Page 113 - Aerodynamics for Engineering Students

96 Aerodynamics for Engineering Students

h

Fig. 2.30

are parallel to the x axis, so v = 0. Therefore Eqn (2.93) implies du/dx = 0, i.e. u is a
function only of y. There is no external pressure field, so Eqn (2.92a) reduces to

a2U
p- = 0 implying u = Cly + C2 (2.106)
dY2
where C1 and C2 are constants of integration. u = 0 and UT when y = 0 and h
respectively, so Eqn (2.106) becomes

where T is the constant viscous shear stress.
This solution approximates well the flow between two concentric cylinders with the
inner one rotating at fmed speed, provided the clearance is small compared with the
cylinder’s radius, R. This is the basis of a viscometer - an instrument for measuring
viscosity, since the torque required to rotate the cylinder at constant speed w is
proportional to T which is given by pwR/h. Thus if the torque and rotational speed
are measured the viscosity can be determined.

2.10.2 Plane Poiseuille flow - pressure-driven channel flow
This also corresponds to the flow between two infinite, plane, parallel surfaces (see
Fig. 2.31). Unlike Couette flow, both surfaces are stationary and flow is produced by
the application of pressure. Thus all the arguments used in Section 2.10.1 to simplify
the Navier-Stokes equations still hold. The only difference is that the pressure term
in Eqn (2.95a) is retained so that it simplifies to

dp 8% 1 dPY2
- - + p- = 0 implying u = --- + clY + c2 (2.108)
dx 8y2 pdx 2
The no-slip condition implies that u = 0 at y = 0 and h, so Eqn (2.108) becomes

(2.109)

Thus the velocity profile is parabolic in shape.
The true Poiseuille flow is found in capillaries with round sections. A very similar
solution can be found for this case in a similar way to Eqn (2.109) that again has