The cross-section of a fluid bearing is shown below. The concentric cylinders are 10 cm in length. The innercylinder has radius 1.9 cm and is stationary whilst the outer cylinder has radius 2 cm and rotates with an angularvelocity 𝜔 = 500 rad/s. The fluid between the cylinders is S.A.E. 10 oil (density 𝜌 = 917 kg/m3 and dynamicviscosity 𝜇 = 0.104 N.s/m2).𝜔 = 500 rad/s 1.9 cm2 cm10 cmfluid in gap(a) By assuming a linear velocity profile of the fluid in the gap between the two cylinders, estimate the(resistive) torque and the power required to maintain constant angular velocity of the outer cylinder.(b) In arriving at the estimate of part (a) above, what assumptions have been made in the modelling.(c) By estimating the Reynolds number, suggest why the calculations might not be appropriate for the sameparameters except with water (density 𝜌 = 1000 kg/m3 and dynamic viscosity 𝜇 = 0.001 N.s/m2) replacingthe oil.

(a) The resistive torque (T) can be calculated using the formula for viscous torque in a cylindrical coordinate system:

T = 2πμL(R2^2 - R1^2)ω / ln(R2/R1)

where: μ = dynamic viscosity = 0.104 N.s/m2 L = length of the cylinder = 10 cm = 0.1 m R2 = outer radius = 2 cm = 0.02 m R1 = inner radius = 1.9 cm = 0.019 m ω = angular velocity = 500 rad/s

Substituting these values into the formula, we can calculate the torque.

The power (P) required to maintain constant angular velocity can be calculated using the formula:

P = Tω

Substitute the calculated torque and given angular velocity into this formula to find the power.

(b) The assumptions made in the modelling are:

• The velocity profile of the fluid in the gap between the two cylinders is linear.
• The fluid is incompressible and Newtonian.
• The flow is fully developed and steady.
• There is no slip at the cylinder walls (no-slip condition).
• The gap between the cylinders is small compared to their radii.

(c) The Reynolds number (Re) can be estimated using the formula:

Re = ρωR2^2 / μ

where: ρ = density of the fluid R2 = outer radius ω = angular velocity μ = dynamic viscosity

Substitute the given values for water into this formula to calculate the Reynolds number. If the Reynolds number is greater than 2000, the flow is likely to be turbulent, and the assumption of a linear velocity profile would not be valid. Therefore, the calculations might not be appropriate for water.