as:
(F 2)

Owing to symmetry ax,, = au.. Just as the components of a vector change
when the coordinate system is changed, so do the components of the stress
tensor. In many engineering applications the direct viscous stresses (axx, ayy)
are negligible compared with the shear stresses. The viscous stress is generated
by fluid motion and cannot exist in a still fluid.
Other surface forces, e.g. surface tension, can be important in some engin-
eering applications.

t
DW6XX1

U,&X1

Fig. 2.20