Critical velocity
The critical velocity of a liquid is that limiting value of its velocity of flow upto which the flow is streamlined and above which the flow becomes turbulent.
The critical velocity vc of a liquid flowing through a tube depends on
(i) coefficient of viscosity of the liquid (η)
(ii) density of the liquid (r)
(iii) diameter of the tube (D)
Let vc = k ηa rb Dc
where k is a dimensionless constant. Writing the above equation in dimensional form, we get
[M0LT-1] = [ML-1T-1]a [ML-3]b [L]c
[M0LT-1] = [Ma + b L-a – 3b + cT – a]
Equating powers of M, L and T, we get
a + b = 0
– a – 3b + c = 1
– a = -1
On solving, we get a = 1, b = -1, c = -1
∴ vc = k η r-1 D-1 = k η / r D
Clearly, the critical velocity vc will be large if η is large, and p and D are small. So, we can conclude that
(i) The flow of liquids of higher viscosity and lower density through narrow pipes tends to be streamlined.
(ii) The flow of liquids of lower viscosity and higher density through broad pipes tends to become turbulent, because in that case the critical velocity will be very small.