Page 23 - Aerodynamics for Engineering Students

6 Aerodynamics for Engineering Students

Very frequently the static pressure is referred to simply as pressure. The term static is
rather misleading. Note that its use does not imply the fluid is at rest.
For large bodies moving or at rest in the fluid, e.g. air, the pressure is not uni-
form over the surface and this gives rise to aerodynamic force or aerostatic force
respectively.
Since a pressure is force per unit area, it has the dimensions

[Force] -k [area] = [MLT-2] t [L2] = [ML-'T-2]
and is expressed in the units of Newtons per square metre or Pascals (Nm-2 or Pa).

Pressure in fluid at rest
Consider a small cubic element containing fluid at rest in a larger bulk of fluid also at
rest. The faces of the cube, assumed conceptually to be made of some thin flexible
material, are subject to continual bombardment by the molecules of the fluid, and
thus experience a force. The force on any face may be resolved into two components,
one acting perpendicular to the face and the other along it, i.e. tangential to it.
Consider for the moment the tangential components only; there are three signifi-
cantly different arrangements possible (Fig. 1.1). The system (a) would cause the
element to rotate and thus the fluid would not be at rest. System (b) would cause
the element to move (upwards and to the right for the case shown) and once more,
the fluid would not be at rest. Since a fluid cannot resist shear stress, but only rate of
change of shear strain (Sections 1.2.6 and 2.7.2) the system (c) would cause the
element to distort, the degree of distortion increasing with time, and the fluid would
not remain at rest.
The conclusion is that a fluid at rest cannot sustain tangential stresses, or con-
versely, that in a fluid at rest the pressure on a surface must act in the direction
perpendicular to that surface.

Pascal% law
Consider the right prism of length Sz into the paper and cross-section ABC, the
angle ABC being a right-angle (Fig. 1.2). The prism is constructed of material of
the same density as a bulk of fluid in which the prism floats at rest with the face
BC horizontal.
Pressurespl,p2 andp3 act on the faces shown and, as proved above, these pressures
act in the direction perpendicular to the respective face. Other pressures act on the
end faces of the prism but are ignored in the present problem. In addition to these
pressures, the weight W of the prism acts vertically downwards. Consider the forces
acting on the wedge which is in equilibrium and at rest.

Fig. 1.1 Fictitious systems of tangential forces in static fluid