some flow geometries, as well as values for Recr. The flow around a sphere
relates, for instance, to a sedimenting particle, where turbulence will develop
in its wake for Re>1.
It may finally be noted that a geometrical constraint that induces
elongational flow if Re<Recr(e.g., a sudden constriction in a tube) tends
locally to depress turbulence if Re>Recr. Turbulence is also depressed by
the presence of large-molar mass polymers or by a high concentration of
dispersed particles in the liquid.
Question 1
Show for simple shear flow that the velocity gradient dv=dyequals the shear rate
dg=dt.
Answer
Consider Figure 5.2. Since the velocity at the bottom of the volume element equals
zero, the velocity gradient dv=dyis given by the velocity at the top divided byL 0 ,
hence byðdd=dtÞ=L 0 ¼dðd=L 0 Þ=dt¼dg=dt.
Question 2
In a Couette apparatus, i.e., between two concentric cylinders, of which the outer one
of radius 10 cm is rotating at 180 revolutions per minute and the inner stationary one
has a radius of 9.5 cm; some triglyceride oil is present. The temperature is 20C. How
long will it take before the temperature is raised to 21C, assuming that no heat loss
from the oil to the environment occurs? Tip: Some useful data are in Table 9.2.
TABLE5.1 Reynolds Numbers (Re) for Various Flow Geometries
Flow geometry Re¼ Value of RecrIn a cylindrical pipe of diameterDDvr=Z 2300
Between flat plates at separation distanced 2 dvr=Z * 2000
Film (thicknessd) flowing over sloping flat plate 4dvr=Z * 10
Flow around a sphereaof diameterddvr=Z * 1
aHerevis the velocity of the sphere relative to the liquid.