Two equally long 12-cm diameter pipelines have the same pressure difference between ends. One carries a ‘Type A’ oil (๐œ‚ = 0.091 Pa s) and the other a ‘Type B’ oil (๐œ‚ = 0.064 Pa s). If ‘Type A’ oil flows at 0.14 m3/s, at which rate will ‘Type B’ oil flow?

To determine the rate at which the 'Type B' oil will flow, we can use the principle of viscosity and the concept of Poiseuille's law.

Poiseuille's law states that the flow rate through a pipe is directly proportional to the pressure difference between the ends and the fourth power of the radius of the pipe, and inversely proportional to the viscosity and length of the pipe.

Given:
Diameter of the pipelines = 12 cm
Radius of the pipelines = 12 cm / 2 = 6 cm = 0.06 m
Viscosity of 'Type A' oil (๐œ‚A) = 0.091 Pa s
Viscosity of 'Type B' oil (๐œ‚B) = 0.064 Pa s
Flow rate of 'Type A' oil = 0.14 m^3/s

To find the flow rate of 'Type B' oil, we need to determine the pressure difference between the ends of the pipelines. Since the pressure difference is the same for both pipelines, we can equate the flow rates using Poiseuille's law.

The formula for flow rate using Poiseuille's law is:
Q = (๐œ‹ * ฮ”P * r^4) / (8 * ๐œ‚ * L)

Where:
Q = Flow rate
๐œ‹ = Pi (approximately 3.14159)
ฮ”P = Pressure difference between the ends of the pipeline
r = Radius of the pipeline
๐œ‚ = Viscosity
L = Length of the pipeline

Let's calculate the flow rate of 'Type B' oil using the given values:

For 'Type A' oil:
Qa = 0.14 m^3/s
๐œ‚A = 0.091 Pa s

For 'Type B' oil:
Qb = ?
๐œ‚B = 0.064 Pa s

Since both pipelines are equally long and have the same pressure difference, we can assume that the length and pressure difference cancel out in the equation.

Qa / Qb = (๐œ‚B * rB^4) / (๐œ‚A * rA^4)

Since the radii of both pipelines are the same:
rA = rB = 0.06 m

Substituting the given values into the equation:
Qb = (Qa * ๐œ‚B * rB^4) / (๐œ‚A * rA^4)
Qb = (0.14 * 0.064 * (0.06)^4) / (0.091 * (0.06)^4)

Simplifying the equation:
Qb = (0.014848) / (0.0002663494)
Qb โ‰ˆ 55.77 m^3/s

Therefore, the flow rate of 'Type B' oil will be approximately 55.77 m^3/s.